Module `phiml.math`

Vectorized operations, tensors with named dimensions.

This package provides a common interface for tensor operations. Is internally uses NumPy, TensorFlow or PyTorch.

Main classes: Tensor, Shape, DType, Extrapolation.

The provided operations are not implemented directly. Instead, they delegate the actual computation to either NumPy, TensorFlow or PyTorch, depending on the configuration. This allows the user to write simulation code once and have it run with various computation backends.

See the documentation at https://tum-pbs.github.io/PhiML/

Sub-modules

phiml.math.extrapolation: Extrapolations are used for padding tensors and sampling coordinates lying outside the tensor bounds. Standard extrapolations are listed as global …
phiml.math.magic: Magic methods allow custom classes to be compatible with various functions defined in phiml.math, analogous to how implementing __hash__ allows …

Global variables

var INF

Floating-point representation of positive infinity.

var NAN

Floating-point representation of NaN (not a number).

var NUMPY

Default backend for NumPy arrays and SciPy objects.

var PI

Value of π to double precision

var f

Automatic mapper for broadcast string formatting of tensors, resulting in tensors of strings. Used with the special -f- syntax.

Examples

>>> from phiml.math import f
>>> -f-f'String containing {tensor1} and {tensor2:.1f}'
# Result is a str tensor containing all dims of tensor1 and tensor2

var math

Convenience alias for the module phiml.math. This way, you can import the module and contained items in one line.

from phiml.math import math, Tensor, wrap, extrapolation, l2_loss

Expand source code

"""
Vectorized operations, tensors with named dimensions.

This package provides a common interface for tensor operations.
Is internally uses NumPy, TensorFlow or PyTorch.

Main classes: `Tensor`, `Shape`, `DType`, `Extrapolation`.

The provided operations are not implemented directly.
Instead, they delegate the actual computation to either NumPy, TensorFlow or PyTorch, depending on the configuration.
This allows the user to write simulation code once and have it run with various computation backends.

See the documentation at https://tum-pbs.github.io/PhiML/
"""

from ..backend._dtype import DType
from ..backend import NUMPY, precision, set_global_precision, get_precision, set_global_default_backend as use

from ._shape import (
    shape, Shape, EMPTY_SHAPE, DimFilter,
    spatial, channel, batch, instance, dual,
    non_batch, non_spatial, non_instance, non_channel, non_dual, non_primal, primal,
    merge_shapes, concat_shapes, IncompatibleShapes,
    enable_debug_checks,
)

from ._magic_ops import (
    slice_ as slice, unstack,
    stack, concat, ncat, tcat, ccat, scat, icat, dcat, expand,
    rename_dims, rename_dims as replace_dims, pack_dims, unpack_dim, flatten,
    b2i, c2b, c2d, i2b, s2b, si2d, d2i, d2s,
    copy_with, replace, find_differences
)

from ._tensors import Tensor, wrap, tensor, layout, native, numpy_ as numpy, reshaped_numpy, Dict, to_dict, from_dict, is_scalar, BROADCAST_FORMATTER as f, save, load

from ._sparse import dense, get_sparsity, get_format, to_format, is_sparse, sparse_tensor, stored_indices, stored_values, tensor_like, matrix_rank

from .extrapolation import Extrapolation, as_extrapolation

from ._ops import (
    choose_backend_t as choose_backend, all_available, convert, seed, to_device,
    reshaped_native, reshaped_tensor, copy, native_call,
    print_ as print,
    slice_off,
    zeros, ones, fftfreq, random_normal, random_normal as randn, random_uniform, random_uniform as rand, random_permutation, pick_random,
    meshgrid, linspace, arange, arange as range, range_tensor,  # creation operators (use default backend)
    zeros_like, ones_like,
    pad,
    swap_axes,  # reshape operations
    sort,
    safe_div,
    where, nonzero, ravel_index,
    sum_ as sum, finite_sum, dsum, isum, ssum, csum,
    mean, finite_mean, dmean, imean, smean, cmean,
    std,
    prod, dprod, sprod, iprod, cprod,
    max_ as max, dmax, smax, imax, cmax, finite_max,
    min_ as min, dmin, smin, imin, cmin, finite_min,
    any_ as any, all_ as all, quantile, median,  # reduce
    at_max, at_min, argmax, argmin,
    dot,
    abs_ as abs, sign,
    round_ as round, ceil, floor,
    maximum, minimum, clip,
    sqrt, exp, erf, log, log2, log10, sigmoid, soft_plus, softmax,
    sin, cos, tan, sinh, cosh, tanh, arcsin, arccos, arctan, arcsinh, arccosh, arctanh, log_gamma, factorial, incomplete_gamma,
    to_float, to_int32, to_int64, to_complex, imag, real, conjugate, angle,
    radians_to_degrees, degrees_to_radians,
    boolean_mask,
    is_finite, is_nan, is_inf, nan_to_0,
    closest_grid_values, grid_sample, scatter, gather,
    histogram,
    fft, ifft, convolve, cumulative_sum,
    dtype, cast,
    close, always_close, assert_close, equal,
    stop_gradient,
    pairwise_differences, pairwise_differences as pairwise_distances, map_pairs,
    with_diagonal,
    eigenvalues, svd,
    contains, count_occurrences, count_intersections,
)

from ._nd import (
    shift, index_shift,
    vec, const_vec, length, length as vec_length, vec_squared, normalize, normalize as vec_normalize,
    dim_mask,
    normalize_to,
    l1_loss, l2_loss, frequency_loss,
    spatial_gradient, laplace,
    neighbor_reduce, neighbor_mean, neighbor_sum, neighbor_max, neighbor_min, at_min_neighbor, at_max_neighbor,
    fourier_laplace, fourier_poisson, abs_square,
    downsample2x, upsample2x, sample_subgrid,
    masked_fill, finite_fill,
    find_closest,
)

from ._trace import matrix_from_function

from ._functional import (
    LinearFunction, jit_compile_linear, jit_compile,
    jacobian, gradient, custom_gradient, print_gradient,
    safe_mul,
    map_types, map_s2b, map_i2b, map_c2b, map_d2b, map_d2c, map_c2d,
    broadcast,
    iterate,
    identity,
    trace_check,
    map_ as map,
    when_available,
    perf_counter,
)

from ._optimize import solve_linear, solve_nonlinear, minimize, Solve, SolveInfo, ConvergenceException, NotConverged, Diverged, SolveTape, factor_ilu

from ._deprecated import clip_length, cross_product, cross_product as cross, rotate_vector, rotation_matrix

import sys as _sys
math = _sys.modules[__name__]
"""Convenience alias for the module `phiml.math`.
This way, you can import the module and contained items in one line.
```
from phiml.math import math, Tensor, wrap, extrapolation, l2_loss
```"""

PI = 3.14159265358979323846
"""Value of π to double precision """
pi = PI  # intentionally undocumented, use PI instead. Exists only as an anlog to numpy.pi

INF = float("inf")
""" Floating-point representation of positive infinity. """
inf = INF  # intentionally undocumented, use INF instead. Exists only as an anlog to numpy.inf


NAN = float("nan")
""" Floating-point representation of NaN (not a number). """
nan = NAN  # intentionally undocumented, use NAN instead. Exists only as an anlog to numpy.nan

NUMPY = NUMPY  # to show up in pdoc
"""Default backend for NumPy arrays and SciPy objects."""

f = f
"""
Automatic mapper for broadcast string formatting of tensors, resulting in tensors of strings.
Used with the special `-f-` syntax.

Examples:
    >>> from phiml.math import f
    >>> -f-f'String containing {tensor1} and {tensor2:.1f}'
    # Result is a str tensor containing all dims of tensor1 and tensor2
"""

__all__ = [key for key in globals().keys() if not key.startswith('_')]

__pdoc__ = {
    'Extrapolation': False,
    'Shape.__init__': False,
    'SolveInfo.__init__': False,
    'TensorDim.__init__': False,
    'ConvergenceException.__init__': False,
    'Diverged.__init__': False,
    'NotConverged.__init__': False,
    'LinearFunction.__init__': False,
}

Functions

def abs(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes ||x||₁. Complex x result in matching precision float values.

Note: The gradient of this operation is undefined for x=0. TensorFlow and PyTorch return 0 while Jax returns 1.

Args

x: Tensor or PhiTreeNode

Returns

Absolute value of x of same type as x.

def abs_square(complex_values: Union[phiml.math._tensors.Tensor, complex]) ‑> phiml.math._tensors.Tensor

Squared magnitude of complex values.

Args

complex_values: complex Tensor

Returns

Tensor: real valued magnitude squared

def all(boolean_value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> phiml.math._tensors.Tensor

Tests whether all entries of boolean_tensor are True along the specified dimensions.

Args

boolean_value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

Returns

Tensor without the reduced dimensions.

def all_available(*values) ‑> bool

Tests if all tensors contained in the given values are currently known and can be read. Placeholder tensors used to trace functions for just-in-time compilation or matrix construction are considered not available, even when they hold example values like with PyTorch's JIT.

Tensors are not available during jit_compile(), jit_compile_linear() or while using TensorFlow's legacy graph mode.

Tensors are typically available when the backend operates in eager mode and is not currently tracing a function.

This can be used instead of the native checks

PyTorch: torch._C._get_tracing_state()
TensorFlow: tf.executing_eagerly()
Jax: isinstance(x, jax.core.Tracer)

Args

values: Tensors to check.

Returns

True if no value is a placeholder or being traced, False otherwise.

def always_close(t1: Union[numbers.Number, phiml.math._tensors.Tensor, bool], t2: Union[numbers.Number, phiml.math._tensors.Tensor, bool], rel_tolerance=1e-05, abs_tolerance=0, equal_nan=False) ‑> bool

Checks whether two tensors are guaranteed to be close() in all values. Unlike close(), this function can be used with JIT compilation and with tensors of incompatible shapes. Incompatible tensors are never close.

If one of the given tensors is being traced, the tensors are only equal if they reference the same native tensor. Otherwise, an element-wise equality check is performed.

Args

t1: First tensor or number to compare.
t2: Second tensor or number to compare.
rel_tolerance: Relative tolerance, only used if neither tensor is traced.
abs_tolerance: Absolute tolerance, only used if neither tensor is traced.
equal_nan: If True, tensors are considered close if they are NaN in the same places.

Returns

bool

def angle(x: ~TensorOrTree) ‑> ~TensorOrTree

Compute the angle of a complex number. This is equal to atan(Im/Re) for most values.

Args

x: Tensor or PhiTreeNode

Returns

Angle of complex number in radians.

def any(boolean_value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> phiml.math._tensors.Tensor

Tests whether any entry of boolean_tensor is True along the specified dimensions.

Args

boolean_value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

Returns

Tensor without the reduced dimensions.

def arange(dim: phiml.math._shape.Shape, start_or_stop: Optional[int] = None, stop: Optional[int] = None, step=1)

Returns evenly spaced values between start and stop. If only one limit is given, 0 is used for the start.

See Also: range_tensor(), linspace(), meshgrid().

Args

dim: Dimension name and type as Shape object. The size of dim is interpreted as stop unless start_or_stop is specified.
start_or_stop: (Optional) int. Interpreted as start if stop is specified as well. Otherwise this is stop.
stop: (Optional) int. stop value.
step: Distance between values.

Returns

Tensor

def arccos(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the inverse of cos(x) of the Tensor or PhiTreeNode x. For real arguments, the result lies in the range [0, π].

def arccosh(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the inverse of cosh(x) of the Tensor or PhiTreeNode x.

def arcsin(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the inverse of sin(x) of the Tensor or PhiTreeNode x. For real arguments, the result lies in the range [-π/2, π/2].

def arcsinh(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the inverse of sinh(x) of the Tensor or PhiTreeNode x.

def arctan(x: ~TensorOrTree, divide_by=None) ‑> ~TensorOrTree

Computes the inverse of tan(x) of the Tensor or PhiTreeNode x.

Args

x: Input. The single-argument arctan() function cannot output π/2 or -π/2 since tan(π/2) is infinite.
divide_by: If specified, computes arctan(x/divide_by) so that it can return π/2 and -π/2. This is equivalent to the common arctan2 function.

def arctanh(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the inverse of tanh(x) of the Tensor or PhiTreeNode x.

def argmax(x: phiml.math._tensors.Tensor, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], index_dim=(indexᶜ=None))

Finds the maximum value along one or multiple dimensions and returns the corresponding index.

Args

x: Tensor
dim: Dimensions along which the maximum should be determined. These are reduced in the operation.
index_dim: Dimension listing the index components for multidimensional argmax.

Returns

Index tensor idx, such that x[idx] = max(x).

def argmin(x: phiml.math._tensors.Tensor, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], index_dim=(indexᶜ=None))

Finds the minimum value along one or multiple dimensions and returns the corresponding index.

Args

x: Tensor
dim: Dimensions along which the minimum should be determined. These are reduced in the operation.
index_dim: Dimension listing the index components for multidimensional argmin.

Returns

Index tensor idx, such that x[idx] = min(x).

def as_extrapolation(obj) ‑> Extrapolation

Creates an Extrapolation from a descriptor object.

Args

obj

Extrapolation specification, one of the following:

Extrapolation
Primitive name as str: periodic, zero, one, zero-gradient, symmetric, symmetric-gradient, antisymmetric, reflect, antireflect
dict containing exactly the keys 'normal' and 'tangential'
dict mapping spatial dimension names to extrapolations

Returns

Extrapolation

def assert_close(*values, rel_tolerance: float = 1e-05, abs_tolerance: float = 0, msg: str = '', verbose: bool = True, equal_nan=True)

Checks that all given tensors have equal values within the specified tolerance. Raises an AssertionError if the values of this tensor are not within tolerance of any of the other tensors.

Does not check that the shapes match as long as they can be broadcast to a common shape.

Args

values: Tensors or native tensors or numbers or sequences of numbers.
rel_tolerance: Relative tolerance.
abs_tolerance: Absolute tolerance.
msg: Optional error message.
verbose: Whether to print conflicting values.
equal_nan: If False, NaN values will always trigger an assertion error.

def at_max(value, key: phiml.math._tensors.Tensor, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>)

Looks up the values of value at the positions where the maximum values in key are located along dim.

Args

value: Tensors or trees from which to lookup and return values. These tensors are indexed at the maximum index in `key´. You can pass arange() (the type) to retrieve the picked indices.
key: Tensor containing at least one dimension of dim. The maximum index of key is determined.
dim: Dimensions along which to compute the maximum of key.

Returns

The values of other_tensors at the positions where the maximum values in value are located along dim.

def at_max_neighbor(values, key_grid: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None, offsets=(0, 1), diagonal=True) ‑> phiml.math._tensors.Tensor

Computes the min of neighboring values in key_grid along each dimension in dims and retrieves the corresponding values from values.

Args

values: Values to look up and return. Tensor or tree structure.
key_grid: Values to compare.
dims: Dimensions along which neighbors should be averaged.
padding: Padding at the upper edges of grid along dims'. If notNone, the result tensor() will have the same shape() as grid`.
offsets: Relative neighbor indices as int. 0 refers to self, negative values to earlier (left) neighbors and positive values to later (right) neighbors.
diagonal: If True, performs sequential reductions along each axis, determining the minimum value along each axis independently. If the values of key_grid depend on values or their position in the grid, this can lead to undesired behavior.

Returns

Tree or Tensor like values.

def at_min(value, key: phiml.math._tensors.Tensor, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>)

Looks up the values of value at the positions where the minimum values in key are located along dim.

Args

value: Tensors or trees from which to lookup and return values. These tensors are indexed at the minimum index in `key´. You can pass arange() (the type) to retrieve the picked indices.
key: Tensor containing at least one dimension of dim. The minimum index of key is determined.
dim: Dimensions along which to compute the minimum of key.

Returns

The values of other_tensors at the positions where the minimum values in value are located along dim.

def at_min_neighbor(values, key_grid: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None, offsets=(0, 1), diagonal=True) ‑> phiml.math._tensors.Tensor

Computes the max of neighboring values in key_grid along each dimension in dims and retrieves the corresponding values from values.

Args

values: Values to look up and return.
key_grid: Values to compare.
dims: Dimensions along which neighbors should be averaged.
padding: Padding at the upper edges of grid along dims'. If notNone, the result tensor() will have the same shape() as grid`.
offsets: Relative neighbor indices as int. 0 refers to self, negative values to earlier (left) neighbors and positive values to later (right) neighbors.
diagonal: If True, performs sequential reductions along each axis, determining the minimum value along each axis independently. If the values of key_grid depend on values or their position in the grid, this can lead to undesired behavior.

Returns

Tree or Tensor like values.

def b2i(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all batch dimensions of value to instance dimensions. See rename_dims().

def batch(*args, **dims: Union[int, str, tuple, list, phiml.math._shape.Shape, ForwardRef('Tensor')])

Returns the batch dimensions of an existing Shape or creates a new Shape with only batch dimensions.

Usage for filtering batch dimensions:

>>> batch_dims = batch(shape)
>>> batch_dims = batch(tensor)

Usage for creating a Shape with only batch dimensions:

>>> batch_shape = batch('undef', batch=2)
(batch=2, undef=None)

Here, the dimension undef is created with an undefined size of None. Undefined sizes are automatically filled in by tensor(), wrap(), stack() and concat().

To create a shape with multiple types, use merge_shapes(), concat_shapes() or the syntax shape1 & shape2.

See Also: channel(), spatial(), instance()

Args

*args

Either

Shape or Tensor to filter or
Names of dimensions with undefined sizes as str.

**dims

Dimension sizes and names. Must be empty when used as a filter operation.

Returns

Shape containing only dimensions of type batch.

def boolean_mask(x, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], mask: phiml.math._tensors.Tensor, preserve_names=False)

Discards values x.dim[i] where mask.dim[i]=False. All dimensions of mask that are not dim are treated as batch dimensions.

Alternative syntax: x.dim[mask].

Implementations:

NumPy: Slicing
PyTorch: masked_select
TensorFlow: tf.boolean_mask
Jax: Slicing

Args

x: Tensor or Sliceable.
dim: Dimension of x to along which to discard slices.
mask: Boolean Tensor marking which values to keep. Must have the dimension dim matching `x´.
preserve_names: This only supports uniform 1D slicing. Batched slicing will remove item names if incompatible.

Returns

Selected values of x as Tensor with dimensions from x and mask.

def broadcast(function=None, dims=<function shape>, range=builtins.range, unwrap_scalars=True)

Function decorator for non-vectorized functions. When passing Tensor arguments to a broadcast function, the function is called once for each slice of the tensor. How tensors are sliced is determined by dims. Decorating a function with broadcast() is equal to passing the function to phi.math.map().

Args

function: Function to broadcast.
dims: Dimensions which should be sliced. function is called once for each element in dims, i.e. dims.volume times. If dims is not specified, all dimensions from the Sliceable values in args and kwargs will be mapped.
range: Optional range function. Can be used to generate tqdm output by passing trange.
unwrap_scalars: If True, passes the contents of scalar Tensors instead of the tensor objects.

Returns

Broadcast function

def c2b(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all channel dimensions of value to batch dimensions. See rename_dims().

def c2d(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all channel dimensions of value to dual dimensions. See rename_dims().

def cast(x: ~MagicType, dtype: Union[phiml.backend._dtype.DType, type]) ‑> ~OtherMagicType

Casts x to a different data type.

Implementations:

NumPy: x.astype()
PyTorch: x.to()
TensorFlow: tf.cast
Jax: jax.numpy.array

See Also: to_float(), to_int32(), to_int64(), to_complex().

Args

x: Tensor
dtype: New data type as DType, e.g. DType(int, 16).

Returns

Tensor with data type dtype()

def ceil(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes ⌈x⌉ of the Tensor or PhiTreeNode x.

def channel(*args, **dims: Union[int, str, tuple, list, phiml.math._shape.Shape, ForwardRef('Tensor')])

Returns the channel dimensions of an existing Shape or creates a new Shape with only channel dimensions.

Usage for filtering channel dimensions:

>>> channel_dims = channel(shape)
>>> channel_dims = channel(tensor)

Usage for creating a Shape with only channel dimensions:

>>> channel_shape = channel('undef', vector=2)
(vector=2, undef=None)

Here, the dimension undef is created with an undefined size of None. Undefined sizes are automatically filled in by tensor(), wrap(), stack() and concat().

To create a shape with multiple types, use merge_shapes(), concat_shapes() or the syntax shape1 & shape2.

See Also: spatial(), batch(), instance()

Args

*args

Either

Shape or Tensor to filter or
Names of dimensions with undefined sizes as str.

**dims

Dimension sizes and names. Must be empty when used as a filter operation.

Returns

Shape containing only dimensions of type channel.

def choose_backend(*values, prefer_default=False) ‑> phiml.backend._backend.Backend

Choose backend for given Tensor or native tensor values. Backends need to be registered to be available, e.g. via init() or set_global_default_backend().

Args

*values: Sequence of Tensors, native tensors or constants.
prefer_default: Whether to always select the default backend if it can work with values, see default_backend().

Returns

The selected phiml.math.backend.Backend

def clip(x: phiml.math._tensors.Tensor, lower_limit: Union[float, phiml.math._tensors.Tensor] = 0, upper_limit: Union[float, phiml.math._tensors.Tensor] = 1)

Limits the values of the Tensor x to lie between lower_limit and upper_limit (inclusive).

def clip_length(vec: phiml.math._tensors.Tensor, min_len=0, max_len=1, vec_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>, eps: Union[float, phiml.math._tensors.Tensor] = None)

Clips the length of a vector to the interval [min_len, max_len] while keeping the direction. Zero-vectors remain zero-vectors.

Args

vec: Tensor
min_len: Lower clipping threshold.
max_len: Upper clipping threshold.
vec_dim: Dimensions to compute the length over. By default, all channel dimensions are used to compute the vector length.
eps: Minimum vector length. Use to avoid inf gradients for zero-length vectors.

Returns

Tensor with same shape as vec().

def close(*tensors, rel_tolerance: Union[float, phiml.math._tensors.Tensor] = 1e-05, abs_tolerance: Union[float, phiml.math._tensors.Tensor] = 0, equal_nan=False, reduce=<function shape>) ‑> bool

Checks whether all tensors have equal values within the specified tolerance.

Does not check that the shapes exactly match. Unlike with always_close(), all shapes must be compatible and tensors with different shapes are reshaped before comparing.

Args

*tensors: At least two Tensor or tensor-like objects or None. The shapes of all tensors must be compatible but not all tensors must have all dimensions. If any argument is None, returns True only if all are None.
rel_tolerance: Relative tolerance
abs_tolerance: Absolute tolerance
equal_nan: If True, tensors are considered close if they are NaN in the same places.

Returns

bool, whether all given tensors are equal to the first tensor within the specified tolerance.

def closest_grid_values(grid: phiml.math._tensors.Tensor, coordinates: phiml.math._tensors.Tensor, extrap: e_.Extrapolation, stack_dim_prefix='closest_', **kwargs)

Finds the neighboring grid points in all directions and returns their values. The result will have 2^d values for each vector in coordinates in d dimensions.

If coordinates does not have a channel dimension with item names, the spatial dims of grid will be used.

Args

grid: grid data. The grid is spanned by the spatial dimensions of the tensor
coordinates: tensor with 1 channel dimension holding vectors pointing to locations in grid index space
extrap: grid extrapolation
stack_dim_prefix: For each spatial dimension dim, stacks lower and upper closest values along dimension stack_dim_prefix+dim.
kwargs: Additional information for the extrapolation.

Returns

Tensor of shape (batch, coord_spatial, grid_spatial=(2, 2,…), grid_channel)

def concat(values: Sequence[~PhiTreeNodeType], dim: Union[str, phiml.math._shape.Shape], expand_values=False, **kwargs) ‑> ~PhiTreeNodeType

Concatenates a sequence of Shapable objects, e.g. Tensor, along one dimension. All values must have the same spatial, instance and channel dimensions and their sizes must be equal, except for dim. Batch dimensions will be added as needed.

Args

values: Tuple or list of Shapable, such as Tensor
dim: Concatenation dimension, must be present in all values. The size along dim is determined from values and can be set to undefined (None). Alternatively, a str of the form 't->name:t' can be specified, where t is on of b d i s c denoting the dimension type. This first packs all dimensions of the input into a new dim with given name and type, then concatenates the values along this dim.
expand_values: If True, will first add missing dimensions to all values, not just batch dimensions. This allows tensors with different dimensions to be concatenated. The resulting tensor will have all dimensions that are present in values.
**kwargs: Additional keyword arguments required by specific implementations. Adding spatial dimensions to fields requires the bounds: Box argument specifying the physical extent of the new dimensions. Adding batch dimensions must always work without keyword arguments.

Returns

Concatenated Tensor

Examples

>>> concat([math.zeros(batch(b=10)), math.ones(batch(b=10))], 'b')
(bᵇ=20) 0.500 ± 0.500 (0e+00...1e+00)

>>> concat([vec(x=1, y=0), vec(z=2.)], 'vector')
(x=1.000, y=0.000, z=2.000) float64

def concat_shapes(*shapes: Union[phiml.math._shape.Shape, Any]) ‑> phiml.math._shape.Shape

Creates a Shape listing the dimensions of all shapes in the given order.

Args

*shapes: Shapes to concatenate. No two shapes must contain a dimension with the same name.

Returns

Combined Shape.

def conjugate(x: ~TensorOrTree) ‑> ~TensorOrTree

Args

x: Real or complex Tensor or PhiTreeNode or native tensor.

Returns

Complex conjugate of x if x is complex, else x.

def const_vec(value: Union[float, phiml.math._tensors.Tensor], dim: Union[phiml.math._shape.Shape, tuple, list, str])

Creates a single-dimension tensor with all values equal to value. value is not converted to the default backend, even when it is a Python primitive.

Args

value: Value for filling the vector.
dim: Either single-dimension non-spatial Shape or Shape consisting of any number of spatial dimensions. In the latter case, a new channel dimension named 'vector' will be created from the spatial shape.

Returns

Tensor

def contains(values: phiml.math._tensors.Tensor, query: phiml.math._tensors.Tensor, feature_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>) ‑> phiml.math._tensors.Tensor

For each query item, checks whether it is contained in values.

Args

values: Data Tensor containing all feature_dims. All non-batch and dims not specified as feature_dims are flattened.
query: Items to count the occurrences of. Must contain all feature_dims.
feature_dims: One item is considered to be the set of all values along feature_dims. The number of items in a tensor is given by all dims except feature_dims.

Returns

Integer Tensor matching query without feature_dims.

def convert(x, backend: phiml.backend._backend.Backend = None, use_dlpack=True)

Convert the native representation of a Tensor or PhiTreeNode to the native format of backend.

Warning: This operation breaks the automatic differentiation chain.

See Also: phiml.math.backend.convert().

Args

x: Tensor to convert. If x is a PhiTreeNode, its variable attributes are converted.
backend: Target backend. If None, uses the current default backend, see phiml.math.backend.default_backend().

Returns

Tensor with native representation belonging to backend.

def convolve(value: phiml.math._tensors.Tensor, kernel: phiml.math._tensors.Tensor, extrapolation: e_.Extrapolation = None, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>) ‑> phiml.math._tensors.Tensor

Computes the convolution of value and kernel along the specified dims.

Dual dims of kernel are reduced against the corresponding primal dims of value. All other primal dims of value are treated as batch.

Args

value: Tensor whose shape includes all spatial dimensions of kernel.
kernel: Tensor used as convolutional filter.
dims: Which dimensions to convolve over. Defaults to all spatial dims.
extrapolation: If not None, pads value so that the result has the same shape as value.

Returns

Tensor with all non-reduced dims of value and additional non-dual dims from kernel.

def copy(value: phiml.math._tensors.Tensor)

Copies the data buffer and encapsulating Tensor object.

Args

value: Tensor to be copied.

Returns

Copy of value.

def copy_with(obj: ~PhiTreeNodeType, **updates) ‑> ~PhiTreeNodeType

Creates a copy of the given PhiTreeNode with updated values as specified in updates.

If obj overrides __with_attrs__, the copy will be created via that specific implementation. Otherwise, the copy() module and setattr will be used.

Args

obj: PhiTreeNode
**updates: Values to be replaced.

Returns

Copy of obj with updated values.

def cos(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes cos(x) of the Tensor or PhiTreeNode x.

def cosh(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes cosh(x) of the Tensor or PhiTreeNode x.

def count_intersections(values: phiml.math._tensors.Tensor, arg_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], list_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function instance>, feature_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>) ‑> phiml.math._tensors.Tensor

Counts the number of elements that are part of each pair of lists.

Args

values:
arg_dims: Dims enumerating the input lists.
list_dims: Dims listing the elements.
feature_dims: Vector dims of one element. Elements are equal if all values along feature_dims are equal.

Returns

Tensor.

def count_occurrences(values: phiml.math._tensors.Tensor, query: phiml.math._tensors.Tensor, feature_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>) ‑> phiml.math._tensors.Tensor

For each query item, counts how often this value occurs in values.

Args

values: Data Tensor containing all feature_dims. All non-batch and dims not specified as feature_dims are flattened.
query: Items to count the occurrences of. Must contain all feature_dims.
feature_dims: One item is considered to be the set of all values along feature_dims. The number of items in a tensor is given by all dims except feature_dims.

Returns

Integer Tensor matching query without feature_dims.

def cross(vec1: phiml.math._tensors.Tensor, vec2: phiml.math._tensors.Tensor) ‑> phiml.math._tensors.Tensor

Computes the cross product of two vectors in 2D.

Args

vec1: Tensor with a single channel dimension called 'vector'
vec2: Tensor with a single channel dimension called 'vector'

Returns

Tensor

def cross_product(vec1: phiml.math._tensors.Tensor, vec2: phiml.math._tensors.Tensor) ‑> phiml.math._tensors.Tensor

Computes the cross product of two vectors in 2D.

Args

vec1: Tensor with a single channel dimension called 'vector'
vec2: Tensor with a single channel dimension called 'vector'

Returns

Tensor

def cumulative_sum(x: phiml.math._tensors.Tensor, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], include_0=False, include_sum=True, index_dim: Union[str, phiml.math._shape.Shape, None] = None)

Performs a cumulative sum of x along dim.

Implementations:

NumPy: cumsum
PyTorch: cumsum
TensorFlow: cumsum
Jax: cumsum

Args

x: Tensor
dim: Dimension along which to sum, as str or Shape. If multiple dims are passed, x the cumulative sum will be computed on the flattened array.
include_0: If True, adds a 0 to the result before the first value.
include_sum: If False, the total sum will be sliced off the result.
index_dim: If given, adds an index dimension for dim.

Returns

Tensor with the same shape as x.

def custom_gradient(f: Callable, gradient: Callable, auxiliary_args: str = '')

Creates a function based on f that uses a custom gradient for the backpropagation pass.

Warning This method can lead to memory leaks if the gradient function is not called. Make sure to pass tensors without gradients if the gradient is not required, see stop_gradient().

Args

f: Forward function mapping Tensor arguments x to a single Tensor output or sequence of tensors y.
gradient: Function to compute the vector-Jacobian product for backpropagation. Will be called as gradient(input_dict, *y, *dy) -> output_dict where input_dict contains all named arguments passed to the forward function and output_dict contains only those parameters for which a gradient is defined.
auxiliary_args: Comma-separated parameter names of arguments that are not relevant to backpropagation.

Returns

Function with similar signature and return values as f. However, the returned function does not support keyword arguments.

def d2i(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all dual dimensions of value to instance dimensions. See rename_dims().

def d2s(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all dual dimensions of value to spatial dimensions. See rename_dims().

def degrees_to_radians(deg: ~TensorOrTree) ‑> ~TensorOrTree

Convert degrees to radians.

def dense(x: phiml.math._tensors.Tensor) ‑> phiml.math._tensors.Tensor

Convert a sparse tensor representation to an equivalent dense one in which all values are explicitly stored contiguously in memory.

Args

x: Any Tensor. Python primitives like float, int or bool will be converted to Tensors in the process.

Returns

Dense tensor.

def dim_mask(all_dims: Union[phiml.math._shape.Shape, tuple, list], dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], mask_dim=(vectorᶜ=None)) ‑> phiml.math._tensors.Tensor

Creates a masked vector with 1 elements for dims and 0 for all other dimensions in all_dims.

Args

all_dims: All dimensions for which the vector should have an entry.
dims: Dimensions marked as 1.
mask_dim: Dimension of the masked vector. Item names are assigned automatically.

Returns

Tensor

def dot(x: phiml.math._tensors.Tensor, x_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], y: phiml.math._tensors.Tensor, y_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None]) ‑> phiml.math._tensors.Tensor

Computes the dot product along the specified dimensions. Contracts x_dims with y_dims by first multiplying the elements and then summing them up.

For one dimension, this is equal to matrix-matrix or matrix-vector multiplication.

The function replaces the traditional dot() / tensordot / matmul / einsum functions.

NumPy: numpy.tensordot, numpy.einsum
PyTorch: torch.tensordot, torch.einsum
TensorFlow: tf.tensordot, tf.einsum
Jax: jax.numpy.tensordot, jax.numpy.einsum

Args

x: First Tensor
x_dims: Dimensions of x to reduce against y
y: Second Tensor
y_dims: Dimensions of y to reduce against x.

Returns

Dot product as Tensor.

def downsample2x(grid: phiml.math._tensors.Tensor, padding: Extrapolation = zero-gradient, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>) ‑> phiml.math._tensors.Tensor

Resamples a regular grid to half the number of spatial sample points per dimension. The grid values at the new points are determined via mean (linear interpolation).

Args

grid: full size grid
padding: grid extrapolation. Used to insert an additional value for odd spatial dims
dims: dims along which down-sampling is applied. If None, down-sample along all spatial dims.
grid: Tensor:
padding: Extrapolation: (Default value = extrapolation.BOUNDARY)
dims: tuple or None: (Default value = None)

Returns

half-size grid

def dtype(x) ‑> phiml.backend._dtype.DType

Returns the data type of x.

Args

x: Tensor or native tensor.

Returns

DType

def dual(*args, **dims: Union[int, str, tuple, list, phiml.math._shape.Shape, ForwardRef('Tensor')])

Returns the dual dimensions of an existing Shape or creates a new Shape with only dual dimensions.

Dual dimensions are assigned the prefix ~ to distinguish them from regular dimensions. This way, a regular and dual dimension of the same name can exist in one Shape.

Dual dimensions represent the input space and are typically only present on matrices or higher-order matrices. Dual dimensions behave like batch dimensions in regular operations, if supported. During matrix multiplication, they are matched against their regular counterparts by name (ignoring the ~ prefix).

Usage for filtering dual dimensions:

>>> dual_dims = dual(shape)
>>> dual_dims = dual(tensor)

Usage for creating a Shape with only dual dimensions:

>>> dual('undef', points=2)
(~undefᵈ=None, ~pointsᵈ=2)

Here, the dimension undef is created with an undefined size of None. Undefined sizes are automatically filled in by tensor(), wrap(), stack() and concat().

To create a shape with multiple types, use merge_shapes(), concat_shapes() or the syntax shape1 & shape2.

See Also: channel(), batch(), spatial()

Args

*args

Either

Shape or Tensor to filter or
Names of dimensions with undefined sizes as str.

**dims

Dimension sizes and names. Must be empty when used as a filter operation.

Returns

Shape containing only dimensions of type dual.

def eigenvalues(matrix: phiml.math._tensors.Tensor, eigen_dim=(eigenvaluesᶜ=None))

Computes the eigenvalues of a square matrix. The matrix columns are listed along dual dimensions and the rows are listed along the corresponding non-dual dimensions. Row dims are matched by name if possible, else all primal dims are used.

Args

matrix: Square matrix. Must have at least one dual dim and corresponding non-dual dim.
eigen_dim: Dimension along which eigenvalues should be listed.

Returns

Tensor listing the eigenvalues along eigen_dim.

def enable_debug_checks()

Once called, additional type checks are enabled. This may result in a noticeable drop in performance.

def equal(*objects, equal_nan=False) ‑> bool

Checks whether all objects are equal.

Args

*objects: Objects to compare. Can be tensors or other objects or None
equal_nan: If all objects are tensor-like, whether to count NaN values as equal.

Returns

bool, whether all given objects are equal to the first one.

def erf(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the error function erf(x) of the Tensor or PhiTreeNode x.

def exp(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes exp(x) of the Tensor or PhiTreeNode x.

def expand(value, *dims: Union[str, phiml.math._shape.Shape], **kwargs)

Adds dimensions to a Tensor or tensor-like object by implicitly repeating the tensor values along the new dimensions. If value already contains any of the new dimensions, a size and type check is performed for these instead.

If any of dims varies along a dimension that is present neither in value nor on dims, it will also be added to value.

This function replaces the usual tile / repeat functions of NumPy, PyTorch, TensorFlow and Jax.

Additionally, it replaces the traditional unsqueeze / expand_dims functions.

Args

value: Shapable, such as Tensor For tree nodes, expands all value attributes by dims or the first variable attribute if no value attributes are set.
*dims: Dimensions to be added as Shape
**kwargs: Additional keyword arguments required by specific implementations. Adding spatial dimensions to fields requires the bounds: Box argument specifying the physical extent of the new dimensions. Adding batch dimensions must always work without keyword arguments.

Returns

Same type as value.

def factor_ilu(matrix: phiml.math._tensors.Tensor, iterations: int, safe=False)

Incomplete LU factorization for dense or sparse matrices.

For sparse matrices, keeps the sparsity pattern of matrix. L and U will be trimmed to the respective areas, i.e. stored upper elements in L will be dropped, unless this would lead to varying numbers of stored elements along a batch dimension.

Args

matrix: Dense or sparse matrix to factor. Currently, compressed sparse matrices are decompressed before running the ILU algorithm.
iterations: (Optional) Number of fixed-point iterations to perform. If not given, will be automatically determined from matrix size and sparsity.
safe: If False (default), only matrices with a rank deficiency of up to 1 can be factored as all values of L and U are uniquely determined. For matrices with higher rank deficiencies, the result includes NaN values. If True, the algorithm runs slightly slower but can factor highly rank-deficient matrices as well. However, then L is undeterdetermined and unused values of L are set to 0. Rank deficiencies of 1 occur frequently in periodic settings but higher ones are rare.

Returns

L: Lower-triangular matrix as Tensor with all diagonal elements equal to 1.
U: Upper-triangular matrix as Tensor.

Examples

>>> matrix = wrap([[-2, 1, 0],
>>>                [1, -2, 1],
>>>                [0, 1, -2]], channel('row'), dual('col'))
>>> L, U = math.factor_ilu(matrix)
>>> math.print(L)
row=0      1.          0.          0.         along ~col
row=1     -0.5         1.          0.         along ~col
row=2      0.         -0.6666667   1.         along ~col
>>> math.print(L @ U, "L @ U")
            L @ U
row=0     -2.   1.   0.  along ~col
row=1      1.  -2.   1.  along ~col
row=2      0.   1.  -2.  along ~col

def factorial(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes factorial(x) of the Tensor or PhiTreeNode x. For floating-point numbers computes the continuous factorial using the gamma function. For integer numbers computes the exact factorial and returns the same integer type. However, this results in integer overflow for inputs larger than 12 (int32) or 19 (int64).

def fft(x: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>) ‑> phiml.math._tensors.Tensor

Performs a fast Fourier transform (FFT) on all spatial dimensions of x.

The inverse operation is ifft().

Implementations:

NumPy: np.fft.fft, numpy.fft.fft2, numpy.fft.fftn
PyTorch: torch.fft.fft
TensorFlow: tf.signal.fft, tf.signal.fft2d, tf.signal.fft3d
Jax: jax.numpy.fft.fft, jax.numpy.fft.fft2 jax.numpy.fft.fft

Args

x: Uniform complex or float Tensor with at least one spatial dimension.
dims: Dimensions along which to perform the FFT. If None, performs the FFT along all spatial dimensions of x.

Returns

Ƒ(x) as complex Tensor

def fftfreq(resolution: phiml.math._shape.Shape, dx: Union[float, phiml.math._tensors.Tensor] = 1, dtype: phiml.backend._dtype.DType = None)

Returns the discrete Fourier transform sample frequencies. These are the frequencies corresponding to the components of the result of math.fft on a tensor of shape resolution.

Args

resolution: Grid resolution measured in cells
dx: Distance between sampling points in real space.
dtype: Data type of the returned tensor (Default value = None)

Returns

Tensor holding the frequencies of the corresponding values computed by math.fft

def find_closest(vectors: phiml.math._tensors.Tensor, query: phiml.math._tensors.Tensor, method='kd', index_dim=(indexᶜ=None))

Finds the closest vector to query from vectors. This is implemented using a k-d tree built from vectors.

Args

vectors

Points to find.

query

Target locations.

method

One of the following:

'dense': compute the pair-wise distances between all vectors and query points, then return the index of the smallest distance for each query point.
'kd' (default): Build a k-d tree from vectors and use it to query all points in query. The tree will be cached if this call is jit-compiled and vectors is constant.

index_dim

Dimension along which components should be listed as Shape. Pass None to get 1D indices as scalars.

Returns

Index tensor idx so that the closest points to query are vectors[idx].

def find_differences(tree1, tree2, compare_tensors_by_id=False, attr_type=<function value_attributes>, tensor_equality=None) ‑> Sequence[Tuple[str, str, Any, Any]]

Compares tree1 and tree2 and returns all differences in the form (difference_description: str, variable_identifier: str, value1, value2).

Args

tree1: Nested tree or leaf
tree2: Nested tree or leaf
compare_tensors_by_id: Whether Tensor objects should be compared by identity or values.
attr_type: What attributes to compare, either value_attributes or variable_attributes.
tensor_equality: Function that compares two tensors for equality. None defaults to equal().

Returns

List of differences, each represented as a tuple.

def finite_fill(values: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, distance: int = 1, diagonal: bool = True, padding=zero-gradient, padding_kwargs: dict = None) ‑> phiml.math._tensors.Tensor

Fills non-finite (NaN, inf, -inf) values from nearby finite values. Extrapolates the finite values of values for distance steps along dims. Where multiple finite values could fill an invalid value, the average is computed.

Args

values: Floating-point Tensor. All non-numeric values (NaN, inf, -inf) are interpreted as invalid.
dims: Dimensions along which to fill invalid values from finite ones.
distance: Number of extrapolation steps, each extrapolating one cell out.
diagonal: Whether to extrapolate values to their diagonal neighbors per step.
padding: Extrapolation of values. Determines whether to extrapolate from the edges as well.
padding_kwargs: Additional keyword arguments to be passed to pad().

Returns

Tensor of same shape as values.

def finite_max(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, default: Union[complex, float] = nan)

Finds the maximum along dim ignoring all non-finite values.

Args

value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

default

Value to use where no finite value was encountered.

Returns

Tensor without the reduced dimensions.

def finite_mean(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, default: Union[complex, float] = nan)

Computes the mean value of all finite values in value along dim.

Args

value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

default

Value to use where no finite value was encountered.

Returns

Tensor without the reduced dimensions.

def finite_min(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, default: Union[complex, float] = nan)

Finds the minimum along dim ignoring all non-finite values.

Args

value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

default

Value to use where no finite value was encountered.

Returns

Tensor without the reduced dimensions.

def finite_sum(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, default: Union[complex, float] = nan)

Sums all finite values in value along dim.

Args

value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

default

Value to use where no finite value was encountered.

Returns

Tensor without the reduced dimensions.

def flatten(value, flat_dim: phiml.math._shape.Shape = (flatⁱ=None), flatten_batch=False, **kwargs)

Returns a Tensor with the same values as value but only a single dimension flat_dim. The order of the values in memory is not changed.

Args

value: Shapable, such as Tensor. If a non-Shaped object or one with an empty Shape is passed, it is returned without alteration.
flat_dim: Dimension name and type as Shape object. The size is ignored.
flatten_batch: Whether to flatten batch dimensions as well. If False, batch dimensions are kept, only onn-batch dimensions are flattened.
**kwargs: Additional keyword arguments required by specific implementations. Adding spatial dimensions to fields requires the bounds: Box argument specifying the physical extent of the new dimensions. Adding batch dimensions must always work without keyword arguments.

Returns

Same type as value.

Examples

>>> flatten(math.zeros(spatial(x=4, y=3)))
(flatⁱ=12) const 0.0

def floor(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes ⌊x⌋ of the Tensor or PhiTreeNode x.

def fourier_laplace(grid: phiml.math._tensors.Tensor, dx: Union[phiml.math._tensors.Tensor, phiml.math._shape.Shape, float, list, tuple], times: int = 1)

Applies the spatial laplace operator to the given tensor with periodic boundary conditions.

Note: The results of fourier_laplace() and laplace() are close but not identical.

This implementation computes the laplace operator in Fourier space. The result for periodic fields is exact, i.e. no numerical instabilities can occur, even for higher-order derivatives.

Args

grid: tensor, assumed to have periodic boundary conditions
dx: distance between grid points, tensor-like, scalar or vector
times: number of times the laplace operator is applied. The computational cost is independent of this parameter.
grid: Tensor:
dx: Tensor or Shape or float or list or tuple:
times: int: (Default value = 1)

Returns

tensor of same shape as tensor()

def fourier_poisson(grid: phiml.math._tensors.Tensor, dx: Union[phiml.math._tensors.Tensor, phiml.math._shape.Shape, float, list, tuple], times: int = 1)

Inverse operation to fourier_laplace().

Args

grid: Tensor:
dx: Tensor or Shape or float or list or tuple:
times: int: (Default value = 1)

Returns:

def frequency_loss(x, frequency_falloff: float = 100, threshold=1e-05, ignore_mean=False, n=2) ‑> phiml.math._tensors.Tensor

Penalizes the squared values in frequency (Fourier) space. Lower frequencies are weighted more strongly then higher frequencies, depending on frequency_falloff.

Args

x: Tensor or PhiTreeNode Values to penalize, typically actual - target.
frequency_falloff: Large values put more emphasis on lower frequencies, 1.0 weights all frequencies equally. Note: The total loss is not normalized. Varying the value will result in losses of different magnitudes.
threshold: Frequency amplitudes below this value are ignored. Setting this to zero may cause infinities or NaN values during backpropagation.
ignore_mean: If True, does not penalize the mean value (frequency=0 component).

Returns

Scalar loss value

def from_dict(dict_: dict, convert=False)

Loads a Tensor or Shape from a serialized form.

Args

dict_: Serialized tensor properties.
convert: Whether to convert the data to the current backend format or keep it as a Numpy array.

Returns

Tensor or Shape.

def gather(values, indices: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = None, pref_index_dim='index')

Gathers the entries of values at positions described by indices. All non-channel dimensions of indices that are part of values but not indexed are treated as batch dimensions.

Args

values: Tensor or phiml.math.matic.PhiTreeNode containing values to gather.
indices: int Tensor. Multidimensional position references in values. Must contain a single channel dimension for the index vector matching the number of dimensions to index. This channel dimension should list the dimension names to index as item names unless explicitly specified as dims.
dims: (Optional) Dimensions indexed by indices. Alternatively, the dimensions can be specified as the item names of the channel dimension of indices. If None and no index item names are specified, will default to all spatial dimensions or all instance dimensions, depending on which ones are present (but not both).
pref_index_dim: In case indices has multiple channel dims, use this dim as the index, treating the others as batch. Has no effect if indices only has one channel dim.

Returns

Tensor with combined batch dimensions, channel dimensions of values and spatial/instance dimensions of indices.

def get_format(x: phiml.math._tensors.Tensor) ‑> str

Returns the sparse storage format of a tensor.

Args

x: Tensor

Returns

One of 'coo', 'csr', 'csc', 'dense'.

def get_precision() ‑> int

Gets the current target floating point precision in bits. The precision can be set globally using set_global_precision() or locally using with precision(p):.

Any Backend method may convert floating point values to this precision, even if the input had a different precision.

Returns

16 for half, 32 for single, 64 for double

def get_sparsity(x: phiml.math._tensors.Tensor)

Fraction of values currently stored on disk for the given Tensor x. For sparse tensors, this is nnz / shape.

This is a lower limit on the number of values that will need to be processed for operations involving x. The actual number is often higher since many operations require data be laid out in a certain format. In these cases, missing values, such as zeros, are filled in before the operation.

The following operations may return tensors whose values are only partially stored:

expand()
phiml.math.pairwise_distance() with max_distance set.
Tracers used in jit_compile_linear()
Stacking any of the above.

Args

x: Tensor

Returns

The number of values that are actually stored on disk. This does not include additional information, such as position information / indices. For sparse matrices, this is equal to the number of nonzero values.

def gradient(f: Callable, wrt: str = None, get_output=True) ‑> Callable

Creates a function which computes the gradient of f.

Example:

def loss_function(x, y):
    prediction = f(x)
    loss = math.l2_loss(prediction - y)
    return loss, prediction

dx = gradient(loss_function, 'x', get_output=False)(x, y)

(loss, prediction), (dx, dy) = gradient(loss_function,
                                        'x,y', get_output=True)(x, y)

Functional gradients are implemented for the following backends:

PyTorch: torch.autograd.grad / torch.autograd.backward
TensorFlow: tf.GradientTape
Jax: jax.grad

When the gradient function is invoked, f is called with tensors that track the gradient. For PyTorch, arg.requires_grad = True for all positional arguments of f.

Args

f: Function to be differentiated. f must return a floating point Tensor with rank zero. It can return additional tensors which are treated as auxiliary data and will be returned by the gradient function if return_values=True. All arguments for which the gradient is computed must be of dtype float or complex.
get_output: Whether the gradient function should also return the return values of f.
wrt: Comma-separated parameter names of f with respect to which the gradient should be computed. If not specified, the gradient will be computed w.r.t. the first positional argument (highly discouraged).

Returns

Function with the same arguments as f that returns the value of f, auxiliary data and gradient of f if get_output=True, else just the gradient of f.

def grid_sample(grid: phiml.math._tensors.Tensor, coordinates: phiml.math._tensors.Tensor, extrap: Union[ForwardRef('e_.Extrapolation'), float, str], **kwargs)

Samples values of grid at the locations referenced by coordinates. Values lying in between sample points are determined via linear interpolation.

If coordinates has a channel dimension, its item names are used to determine the grid dimensions of grid. Otherwise, the spatial dims of grid will be used.

For values outside the valid bounds of grid (coord < 0 or coord > grid.shape - 1), extrap is used to determine the neighboring grid values. If the extrapolation does not support resampling, the grid is padded by one cell layer before resampling. In that case, values lying further outside will not be sampled according to the extrapolation.

Args

grid: Grid with at least one spatial dimension and no instance dimensions.
coordinates: Coordinates with a single channel dimension called 'vector'. The size of the vector dimension must match the number of spatial dimensions of grid.
extrap: Extrapolation used to determine the values of grid outside its valid bounds.
kwargs: Additional information for the extrapolation.

Returns

Tensor with channel dimensions of grid, spatial and instance dimensions of coordinates and combined batch dimensions.

def histogram(values: phiml.math._tensors.Tensor, bins: phiml.math._shape.Shape = (binsˢ=30), weights=1, same_bins: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = None)

Compute a histogram of a distribution of values.

Important Note: In its current implementation, values outside the range of bins may or may not be added to the outermost bins.

Args

values: Tensor listing the values to be binned along spatial or instance dimensions. `values´ may not contain channel or dual dimensions.
bins: Either Shape specifying the number of equally-spaced bins to use or bin edge positions as Tensor with a spatial or instance dimension.
weights: Tensor assigning a weight to every value in values that will be added to the bin, default 1.
same_bins: Only used if bins is given as a Shape. Use the same bin sizes and positions across these batch dimensions. By default, bins will be chosen independently for each example.

Returns

hist: Tensor containing all batch dimensions and the bins dimension with dtype matching weights.
bin_edges: Tensor
bin_center: Tensor

def i2b(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all instance dimensions of value to batch dimensions. See rename_dims().

def identity(x)

Identity function for one argument. Vararg functions cannot be transformed as the argument names are unknown.

Args

x: Positional argument.

Returns

x

def ifft(k: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>)

Inverse of fft().

Args

k: Complex or float Tensor with at least one spatial dimension.
dims: Dimensions along which to perform the inverse FFT. If None, performs the inverse FFT along all spatial dimensions of k.

Returns

Ƒ^-1(k) as complex Tensor

def imag(x: ~TensorOrTree) ‑> ~TensorOrTree

Returns the imaginary part of x. If x does not store complex numbers, returns a zero tensor with the same shape and dtype as this tensor.

Args

x: Tensor or PhiTreeNode or native tensor.

Returns

Imaginary component of x if x is complex, zeros otherwise.

def incomplete_gamma(a: ~TensorOrTree, x: ~TensorOrTree, upper=False, regularized=True) ‑> ~TensorOrTree

Computes the incomplete gamma function.

Args

a: Positive parameter, Tensor or tree.
x: Non-negative argument, Tensor or tree.
upper: Whether to complete the upper integral (x to infinity) or the lower integral (0 to x).
regularized: Whether the integral is divided by Γ(a).

def index_shift(x: phiml.math._tensors.Tensor, offsets: Sequence[Union[int, phiml.math._tensors.Tensor]], padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None) ‑> List[phiml.math._tensors.Tensor]

Returns shifted versions of x according to offsets where each offset is an int vector indexing some dimensions of x.

See Also: shift(), neighbor_reduce().

Args

x: Input grid-like Tensor.
offsets: Sequence of offset vectors. Each offset is an int vector indexing some dimensions of x. Offsets can have different subsets of the dimensions of x. Missing dimensions count as 0. The value 0 can also be passed as a zero-shift.
padding: Padding to be performed at the boundary so that the shifted versions have the same size as x. Must be one of the following: Extrapolation, Tensor or number for constant extrapolation, name of extrapolation as str. Can be set to None to disable padding. Then the result tensors will be smaller than x.

Returns

list of shifted tensors. The number of return tensors is equal to the number of offsets.

def instance(*args, **dims: Union[int, str, tuple, list, phiml.math._shape.Shape, ForwardRef('Tensor')])

Returns the instance dimensions of an existing Shape or creates a new Shape with only instance dimensions.

Usage for filtering instance dimensions:

>>> instance_dims = instance(shape)
>>> instance_dims = instance(tensor)

Usage for creating a Shape with only instance dimensions:

>>> instance_shape = instance('undef', points=2)
(points=2, undef=None)

Here, the dimension undef is created with an undefined size of None. Undefined sizes are automatically filled in by tensor(), wrap(), stack() and concat().

To create a shape with multiple types, use merge_shapes(), concat_shapes() or the syntax shape1 & shape2.

See Also: channel(), batch(), spatial()

Args

*args

Either

Shape or Tensor to filter or
Names of dimensions with undefined sizes as str.

**dims

Dimension sizes and names. Must be empty when used as a filter operation.

Returns

Shape containing only dimensions of type instance.

def is_finite(x: ~TensorOrTree) ‑> ~TensorOrTree

Returns a Tensor or PhiTreeNode matching x with values True where x has a finite value and False otherwise.

def is_inf(x: ~TensorOrTree) ‑> ~TensorOrTree

Returns a Tensor or PhiTreeNode matching x with values True where x is +inf or -inf and False otherwise.

def is_nan(x: ~TensorOrTree) ‑> ~TensorOrTree

Returns a Tensor or PhiTreeNode matching x with values True where x is NaN and False otherwise.

def is_scalar(value) ‑> bool

Checks whether value has no dimensions.

Args

value: Tensor or Python primitive or native tensor.

Returns

bool

def is_sparse(x: phiml.math._tensors.Tensor)

Checks whether a tensor is represented in COO, CSR or CSC format. If the tensor is neither sparse nor dense, this function raises an error.

Args

x: Tensor to test.

Returns

True if x is sparse, False if x is dense.

Raises

AssertionError if x is neither sparse nor fully dense.

def iterate(map_function: Callable, iterations: Union[int, phiml.math._shape.Shape], *x0, f_kwargs: dict = None, range: Callable = builtins.range, measure: Callable = None, substeps: int = 1, **f_kwargs_)

Repeatedly call function, passing the previous output as the next input.

If the function outputs more values than the number of arguments in x0, only the first len(x0) ones are passed to map_function. However, all outputs will be returned by iterate().

Args

map_function: Function to call. Must be callable as f(x0, **f_kwargs) and f(f(x0, **f_kwargs), **f_kwargs).
iterations: Number of iterations as int or single-dimension Shape. If int, returns the final output of map_function. If Shape, returns the trajectory (x0 and all outputs of map_function), stacking the values along this dimension.
x0: Initial positional arguments for map_function. Values that are initially None are not stacked with the other values if iterations is a Shape.
range: Range function. Can be used to generate tqdm output by passing trange.
measure: Function without arguments to call at the start and end (and in between if isinstance(iterations, Shape)) calls to map_function. The measure of each call to map_function is measure() after minus measure() before the call.
substeps: If > 1, iterates the function multiple times for each recorded step. The returned trajectories as well as measurements only record the large steps, not the sub-steps. The arange() is also only used on large steps, not sub-steps.
f_kwargs: Additional keyword arguments to be passed to map_function. These arguments can be of any type.
f_kwargs_: More keyword arguments.

Returns

final_or_trajectory: Stacked trajectory or final output of map_function, depending on iterations.
measured: Only if measure was specified, returns the measured value or trajectory tensor.

def jacobian(f: Callable, wrt: str = None, get_output=True) ‑> Callable

Creates a function which computes the Jacobian matrix of f. For scalar functions, consider using gradient() instead.

Example:

def f(x, y):
    prediction = f(x)
    loss = math.l2_loss(prediction - y)
    return loss, prediction

dx = jacobian(loss_function, wrt='x', get_output=False)(x, y)

(loss, prediction), (dx, dy) = jacobian(loss_function,
                                    wrt='x,y', get_output=True)(x, y)

Functional gradients are implemented for the following backends:

PyTorch: torch.autograd.grad / torch.autograd.backward
TensorFlow: tf.GradientTape
Jax: jax.grad

When the gradient function is invoked, f is called with tensors that track the gradient. For PyTorch, arg.requires_grad = True for all positional arguments of f.

Args

f: Function to be differentiated. f must return a floating point Tensor with rank zero. It can return additional tensors which are treated as auxiliary data and will be returned by the gradient function if return_values=True. All arguments for which the gradient is computed must be of dtype float or complex.
get_output: Whether the gradient function should also return the return values of f.
wrt: Comma-separated parameter names of f with respect to which the gradient should be computed. If not specified, the gradient will be computed w.r.t. the first positional argument (highly discouraged).

Returns

Function with the same arguments as f that returns the value of f, auxiliary data and Jacobian of f if get_output=True, else just the Jacobian of f.

def jit_compile(f: Callable = None, auxiliary_args: str = '', forget_traces: bool = None) ‑> Callable

Compiles a graph based on the function f. The graph compilation is performed just-in-time (jit), e.g. when the returned function is called for the first time.

The traced function will compute the same result as f but may run much faster. Some checks may be disabled in the compiled function.

Can be used as a decorator:

@math.jit_compile
def my_function(x: math.Tensor) -> math.Tensor:

Invoking the returned function may invoke re-tracing / re-compiling f after the first call if either

it is called with a different number of arguments,
the tensor arguments have different dimension names or types (the dimension order also counts),
any Tensor arguments require a different backend than previous invocations,
PhiTreeNode positional arguments do not match in non-variable properties.

Compilation is implemented for the following backends:

PyTorch: torch.jit.trace
TensorFlow: tf.function
Jax: jax.jit

Jit-compilations cannot be nested, i.e. you cannot call jit_compile() while another function is being compiled. An exception to this is jit_compile_linear() which can be called from within a jit-compiled function.

Args

f: Function to be traced. All positional arguments must be of type Tensor or PhiTreeNode returning a single Tensor or PhiTreeNode.
auxiliary_args: Comma-separated parameter names of arguments that are not relevant to backpropagation.
forget_traces: If True, only remembers the most recent compiled instance of this function. Upon tracing with new instance (due to changed shapes or auxiliary args), deletes the previous traces.

Returns

Function with similar signature and return values as f.

def jit_compile_linear(f: Callable[[~X], ~Y] = None, auxiliary_args: str = None, forget_traces: bool = None) ‑> phiml.math._functional.LinearFunction[~X, ~Y]

Compile an optimized representation of the linear function f. For backends that support sparse tensors, a sparse matrix will be constructed for f.

Can be used as a decorator:

@math.jit_compile_linear
def my_linear_function(x: math.Tensor) -> math.Tensor:

Unlike jit_compile(), jit_compile_linear() can be called during a regular jit compilation.

Args

f: Function that is linear in its positional arguments. All positional arguments must be of type Tensor and f must return a Tensor.
auxiliary_args: Which parameters f is not linear in. These arguments are treated as conditioning arguments and will cause re-tracing on change.
forget_traces: If True, only remembers the most recent compiled instance of this function. Upon tracing with new instance (due to changed shapes or auxiliary args), deletes the previous traces.

Returns

LinearFunction with similar signature and return values as f.

def l1_loss(x, reduce: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> phiml.math._tensors.Tensor

Computes ∑_i ||x_i||₁, summing over all non-batch dimensions.

Args

x: Tensor or PhiTreeNode or 0D or 1D native tensor. For PhiTreeNode objects, only value the sum over all value attributes is computed.
reduce: Dimensions to reduce as DimFilter.

Returns

loss: Tensor

def l2_loss(x, reduce: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> phiml.math._tensors.Tensor

Computes ∑_i ||x_i||₂² / 2, summing over all non-batch dimensions.

Args

x: Tensor or PhiTreeNode or 0D or 1D native tensor. For PhiTreeNode objects, only value the sum over all value attributes is computed.
reduce: Dimensions to reduce as DimFilter.

Returns

loss: Tensor

def laplace(x: phiml.math._tensors.Tensor, dx: Union[float, phiml.math._tensors.Tensor] = 1, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = zero-gradient, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, weights: phiml.math._tensors.Tensor = None, padding_kwargs: dict = None)

Spatial Laplace operator as defined for scalar fields. If a vector field is passed, the laplace is computed component-wise.

Args

x: n-dimensional field of shape (batch, spacial dimensions…, components)
dx: scalar or 1d tensor
padding: Padding mode. Must be one of the following: Extrapolation, Tensor or number for constant extrapolation, name of extrapolation as str.
dims: The second derivative along these dimensions is summed over
weights: (Optional) Multiply the axis terms by these factors before summation. Must be a Tensor with a single channel dimension that lists all laplace dims by name.
padding_kwargs: Additional keyword arguments to be passed to pad().

Returns

Tensor of same shape as x

def layout(objects, *shape: Union[str, phiml.math._shape.Shape]) ‑> phiml.math._tensors.Tensor

Wraps a Python tree in a Tensor, allowing elements to be accessed via dimensions. A python tree is a structure of nested tuple, list, dict and leaf objects where leaves can be any Python object.

All keys of dict containers must be of type str. The keys are automatically assigned as item names along that dimension unless conflicting with other elements.

Strings may also be used as containers.

Example:

>>> t = layout({'a': 'text', 'b': [0, 1]}, channel('dict,inner'))
>>> t.inner[1].dict['a'].native()
'e'

Args

objects: PyTree of list or tuple.
*shape: Tensor dimensions

Returns

Tensor. Calling Tensor.native() on the returned tensor will return objects.

def length(vec: phiml.math._tensors.Tensor, vec_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>, eps: Union[float, phiml.math._tensors.Tensor] = None)

Computes the vector length of vec().

Args

eps: Minimum valid vector length. Use to avoid inf gradients for zero-length vectors. Lengths shorter than eps are set to 0.

def linspace(start: Union[float, phiml.math._tensors.Tensor, tuple, list], stop: Union[float, phiml.math._tensors.Tensor, tuple, list], dim: phiml.math._shape.Shape) ‑> phiml.math._tensors.Tensor

Returns number evenly spaced numbers between start and stop along dim.

If dim contains multiple dimensions, evenly spaces values along each dimension, then stacks the result along a new channel dimension called vector.

Args

start: First value, int or Tensor.
stop: Last value, int or Tensor.
dim: Linspace dimension of integer size. The size determines how many values to linearly space between start and stop. The values will be laid out along dim.

Returns

Tensor

Examples

>>> math.linspace(0, 1, spatial(x=5))
(0.000, 0.250, 0.500, 0.750, 1.000) along xˢ

>>> math.linspace(0, (-1, 1), spatial(x=3))
(0.000, 0.000); (-0.500, 0.500); (-1.000, 1.000) (xˢ=3, vectorᶜ=2)

def load(file: str)

Loads a Tensor or tree from a file previously written using save().

All tensors are restored as NumPy arrays, not the backend-specific tensors they may have been written as. Use convert() to convert all or some of the tensors to a different backend.

Args

file: File to read.

Returns

Same type as what was written.

def log(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the natural logarithm of the Tensor or PhiTreeNode x.

def log10(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes log(x) of the Tensor or PhiTreeNode x with base 10.

def log2(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes log(x) of the Tensor or PhiTreeNode x with base 2.

def log_gamma(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes log(gamma(x)) of the Tensor or PhiTreeNode x.

def map(function: Callable[..., ~Y], *args, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function shape>, range=builtins.range, unwrap_scalars=True, expand_results=False, **kwargs) ‑> Union[None, phiml.math._tensors.Tensor, ~Y]

Calls function on slices of the arguments and returns the stacked result.

Args

function: Function to be called on slices of args and kwargs. Must return one or multiple values that can be stacked. None may be returned but if any return value is None, all calls to function must return None in that position.
*args: Positional arguments for function. Values that are Sliceable will be sliced along dims.
**kwargs: Keyword arguments for function. Values that are Sliceable will be sliced along dims.
dims: Dimensions which should be sliced. function is called once for each element in dims, i.e. dims.volume times. If dims is not specified, all dimensions from the Sliceable values in args and kwargs will be mapped. Pass object to map only objects, not tensors of primitives (dtype.kind == object). This will select only layout()-type dimensions.
range: Optional range function. Can be used to generate tqdm output by passing trange.
unwrap_scalars: If True, passes the contents of scalar Tensors instead of the tensor objects.

Returns

Tensor of same shape as value.

def map_pairs(map_function: Callable, values: phiml.math._tensors.Tensor, connections: phiml.math._tensors.Tensor)

Evaluates map_function on all pairs of elements present in the sparsity pattern of connections.

Args

map_function: Function with signature (Tensor, Tensor) -> Tensor.
values: Values to evaluate map_function on. Needs to have a spatial or instance dimension but must not have a dual dimension.
connections: Sparse tensor.

Returns

Tensor with the sparse dimensions of connections and all non-instance dimensions returned by map_function.

def map_types(f: Callable, dims: Union[phiml.math._shape.Shape, tuple, list, str, Callable], dim_type: Union[str, Callable]) ‑> Callable

Wraps a function to change the dimension types of its Tensor and PhiTreeNode arguments.

Args

f: Function to wrap.
dims: Concrete dimensions or dimension type, such as spatial() or batch(). These dimensions will be mapped to dim_type for all positional function arguments.
dim_type: Dimension type, such as spatial() or batch(). f will be called with dimensions remapped to this type.

Returns

Function with signature matching f.

def masked_fill(values: phiml.math._tensors.Tensor, valid: phiml.math._tensors.Tensor, distance: int = 1) ‑> Tuple[phiml.math._tensors.Tensor, phiml.math._tensors.Tensor]

Extrapolates the values of values which are marked by the nonzero values of valid for distance steps in all spatial directions. Overlapping extrapolated values get averaged. Extrapolation also includes diagonals.

Args

values: Tensor which holds the values for extrapolation
valid: Tensor with same size as x marking the values for extrapolation with nonzero values
distance: Number of extrapolation steps

Returns

values: Extrapolation result
valid: mask marking all valid values after extrapolation

def matrix_from_function(f: Callable, *args, auxiliary_args=None, auto_compress=False, sparsify_batch=None, separate_independent=False, **kwargs) ‑> Tuple[phiml.math._tensors.Tensor, phiml.math._tensors.Tensor]

Trace a linear function and construct a matrix. Depending on the functional form of f, the returned matrix may be dense or sparse.

Args

f: Function to trace.
*args: Arguments for f.
auxiliary_args: Arguments in which the function is not linear. These parameters are not traced but passed on as given in args and kwargs.
auto_compress: If True, returns a compressed matrix if supported by the backend.
sparsify_batch: If False, the matrix will be batched. If True, will create dual dimensions for the involved batch dimensions. This will result in one large matrix instead of a batch of matrices.
**kwargs: Keyword arguments for f.

Returns

matrix: Matrix representing the linear dependency of the output f on the input of f. Input dimensions will be dual() dimensions of the matrix while output dimensions will be regular.
bias: Bias for affine functions or zero-vector if the function is purely linear.

def matrix_rank(matrix: phiml.math._tensors.Tensor) ‑> phiml.math._tensors.Tensor

Approximates the rank of a matrix. The tolerances used depend on the current precision.

Args

matrix: Sparse or dense matrix, i.e. Tensor with primal and dual dims.

Returns

Matrix rank.

def max(value: ~TensorOrTree, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, key: phiml.math._tensors.Tensor = None) ‑> ~TensorOrTree

Determines the maximum value of values along the specified dimensions.

Args

value

(Sparse) Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

key

Optional comparison values. If specified, returns the value where key is maximal, see at_max().

Returns

Tensor without the reduced dimensions.

def maximum(x: Union[float, phiml.math._tensors.Tensor], y: Union[float, phiml.math._tensors.Tensor], allow_none=False)

Computes the element-wise maximum of x and y.

def mean(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, weight: Union[phiml.math._tensors.Tensor, list, tuple] = None, where_no_weight=nan, epsilon=1e-10) ‑> phiml.math._tensors.Tensor

Computes the mean over values along the specified dimensions.

Args

value

(Sparse) Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

weight

Optionally perform a weighted mean operation. Must broadcast to value.

where_no_weight

Value to use when the sum of all weights are smaller than epsilon.

epsilon

Only if where_no_weight. Threshold for using where_no_weight.

Returns

Tensor without the reduced dimensions.

def median(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>)

Reduces dim of value by picking the median value. For odd dimension sizes (ambigous choice), the linear average of the two median values is computed.

Currently implemented via quantile().

Args

value

Tensor

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

Returns

Tensor

def merge_shapes(*objs: Union[phiml.math._shape.Shape, Any], order=(<function batch>, <function dual>, <function instance>, <function spatial>, <function channel>), allow_varying_sizes=False)

Combines shapes into a single Shape, grouping dimensions by type. If dimensions with equal names are present in multiple shapes, their types and sizes must match.

The shorthand shape1 & shape2 merges shapes with check_exact=[spatial].

Args

*objs: Shape or Shaped objects to combine.
order: Dimension type order as tuple of type filters (channel(), batch(), spatial() or instance()). Dimensions are grouped by type while merging.

Returns

Merged Shape

Raises

IncompatibleShapes if the shapes are not compatible

def meshgrid(dims: Union[Callable, phiml.math._shape.Shape] = <function spatial>, stack_dim=(vectorᶜ=None), **dimensions: Union[int, phiml.math._tensors.Tensor, tuple, list, Any]) ‑> phiml.math._tensors.Tensor

Generate a mesh-grid Tensor from keyword dimensions.

Args

**dimensions: Mesh-grid dimensions, mapping names to values. Values may be int, 1D Tensor or 1D native tensor.
dims: Dimension type of mesh-grid dimensions, one of spatial(), channel(), batch(), instance().
stack_dim: Channel dim along which grids are stacked. This is optional for 1D mesh-grids. In that case returns a Tensor without a stack dim if None or an empty Shape is passed.

Returns

Mesh-grid Tensor with the dimensions of dims / dimensions and stack_dim.

Examples

>>> math.meshgrid(x=2, y=2)
(xˢ=2, yˢ=2, vectorᶜ=x,y) 0.500 ± 0.500 (0e+00...1e+00)

>>> math.meshgrid(x=2, y=(-1, 1))
(xˢ=2, yˢ=2, vectorᶜ=x,y) 0.250 ± 0.829 (-1e+00...1e+00)

>>> math.meshgrid(x=2, stack_dim=None)
(0, 1) along xˢ

def min(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, key: phiml.math._tensors.Tensor = None) ‑> phiml.math._tensors.Tensor

Determines the minimum value of values along the specified dimensions.

Args

value

(Sparse) Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

key

Optional comparison values. If specified, returns the value where key is minimal, see at_min().

Returns

Tensor without the reduced dimensions.

def minimize(f: Callable[[~X], ~Y], solve: phiml.math._optimize.Solve[~X, ~Y]) ‑> ~X

Finds a minimum of the scalar function f(x). The method argument of solve determines which optimizer is used. All optimizers supported by scipy.optimize.minimize are supported, see https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html . Additionally a gradient descent solver with adaptive step size can be used with method='GD'.

math.minimize() is limited to backends that support jacobian(), i.e. PyTorch, TensorFlow and Jax.

To obtain additional information about the performed solve, use a SolveTape.

Args

f: Function whose output is subject to minimization. All positional arguments of f are optimized and must be Tensor or PhiTreeNode. If solve.x0 is a tuple or list, it will be passed to f as varargs, f(*x0). To minimize a subset of the positional arguments, define a new (lambda) function depending only on those. The first return value of f must be a scalar float Tensor or PhiTreeNode.
solve: Solve object to specify method type, parameters and initial guess for x.

Returns

x: solution, the minimum point x.

Raises

NotConverged: If the desired accuracy was not be reached within the maximum number of iterations.
Diverged: If the optimization failed prematurely.

def minimum(x: Union[float, phiml.math._tensors.Tensor], y: Union[float, phiml.math._tensors.Tensor], allow_none=False)

Computes the element-wise minimum of x and y.

def nan_to_0(x: ~TensorOrTree) ‑> ~TensorOrTree

Replaces all NaN values in x with 0.

def native(value: Union[phiml.math._tensors.Tensor, numbers.Number, tuple, list, Any])

Returns the native tensor representation of value. If value is a Tensor, this is equal to calling Tensor.native(). Otherwise, checks that value is a valid tensor object and returns it.

Args

value: Tensor or native tensor or tensor-like.

Returns

Native tensor representation

Raises

ValueError if the tensor cannot be transposed to match target_shape

def native_call(f: Callable, *inputs: phiml.math._tensors.Tensor, channels_last=None, channel_dim='vector', spatial_dim=None)

Calls f with the native representations of the inputs tensors in standard layout and returns the result as a Tensor.

All inputs are converted to native tensors (including precision cast) depending on channels_last:

channels_last=True: Dimension layout (total_batch_size, spatial_dims…, total_channel_size)
channels_last=False: Dimension layout (total_batch_size, total_channel_size, spatial_dims…)

All batch dimensions are compressed into a single dimension with total_batch_size = input.shape.batch.volume. The same is done for all channel dimensions.

Additionally, missing batch and spatial dimensions are added so that all inputs have the same batch and spatial shape.

Args

f: Function to be called on native tensors of inputs. The function output must have the same dimension layout as the inputs, unless overridden by spatial_dim, and the batch size must be identical.
*inputs: Uniform Tensor arguments
channels_last: (Optional) Whether to put channels as the last dimension of the native representation. If None, the channels are put in the default position associated with the current backend, see phiml.math.backend.Backend.prefers_channels_last().
channel_dim: Name of the channel dimension of the result.
spatial_dim: Name of the spatial dimension of the result.

Returns

Tensor with batch and spatial dimensions of inputs, unless overridden by spatial_dim, and single channel dimension channel_dim.

def ncat(values: Sequence[~PhiTreeNodeType], dim: phiml.math._shape.Shape, expand_values=False) ‑> ~PhiTreeNodeType

Concatenate named components along dim.

Args

values: Each value can contain multiple components of dim if dim is present in its shape. Else, it is interpreted as a single component whose name will be determined from the leftover item names of dim.
dim: Single dimension that has item names matching components of values.
expand_values: If True, will add all missing dimensions to values, not just batch dimensions. This allows tensors with different dimensions to be concatenated. The resulting tensor will have all dimensions that are present in values. If False, this may return a non-numeric object instead.

Returns

Same type as any value from values.

def neighbor_max(grid: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None) ‑> phiml.math._tensors.Tensor

neighbor_reduce() with reduce_fun set to max_().

def neighbor_mean(grid: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None) ‑> phiml.math._tensors.Tensor

neighbor_reduce() with reduce_fun set to mean().

def neighbor_min(grid: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None) ‑> phiml.math._tensors.Tensor

neighbor_reduce() with reduce_fun set to min_().

def neighbor_reduce(reduce_fun: Callable, grid: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None, padding_kwargs: dict = None) ‑> phiml.math._tensors.Tensor

Computes the sum/mean/min/max/prod/etc. of two neighboring values along each dimension in dim. The result tensor has one entry less than grid in each averaged dimension unless padding is specified.

With two dims, computes the mean of 4 values, in 3D, the mean of 8 values.

Args

reduce_fun: Reduction function, such as sum_(), mean(), max_(), min_(), prod().
grid: Values to reduce.
dims: Dimensions along which neighbors should be reduced.
padding: Padding at the upper edges of grid along dims'. If notNone, the result tensor() will have the same shape() as grid`.
padding_kwargs: Additional keyword arguments to be passed to pad().

Returns

Tensor

def neighbor_sum(grid: phiml.math._tensors.Tensor, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = None) ‑> phiml.math._tensors.Tensor

neighbor_reduce() with reduce_fun set to sum_().

def non_batch(obj) ‑> phiml.math._shape.Shape

Returns the non-batch dimensions of an object.

Args

obj: Shape or object with a valid shape() property.

Returns

Shape

def non_channel(obj) ‑> phiml.math._shape.Shape

Returns the non-channel dimensions of an object.

Args

obj: Shape or object with a valid shape() property.

Returns

Shape

def non_dual(obj) ‑> phiml.math._shape.Shape

Returns the non-dual dimensions of an object.

Args

obj: Shape or object with a valid shape() property.

Returns

Shape

def non_instance(obj) ‑> phiml.math._shape.Shape

Returns the non-instance dimensions of an object.

Args

obj: Shape or object with a valid shape() property.

Returns

Shape

def non_primal(obj) ‑> phiml.math._shape.Shape

Returns the batch and dual dimensions of an object.

Args

obj: Shape or object with a valid shape() property.

Returns

Shape

def non_spatial(obj) ‑> phiml.math._shape.Shape

Returns the non-spatial dimensions of an object.

Args

obj: Shape or object with a valid shape() property.

Returns

Shape

def nonzero(value: phiml.math._tensors.Tensor, list_dim: Union[phiml.math._shape.Shape, str, int] = (nonzeroⁱ=None), index_dim: phiml.math._shape.Shape = (vectorᶜ=None), element_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>, list_dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>, preserve_names=False)

Get spatial indices of non-zero / True values.

Batch dimensions are preserved by this operation. If channel dimensions are present, this method returns the indices where any component is nonzero.

Implementations:

NumPy: numpy.argwhere
PyTorch: torch.nonzero
TensorFlow: tf.where(tf.not_equal(values, 0))
Jax: jax.numpy.nonzero

Args

value: spatial tensor to find non-zero / True values in.
list_dim: Dimension listing non-zero values. If size specified, lists only the first size non-zero values. Special case: For retrieving only the first non-zero value, you may pass 1 instead of a Shape of size 1.
index_dim: Index dimension.
element_dims: Dims listing components of one value. A value is only considered zero if all components are 0.
list_dims: Dims in which non-zero elements are searched. These will be stored in the item names of index_dim.

Returns

Tensor of shape (batch dims…, list_dim=#non-zero, index_dim=value.shape.spatial_rank)

def normalize(vec: phiml.math._tensors.Tensor, vec_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>, epsilon=None, allow_infinite=False, allow_zero=False)

Normalizes the vectors in vec(). If vec_dim is None, the combined channel dimensions of vec() are interpreted as a vector.

Args

vec: Tensor to normalize.
vec_dim: Dimensions to normalize over. By default, all channel dimensions are used to compute the vector length.
epsilon: (Optional) Zero-length threshold. Vectors shorter than this length yield the unit vector (1, 0, 0, …). If not specified, the zero-vector yields NaN as it cannot be normalized.
allow_infinite: Allow infinite components in vectors. These vectors will then only points towards the infinite components.
allow_zero: Whether to return zero vectors for inputs smaller epsilon instead of a unit vector.

def normalize_to(target: phiml.math._tensors.Tensor, source: Union[float, phiml.math._tensors.Tensor], epsilon=1e-05)

Multiplies the target so that its sum matches the source.

Args

target: Tensor
source: Tensor or constant
epsilon: Small number to prevent division by zero.

Returns

Normalized tensor of the same shape as target

def numpy(value: Union[phiml.math._tensors.Tensor, numbers.Number, tuple, list, Any])

Converts value to a numpy.ndarray where value must be a Tensor, backend tensor or tensor-like. If value is a Tensor, this is equal to calling Tensor.numpy().

Note: Using this function breaks the autograd chain. The returned tensor is not differentiable. To get a differentiable tensor, use Tensor.native() instead.

Transposes the underlying tensor to match the name order and adds singleton dimensions for new dimension names. If a dimension of the tensor is not listed in order, a ValueError is raised.

If value is a NumPy array, it may be returned directly.

Returns

NumPy representation of value

Raises

ValueError if the tensor cannot be transposed to match target_shape

def ones(*shape: phiml.math._shape.Shape, dtype: Union[phiml.backend._dtype.DType, tuple, type] = None) ‑> phiml.math._tensors.Tensor

Define a tensor with specified shape with value 1.0/ 1 / True everywhere.

This method may not immediately allocate the memory to store the values.

Args

*shape: This (possibly empty) sequence of Shapes is concatenated, preserving the order.
dtype: Data type as DType object. Defaults to float matching the current precision setting.

Returns

Tensor

def ones_like(value: phiml.math._tensors.Tensor) ‑> phiml.math._tensors.Tensor

Create a Tensor containing only 1.0 / 1 / True with the same shape and dtype as obj.

def pack_dims(value, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], packed_dim: Union[str, phiml.math._shape.Shape], pos: Optional[int] = None, **kwargs)

Compresses multiple dimensions into a single dimension by concatenating the elements. Elements along the new dimensions are laid out according to the order of dims. If the order of dims differs from the current dimension order, the tensor is transposed accordingly. This function replaces the traditional reshape for these cases.

The type of the new dimension will be equal to the types of dims. If dims have varying types, the new dimension will be a batch dimension.

If none of dims exist on value, packed_dim will be added only if it is given with a definite size and value is not a primitive type.

Args

value: Shapable, such as Tensor.
dims: Dimensions to be compressed in the specified order.
packed_dim: Single-dimension Shape.
pos: Index of new dimension. None for automatic, -1 for last, 0 for first.
**kwargs: Additional keyword arguments required by specific implementations. Adding spatial dimensions to fields requires the bounds: Box argument specifying the physical extent of the new dimensions. Adding batch dimensions must always work without keyword arguments.

Returns

Same type as value.

Examples

>>> pack_dims(math.zeros(spatial(x=4, y=3)), spatial, instance('points'))
(pointsⁱ=12) const 0.0

def pad(value: phiml.math._tensors.Tensor, widths: Union[dict, tuple, list], mode: Union[ForwardRef('e_.Extrapolation'), phiml.math._tensors.Tensor, numbers.Number, str, dict] = 0, **kwargs) ‑> phiml.math._tensors.Tensor

Pads a tensor along the specified dimensions, determining the added values using the given extrapolation. Unlike Extrapolation.pad(), this function can handle negative widths which slice off outer values.

Args

value

Tensor to be padded

widths

Number of values to add at the edge of value. Negative values can be used to slice off edge values. Must be one of the following:

tuple containing (lower: int, upper: int). This will pad all non-batch dimensions by lower and upper at the lower and upper edge, respectively.
dict mapping dim: str -> (lower: int, upper: int)
Sequence of slicing dicts. This will add all values specified by the slicing dicts and is the inverse operation to slice_off(). Exactly one value in each slicing dict must be a slice_() object.

mode

Padding mode used to determine values added from positive widths. Must be one of the following: Extrapolation, Tensor or number for constant extrapolation, name of extrapolation as str.

kwargs

Additional padding arguments. These are ignored by the standard extrapolations defined in phiml.math.extrapolation but can be used to pass additional contextual information to custom extrapolations.

Returns

Padded Tensor

Examples

>>> math.pad(math.ones(spatial(x=10, y=10)), {'x': (1, 1), 'y': (2, 1)}, 0)
(xˢ=12, yˢ=13) 0.641 ± 0.480 (0e+00...1e+00)

>>> math.pad(math.ones(spatial(x=10, y=10)), {'x': (1, -1)}, 0)
(xˢ=10, yˢ=10) 0.900 ± 0.300 (0e+00...1e+00)

def pairwise_differences(positions: phiml.math._tensors.Tensor, max_distance: Union[float, phiml.math._tensors.Tensor] = None, format: Union[str, phiml.math._tensors.Tensor] = 'dense', domain: Optional[Tuple[phiml.math._tensors.Tensor, phiml.math._tensors.Tensor]] = None, periodic: Union[bool, phiml.math._tensors.Tensor] = False, method: str = 'auto', default: float = nan, avg_neighbors=8.0) ‑> phiml.math._tensors.Tensor

Computes the distance matrix containing the pairwise position differences between each pair of points. The matrix will consist of the channel and batch dimension of positions and the primal dimensions plus their dual counterparts, spanning the matrix. Points that are further apart than max_distance (if specified) are assigned an invalid value given by default. The diagonal of the matrix (self-distance) consists purely of zero-vectors and is always stored explicitly. The neighbors of the positions are listed along the dual dimension(s) of the matrix, and vectors point towards the neighbors.

This function can operate in dense mode or sparse mode, depending on format. If format=='dense' or a dense Tensor, all possible pair-wise distances are considered and a full-rank tensor is returned. The value of method is ignored in that case.

Otherwise, if format is a sparse format identifier or sparse Tensor, only a subset of distances is considered, depending on method. In this case, the result is a sparse matrix with the same dimensionos as the dense tensor would have had.

JIT behavior: This function can be JIT compiled with all backends. However, as the exact number of neighbors is unknown beforehand, all sparse methods rely on a variable-size buffer. PyTorch and TensorFlow allow variable shapes and behave the same way with JIT compilation as without. JAX, however, requires all tensor shapes to be known beforehand. This function will guess the required buffer size based on avg_neighbors and track the actually required sizes. When using jit_compile(), this will automatically trigger a re-tracing when a buffer overflow is detected. User calling jax.jit manually must retrieve these sizes from the buffer API and implement buffer overflow handling.

Args

positions

Tensor. Channel dimensions are interpreted as position components. Instance and spatial dimensions list nodes.

max_distance

Scalar or Tensor specifying a max_radius for each point separately. Can contain additional batch dimensions but spatial/instance dimensions must match positions if present. If not specified, uses an infinite cutoff radius, i.e. all points will be considered neighbors.

format

Matrix format as str or concrete sparsity pattern as Tensor. Allowed strings are 'dense'','sparse', 'csr', 'coo', 'csc'`. When a Tensor is passed, it needs to have all instance and spatial dims as positions as well as corresponding dual dimensions. The distances will be evaluated at all stored entries of the format tensor.

domain

Lower and upper corner of the bounding box. All positions must lie within this box. This must be specified to use with periodic boundaries.

periodic

Which domain boundaries should be treated as periodic, i.e. particles on opposite sides are neighbors. Can be specified as a bool for all sides or as a vector-valued boolean Tensor to specify periodicity by direction.

default

Value for distances greater than max_distance. Only for dense distance matrices.

method

Neighbor search algorithm; only used if format is a sparse format or Tensor. The default, 'auto' lets the runtime decide on the best method. Supported methods:

'sparse': GPU-supported hash grid implementation with fully sparse connectivity.
'scipy-kd': SciPy's kd-tree implementation.

avg_neighbors

Expected average number of neighbors. This is only relevant for hash grid searches, where it influences the default buffer sizes.

Returns

Distance matrix as sparse or dense Tensor, depending on format. For each spatial/instance dimension in positions, the matrix also contains a dual dimension of the same name and size. The matrix also contains all batch dimensions of positions and the channel dimension of positions.

Examples

>>> pos = vec(x=0, y=tensor([0, 1, 2.5], instance('particles')))
>>> dx = pairwise_differences(pos, format='dense', max_distance=2)
>>> dx.particles[0]
(x=0.000, y=0.000); (x=0.000, y=1.000); (x=0.000, y=0.000) (~particlesᵈ=3, vectorᶜ=x,y)

def pairwise_distances(positions: phiml.math._tensors.Tensor, max_distance: Union[float, phiml.math._tensors.Tensor] = None, format: Union[str, phiml.math._tensors.Tensor] = 'dense', domain: Optional[Tuple[phiml.math._tensors.Tensor, phiml.math._tensors.Tensor]] = None, periodic: Union[bool, phiml.math._tensors.Tensor] = False, method: str = 'auto', default: float = nan, avg_neighbors=8.0) ‑> phiml.math._tensors.Tensor

Args

positions

Tensor. Channel dimensions are interpreted as position components. Instance and spatial dimensions list nodes.

max_distance

format

domain

Lower and upper corner of the bounding box. All positions must lie within this box. This must be specified to use with periodic boundaries.

periodic

default

Value for distances greater than max_distance. Only for dense distance matrices.

method

Neighbor search algorithm; only used if format is a sparse format or Tensor. The default, 'auto' lets the runtime decide on the best method. Supported methods:

'sparse': GPU-supported hash grid implementation with fully sparse connectivity.
'scipy-kd': SciPy's kd-tree implementation.

avg_neighbors

Expected average number of neighbors. This is only relevant for hash grid searches, where it influences the default buffer sizes.

Returns

Examples

>>> pos = vec(x=0, y=tensor([0, 1, 2.5], instance('particles')))
>>> dx = pairwise_differences(pos, format='dense', max_distance=2)
>>> dx.particles[0]
(x=0.000, y=0.000); (x=0.000, y=1.000); (x=0.000, y=0.000) (~particlesᵈ=3, vectorᶜ=x,y)

def perf_counter(wait_for_tensor, *wait_for_tensors: phiml.math._tensors.Tensor) ‑> phiml.math._tensors.Tensor

Get the time (time.perf_counter()) at which all wait_for_tensors are computed. If all tensors are already available, returns the current time.perf_counter().

Args

wait_for_tensor: Tensor that need to be computed before the time is measured.
*wait_for_tensors: Additional tensors that need to be computed before the time is measured.

Returns

Time at which all wait_for_tensors are ready as a scalar Tensor.

def pick_random(value: ~TensorOrTree, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], count: Union[int, phiml.math._shape.Shape, None] = 1, weight: Optional[phiml.math._tensors.Tensor] = None) ‑> ~TensorOrTree

Pick one or multiple random entries from value.

Args

value: Tensor or tree. When containing multiple tensors, the corresponding entries are picked on all tensors that have dim. You can pass arange() (the type) to retrieve the picked indices.
dim: Dimension along which to pick random entries. Shape with one dim.
count: Number of entries to pick. When specified as a Shape, lists picked values along count instead of dim.
weight: Probability weight of each item along dim. Will be normalized to sum to 1.

Returns

Tensor or tree equal to value.

def precision(floating_point_bits: int)

Sets the floating point precision for the local context.

Usage: with precision(p):

This overrides the global setting, see set_global_precision().

Args

floating_point_bits: 16 for half, 32 for single, 64 for double

def primal(obj) ‑> phiml.math._shape.Shape

Returns the instance, spatial and channel dimensions of an object.

Args

obj: Shape or object with a valid shape() property.

Returns

Shape

def print(obj: Union[phiml.math._tensors.Tensor, PhiTreeNode, numbers.Number, tuple, list, None] = None, name: str = '')

Print a tensor with no more than two spatial dimensions, slicing it along all batch and channel dimensions.

Unlike NumPy's array printing, the dimensions are sorted. Elements along the alphabetically first dimension is printed to the right, the second dimension upward. Typically, this means x right, y up.

Args

obj: tensor-like
name: name of the tensor

Returns:

def print_gradient(value: phiml.math._tensors.Tensor, name='', detailed=False) ‑> phiml.math._tensors.Tensor

Prints the gradient vector of value when computed. The gradient at value is the vector-Jacobian product of all operations between the output of this function and the loss value.

The gradient is not printed in jit mode, see jit_compile().

Example

def f(x):
    x = math.print_gradient(x, 'dx')
    return math.l1_loss(x)

math.jacobian(f)(math.ones(x=6))

Args

value: Tensor for which the gradient may be computed later.
name: (Optional) Name to print along with the gradient values
detailed: If False, prints a short summary of the gradient tensor.

Returns

identity()(value) which when differentiated, prints the gradient vector.

def prod(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> phiml.math._tensors.Tensor

Multiplies values along the specified dimensions.

Args

value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

Returns

Tensor without the reduced dimensions.

def quantile(value: phiml.math._tensors.Tensor, quantiles: Union[float, phiml.math._tensors.Tensor, tuple, list], dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>)

Compute the q-th quantile of value along dim for each q in quantiles.

Implementations:

NumPy: quantile
PyTorch: quantile
TensorFlow: tfp.stats.percentile
Jax: quantile

Args

value

Tensor

quantiles

Single quantile or tensor of quantiles to compute. Must be of type float, tuple, list or Tensor.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to reduce the sequence of Tensors

Returns

Tensor with dimensions of quantiles and non-reduced dimensions of value.

def radians_to_degrees(rad: ~TensorOrTree) ‑> ~TensorOrTree

Convert degrees to radians.

def rand(*shape: phiml.math._shape.Shape, low: Union[float, phiml.math._tensors.Tensor] = 0, high: Union[float, phiml.math._tensors.Tensor] = 1, dtype: Union[phiml.backend._dtype.DType, tuple, type] = None) ‑> phiml.math._tensors.Tensor

Creates a Tensor with the specified shape, filled with random values sampled from a uniform distribution.

Args

*shape: This (possibly empty) sequence of Shapes is concatenated, preserving the order.
dtype: (optional) DType or (kind, bits). The dtype kind must be one of float, int, complex. If not specified, a float tensor with the current default precision is created, see get_precision().
low: Minimum value, included.
high: Maximum value, excluded.

Returns

Tensor

def randn(*shape: phiml.math._shape.Shape, dtype: Union[phiml.backend._dtype.DType, tuple, type] = None) ‑> phiml.math._tensors.Tensor

Creates a Tensor with the specified shape, filled with random values sampled from a normal / Gaussian distribution.

Implementations:

Args

*shape: This (possibly empty) sequence of Shapes is concatenated, preserving the order.
dtype: (optional) floating point DType. If None, a float tensor with the current default precision is created, see get_precision().

Returns

Tensor

def random_normal(*shape: phiml.math._shape.Shape, dtype: Union[phiml.backend._dtype.DType, tuple, type] = None) ‑> phiml.math._tensors.Tensor

Creates a Tensor with the specified shape, filled with random values sampled from a normal / Gaussian distribution.

Implementations:

Args

*shape: This (possibly empty) sequence of Shapes is concatenated, preserving the order.
dtype: (optional) floating point DType. If None, a float tensor with the current default precision is created, see get_precision().

Returns

Tensor

def random_permutation(*shape: Union[phiml.math._shape.Shape, Any], dims=<function non_batch>, index_dim=(indexᶜ=None)) ‑> phiml.math._tensors.Tensor

Generate random permutations of the integers between 0 and the size of shape().

When multiple dims are given, the permutation is randomized across all of them and tensor of multi-indices is returned.

Batch dims result in batches of permutations.

Args

*shape: Shape of the result tensor, including dims and batches.
*dims: Sequence dims for an individual permutation. The total Shape.volume defines the maximum integer. All other dims from shape() are treated as batch.

Returns

Tensor

def random_uniform(*shape: phiml.math._shape.Shape, low: Union[float, phiml.math._tensors.Tensor] = 0, high: Union[float, phiml.math._tensors.Tensor] = 1, dtype: Union[phiml.backend._dtype.DType, tuple, type] = None) ‑> phiml.math._tensors.Tensor

Creates a Tensor with the specified shape, filled with random values sampled from a uniform distribution.

Args

*shape: This (possibly empty) sequence of Shapes is concatenated, preserving the order.
dtype: (optional) DType or (kind, bits). The dtype kind must be one of float, int, complex. If not specified, a float tensor with the current default precision is created, see get_precision().
low: Minimum value, included.
high: Maximum value, excluded.

Returns

Tensor

def range(dim: phiml.math._shape.Shape, start_or_stop: Optional[int] = None, stop: Optional[int] = None, step=1)

Returns evenly spaced values between start and stop. If only one limit is given, 0 is used for the start.

See Also: range_tensor(), linspace(), meshgrid().

Args

dim: Dimension name and type as Shape object. The size of dim is interpreted as stop unless start_or_stop is specified.
start_or_stop: (Optional) int. Interpreted as start if stop is specified as well. Otherwise this is stop.
stop: (Optional) int. stop value.
step: Distance between values.

Returns

Tensor

def range_tensor(*shape: phiml.math._shape.Shape)

Returns a Tensor with given shape() containing the linear indices of each element. For 1D tensors, this equivalent to arange() with step=1.

Args

shape: Tensor shape.

Returns

Tensor

def ravel_index(index: phiml.math._tensors.Tensor, resolution: phiml.math._shape.Shape, dim=<function channel>, mode='undefined') ‑> phiml.math._tensors.Tensor

Computes a scalar index from a vector index.

Args

index: Tensor with one channel dim.
resolution: Shape
mode: 'undefined', 'periodic', 'clamp' or an int to use for all invalid indices.

Returns

Tensor

def real(x: ~TensorOrTree) ‑> ~TensorOrTree

Args

x: Tensor or PhiTreeNode or native tensor.

Returns

Real component of x.

def rename_dims(value: ~PhiTreeNodeType, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], names: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], **kwargs) ‑> ~PhiTreeNodeType

Change the name and optionally the type of some dimensions of value.

Dimensions that are not present on value will be ignored. The corresponding new dimensions given by names will not be added.

Args

value

Shape or Tensor or Shapable.

dims

Existing dimensions of value as comma-separated str, tuple, list, Shape or filter function.

names

Either

Sequence of names matching dims as tuple, list or str. This replaces only the dimension names but leaves the types untouched.
Shape matching dims to replace names and types.
Dimension type function to replace only types.

**kwargs

Additional keyword arguments required by specific implementations. Adding spatial dimensions to fields requires the bounds: Box argument specifying the physical extent of the new dimensions. Adding batch dimensions must always work without keyword arguments.

Returns

Same type as value.

def replace(obj: ~PhiTreeNodeType, **updates) ‑> ~PhiTreeNodeType

Creates a copy of the given PhiTreeNode with updated values as specified in updates.

If obj overrides __with_attrs__, the copy will be created via that specific implementation. Otherwise, the copy() module and setattr will be used.

Args

obj: PhiTreeNode
**updates: Values to be replaced.

Returns

Copy of obj with updated values.

def replace_dims(value: ~PhiTreeNodeType, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], names: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], **kwargs) ‑> ~PhiTreeNodeType

Change the name and optionally the type of some dimensions of value.

Dimensions that are not present on value will be ignored. The corresponding new dimensions given by names will not be added.

Args

value

Shape or Tensor or Shapable.

dims

Existing dimensions of value as comma-separated str, tuple, list, Shape or filter function.

names

Either

Sequence of names matching dims as tuple, list or str. This replaces only the dimension names but leaves the types untouched.
Shape matching dims to replace names and types.
Dimension type function to replace only types.

**kwargs

Returns

Same type as value.

def reshaped_native(value: phiml.math._tensors.Tensor, groups: Union[tuple, list], force_expand: Any = True, to_numpy=False)

Returns a native representation of value where dimensions are laid out according to groups.

See Also: native(), pack_dims(), reshaped_tensor(), reshaped_numpy().

Args

value

Tensor

groups

tuple or list of dimensions to be packed into one native dimension. Each entry must be one of the following:

str: the name of one dimension that is present on value.
Shape: Dimensions to be packed. If force_expand, missing dimensions are first added, otherwise they are ignored.
Filter function: Packs all dimensions of this type that are present on value.
Ellipsis …: Packs all remaining dimensions into this slot. Can only be passed once.
None or (): Adds a singleton dimension.

Collections of or comma-separated dims may also be used but only if all dims are present on value.

force_expand

bool or sequence of dimensions. If True, repeats the tensor along missing dimensions. If False, puts singleton dimensions where possible. If a sequence of dimensions is provided, only forces the expansion for groups containing those dimensions.

to_numpy

If True, converts the native tensor to a numpy.ndarray.

Returns

Native tensor with dimensions matching groups.

def reshaped_numpy(value: phiml.math._tensors.Tensor, groups: Union[tuple, list], force_expand: Any = True) ‑> numpy.ndarray

Returns the NumPy representation of value where dimensions are laid out according to groups.

See Also: numpy_(), reshaped_native(), pack_dims(), reshaped_tensor().

Args

value: Tensor
groups: Sequence of dimension names as str or groups of dimensions to be packed_dim as Shape.
force_expand: bool or sequence of dimensions. If True, repeats the tensor along missing dimensions. If False, puts singleton dimensions where possible. If a sequence of dimensions is provided, only forces the expansion for groups containing those dimensions.

Returns

NumPy ndarray with dimensions matching groups.

def reshaped_tensor(value: Any, groups: Union[tuple, list], check_sizes=False, convert=True)

Creates a Tensor from a native tensor or tensor-like whereby the dimensions of value are split according to groups.

See Also: tensor(), reshaped_native(), unpack_dim().

Args

value: Native tensor or tensor-like.
groups: Sequence of dimension groups to be packed_dim as tuple[Shape] or list[Shape].
check_sizes: If True, group sizes must match the sizes of value exactly. Otherwise, allows singleton dimensions.
convert: If True, converts the data to the native format of the current default backend. If False, wraps the data in a Tensor but keeps the given data reference if possible.

Returns

Tensor with all dimensions from groups

def rotate_vector(vector: phiml.math._tensors.Tensor, angle: Union[float, phiml.math._tensors.Tensor, None], invert=False, dim='vector') ‑> phiml.math._tensors.Tensor

Rotates vector around the origin.

Args

vector: n-dimensional vector with exactly one channel dimension
angle: Euler angle(s) or rotation matrix. None is interpreted as no rotation.
invert: Whether to apply the inverse rotation.

Returns

Rotated vector as Tensor

def rotation_matrix(x: Union[float, phiml.math._tensors.Tensor, None], matrix_dim=(vectorᶜ=None)) ‑> Optional[phiml.math._tensors.Tensor]

Create a 2D or 3D rotation matrix from the corresponding angle(s).

Args

x:
2D: scalar angle
3D: Either vector pointing along the rotation axis with rotation angle as length or Euler angles.
Euler angles need to be laid out along a angle() channel dimension with dimension names listing the spatial dimensions.
E.g. a 90° rotation about the z-axis is represented by vec('angles', x=0, y=0, z=PI/2).
If a rotation matrix is passed for angle(), it is returned without modification.
matrix_dim: Matrix dimension for 2D rotations. In 3D, the channel dimension of angle is used.

Returns

Matrix containing matrix_dim in primal and dual form as well as all non-channel dimensions of x.

def round(x: ~TensorOrTree) ‑> ~TensorOrTree

Rounds the Tensor or PhiTreeNode x to the closest integer.

def s2b(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all spatial dimensions of value to batch dimensions. See rename_dims().

def safe_div(x: Union[numbers.Number, phiml.math._tensors.Tensor], y: Union[numbers.Number, phiml.math._tensors.Tensor])

Computes x/y with the Tensors x and y but returns 0 where y=0.

def safe_mul(x: Union[numbers.Number, phiml.math._tensors.Tensor], y: Union[numbers.Number, phiml.math._tensors.Tensor])

Multiplication for tensors with non-finite values. Computes x·y in the forward pass but drops gradient contributions from infinite and NaN values.

def sample_subgrid(grid: phiml.math._tensors.Tensor, start: phiml.math._tensors.Tensor, size: phiml.math._shape.Shape) ‑> phiml.math._tensors.Tensor

Samples a sub-grid from grid with equal distance between sampling points. The values at the new sample points are determined via linear interpolation.

Args

grid: Tensor to be resampled. Values are assumed to be sampled at cell centers.
start: Origin point of sub-grid within grid, measured in number of cells. Must have a single dimension called vector. Example: start=(1, 0.5) would slice off the first grid point in dim 1 and take the mean of neighbouring points in dim 2. The order of dims must be equal to size and grid.shape.spatial.
size: Resolution of the sub-grid. Must not be larger than the resolution of grid. The order of dims must be equal to start and grid.shape.spatial.

Returns

Sub-grid as Tensor

def save(file: str, obj)

Saves a Tensor or tree using NumPy. This function converts all tensors contained in obj to NumPy tensors before storing. Each tensor is given a name corresponding to its path within obj, allowing reading only specific arrays from the file later on. Pickle is used for structures, but no reference to Tensor or its sub-classes is included.

Args

file: Target file, will be stored as .npz.
obj: Tensor or tree to store.

def scatter(base_grid: Union[phiml.math._tensors.Tensor, phiml.math._shape.Shape], indices: Union[phiml.math._tensors.Tensor, dict], values: Union[float, phiml.math._tensors.Tensor], mode: Union[str, Callable] = 'update', outside_handling: str = 'check', indices_gradient=False, default=None, treat_as_batch=None)

Scatters values into base_grid at indices. instance dimensions of indices and/or values are reduced during scattering. Depending on mode, this method has one of the following effects:

mode='update': Replaces the values of base_grid at indices by values. The result is undefined if indices contains duplicates.
mode='add': Adds values to base_grid at indices. The values corresponding to duplicate indices are accumulated.
mode='mean': Replaces the values of base_grid at indices by the mean of all values with the same index.

Implementations:

NumPy: Slice assignment / numpy.add.at
PyTorch: torch.scatter, torch.scatter_add
TensorFlow: tf.tensor_scatter_nd_add, tf.tensor_scatter_nd_update
Jax: jax.lax.scatter_add, jax.lax.scatter

Args

base_grid

Tensor into which values are scattered.

indices

Tensor of n-dimensional indices at which to place values. Must have a single channel dimension with size matching the number of spatial dimensions of base_grid. This dimension is optional if the spatial rank is 1. Must also contain all scatter_dims.

values

Tensor of values to scatter at indices.

mode

Scatter mode as str or function. Supported modes are 'add', 'mean', 'update', 'max', 'min', 'prod', 'any', 'all'. The corresponding functions are the built-in sum_(), max´,min, as well as the reduce functions in phiml.math`.

outside_handling

Defines how indices lying outside the bounds of base_grid are handled.

'check': Raise an error if any index is out of bounds.
'discard': Outside indices are ignored.
'clamp': Outside indices are projected onto the closest point inside the grid.
'undefined': All points are expected to lie inside the grid. Otherwise an error may be thrown or an undefined tensor may be returned.

indices_gradient

Whether to allow the gradient of this operation to be backpropagated through indices.

default

Default value to use for bins into which no value is scattered. By default, NaN is used for the modes update and mean(), 0 for sum_(), inf for min and -inf for max. This will upgrade the data type to float if necessary.

treat_as_batch

Dimensions which should be treated like dims by this operation. This can be used for scattering vectors along instance dims into a grid. Normally, instance dims on values and indices would not be matched to base_grid but when treated as batch, they will be.

Returns

Copy of base_grid with updated values at indices.

def seed(seed: int)

Sets the current seed of all backends and the built-in random package.

Calling this function with a fixed value at the start of an application yields reproducible results as long as the same backend is used.

Args

seed: Seed to use.

def set_global_precision(floating_point_bits: int)

Sets the floating point precision of DYNAMIC_BACKEND which affects all registered backends.

If floating_point_bits is an integer, all floating point tensors created henceforth will be of the corresponding data type, float16, float32 or float64. Operations may also convert floating point values to this precision, even if the input had a different precision.

If floating_point_bits is None, new tensors will default to float32 unless specified otherwise. The output of math operations has the same precision as its inputs.

Args

floating_point_bits: one of (16, 32, 64, None)

def shape(obj, allow_unshaped=False) ‑> phiml.math._shape.Shape

If obj is a Tensor or Shaped, returns its shape. If obj is a Shape, returns obj.

This function can be passed as a dim argument to an operation to specify that it should act upon all dimensions.

Args

obj: Tensor or Shape or Shaped
allow_unshaped: If True, returns an empty shape for unsupported objects, else raises a ValueError.

Returns

Shape

def shift(x: phiml.math._tensors.Tensor, offsets: Sequence[int], dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = zero-gradient, stack_dim: Union[str, phiml.math._shape.Shape, None] = (shiftᶜ=None), extend_bounds=0, padding_kwargs: dict = None) ‑> List[phiml.math._tensors.Tensor]

Shift the tensor x by a fixed offset, using padding for edge values.

This is similar to numpy.roll() but with major differences:

Values shifted in from the boundary are defined by padding.
Positive offsets represent negative shifts.
Support for multi-dimensional shifts

See Also: index_shift(), neighbor_reduce().

Args

x: Input grid-like Tensor.
offsets: tuple listing shifts to compute, each must be an int. One Tensor will be returned for each entry.
dims: Dimensions along which to shift, defaults to all spatial dims of x.
padding: Padding to be performed at the boundary so that the shifted versions have the same size as x. Must be one of the following: Extrapolation, Tensor or number for constant extrapolation, name of extrapolation as str. Can be set to None to disable padding. Then the result tensors will be smaller than x.
stack_dim: Dimension along which the components corresponding to each dim in dims should be stacked. This can be set to None only if dims is a single dimension.
extend_bounds: Number of cells by which to pad the tensors in addition to the number required to maintain the size of x. Can only be used with a valid padding.
padding_kwargs: Additional keyword arguments to be passed to pad().

Returns

list of shifted tensors. The number of return tensors is equal to the number of offsets.

def si2d(value: ~PhiTreeNodeType) ‑> ~PhiTreeNodeType

Change the type of all spatial and instance dimensions of value to dual dimensions. See rename_dims().

def sigmoid(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes the sigmoid function of the Tensor or PhiTreeNode x.

def sign(x: ~TensorOrTree) ‑> ~TensorOrTree

The sign of positive numbers is 1 and -1 for negative numbers. The sign of 0 is undefined.

Args

x: Tensor or PhiTreeNode

Returns

Tensor or PhiTreeNode matching x.

def sin(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes sin(x) of the Tensor or PhiTreeNode x.

def sinh(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes sinh(x) of the Tensor or PhiTreeNode x.

def slice(value: ~PhiTreeNodeType, slices: Union[Dict[str, Union[int, slice_(), str, tuple, list, Any]], Any]) ‑> ~PhiTreeNodeType

Slices a Tensor or PhiTreeNode along named dimensions.

Args

value

Tensor or PhiTreeNode or Number or None.

slices

dict mapping dimension names to slices. A slice can be one of the following:

An index (int)
A range (slice_())
An item name (str)
Multiple item names (comma-separated str)
Multiple indices or item names (tuple or list)

Returns

Tensor or PhiTreeNode of the same type as value.

Examples

>>> math.slice([vec(x=0, y=1), vec(x=2, y=3)], {'vector': 'y'})
[1, 3]

def slice_off(x, *slices: Dict[str, Union[slice_(), int, str]])

Args

x: Any instance of Shapable

*slices: Returns:

def soft_plus(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes softplus(x) of the Tensor or PhiTreeNode x.

def softmax(x, reduce: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None])

Compute the softmax of x over any dimension. The softmax is e^x / ∑ e^x .

def solve_linear(f: Union[Callable[[~X], ~Y], phiml.math._tensors.Tensor], y: ~Y, solve: phiml.math._optimize.Solve[~X, ~Y], *f_args, grad_for_f=False, f_kwargs: dict = None, **f_kwargs_) ‑> ~X

Solves the system of linear equations f(x) = y and returns x. This method will use the solver specified in solve. The following method identifiers are supported by all backends:

'auto': Automatically choose a solver
'CG': Conjugate gradient, only for symmetric and positive definite matrices.
'CG-adaptive': Conjugate gradient with adaptive step size, only for symmetric and positive definite matrices.
'biCG' or 'biCG-stab(0)': Biconjugate gradient
'biCG-stab' or 'biCG-stab(1)': Biconjugate gradient stabilized, first order
'biCG-stab(2)', 'biCG-stab(4)', …: Biconjugate gradient stabilized, second or higher order
'scipy-direct': SciPy direct solve always run oh the CPU using scipy.sparse.linalg.spsolve.
'scipy-CG', 'scipy-GMres', 'scipy-biCG', 'scipy-biCG-stab', 'scipy-CGS', 'scipy-QMR', 'scipy-GCrotMK': SciPy iterative solvers always run oh the CPU, both in eager execution and JIT mode.

For maximum performance, compile f using jit_compile_linear() beforehand. Then, an optimized representation of f (such as a sparse matrix) will be used to solve the linear system.

Caution: The matrix construction may potentially be performed each time solve_linear() is called if auxiliary arguments change. To prevent this, jit-compile the function that makes the call to solve_linear().

To obtain additional information about the performed solve, perform the solve within a SolveTape context. The used implementation can be obtained as SolveInfo.method.

The gradient of this operation will perform another linear solve with the parameters specified by Solve.gradient_solve.

See Also: solve_nonlinear(), jit_compile_linear().

Args

f

One of the following:

Linear function with Tensor or PhiTreeNode first parameter and return value. f can have additional auxiliary arguments and return auxiliary values.
Dense matrix (Tensor with at least one dual dimension)
Sparse matrix (Sparse Tensor with at least one dual dimension)
Native tensor (not yet supported)

y

Desired output of f(x) as Tensor or PhiTreeNode.

solve

Solve object specifying optimization method, parameters and initial guess for x.

*f_args

Positional arguments to be passed to f after solve.x0. These arguments will not be solved for. Supports vararg mode or pass all arguments as a tuple.

f_kwargs

Additional keyword arguments to be passed to f. These arguments are treated as auxiliary arguments and can be of any type.

Returns

x: solution of the linear system of equations f(x) = y as Tensor or PhiTreeNode.

Raises

NotConverged: If the desired accuracy was not be reached within the maximum number of iterations.
Diverged: If the solve failed prematurely.

def solve_nonlinear(f: Callable, y, solve: phiml.math._optimize.Solve) ‑> phiml.math._tensors.Tensor

Solves the non-linear equation f(x) = y by minimizing the norm of the residual.

This method is limited to backends that support jacobian(), currently PyTorch, TensorFlow and Jax.

To obtain additional information about the performed solve, use a SolveTape.

See Also: minimize(), solve_linear().

Args

f: Function whose output is optimized to match y. All positional arguments of f are optimized and must be Tensor or PhiTreeNode. The output of f must match y.
y: Desired output of f(x) as Tensor or PhiTreeNode.
solve: Solve object specifying optimization method, parameters and initial guess for x.

Returns

x: Solution fulfilling f(x) = y within specified tolerance as Tensor or PhiTreeNode.

Raises

NotConverged: If the desired accuracy was not be reached within the maximum number of iterations.
Diverged: If the solve failed prematurely.

def sort(x: phiml.math._tensors.Tensor, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> phiml.math._tensors.Tensor

Sort the values of x along dim. In order to sort a flattened array, use pack_dims() first.

Args

x: Tensor
dim: Dimension to sort. If not present, sorting will be skipped. Defaults to non-batch dim.

Returns

Sorted Tensor or x if x is constant along dims.

def sparse_tensor(indices: Optional[phiml.math._tensors.Tensor], values: Union[numbers.Number, phiml.math._tensors.Tensor], dense_shape: phiml.math._shape.Shape, can_contain_double_entries=True, indices_sorted=False, format=None, indices_constant: bool = True) ‑> phiml.math._tensors.Tensor

Construct a sparse tensor that stores values at the corresponding indices and is 0 everywhere else. In addition to the sparse dimensions indexed by indices, the tensor inherits all batch and channel dimensions from values.

Args

indices

Tensor encoding the positions of stored values. It can either list the individual stored indices (COO format) or encode only part of the index while containing other dimensions directly (compact format).

For COO, it has the following dimensions:

One instance dimension exactly matching the instance dimension on values. It enumerates the positions of stored entries.
One channel dimension. Its item names must match the dimension names of dense_shape but the order can be arbitrary.
Any number of batch dimensions

You may pass None to create a sparse tensor with no entries.

values

Tensor containing the stored values at positions given by indices. It has the following dimensions:

One instance dimension exactly matching the instance dimension on indices. It enumerates the values of stored entries.
Any number of channel dimensions if multiple values are stored at each index.
Any number of batch dimensions

dense_shape

Dimensions listed in indices. The order can differ from the item names of indices.

can_contain_double_entries

Whether some indices might occur more than once. If so, values at the same index will be summed.

indices_sorted

Whether the indices are sorted in ascending order given the dimension order of the item names of indices.

indices_constant

Whether the positions of the non-zero values are fixed. If True, JIT compilation will not create a placeholder for indices.

format

Sparse format in which to store the data, such as 'coo' or 'csr'. See get_format(). If None, uses the format in which the indices were given.

Returns

Sparse Tensor with the specified format.

def spatial(*args, **dims: Union[int, str, tuple, list, phiml.math._shape.Shape, ForwardRef('Tensor')])

Returns the spatial dimensions of an existing Shape or creates a new Shape with only spatial dimensions.

Usage for filtering spatial dimensions:

>>> spatial_dims = spatial(shape)
>>> spatial_dims = spatial(tensor)

Usage for creating a Shape with only spatial dimensions:

>>> spatial_shape = spatial('undef', x=2, y=3)
(x=2, y=3, undef=None)

Here, the dimension undef is created with an undefined size of None. Undefined sizes are automatically filled in by tensor(), wrap(), stack() and concat().

To create a shape with multiple types, use merge_shapes(), concat_shapes() or the syntax shape1 & shape2.

See Also: channel(), batch(), instance()

Args

*args

Either

Shape or Tensor to filter or
Names of dimensions with undefined sizes as str.

**dims

Dimension sizes and names. Must be empty when used as a filter operation.

Returns

Shape containing only dimensions of type spatial.

def spatial_gradient(grid: phiml.math._tensors.Tensor, dx: Union[float, phiml.math._tensors.Tensor] = 1, difference: str = 'central', padding: Union[Extrapolation, float, phiml.math._tensors.Tensor, str, None] = zero-gradient, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, stack_dim: Union[str, phiml.math._shape.Shape, None] = (gradientᶜ=None), pad=0, padding_kwargs: dict = None) ‑> phiml.math._tensors.Tensor

Calculates the spatial_gradient of a scalar channel from finite differences. The spatial_gradient vectors are in reverse order, lowest dimension first.

Args

grid: grid values
dims: (Optional) Dimensions along which the spatial derivative will be computed. sequence of dimension names
dx: Physical distance between grid points, float or Tensor. When passing a vector-valued Tensor, the dx values should be listed along stack_dim, matching dims.
difference: type of difference, one of ('forward', 'backward', 'central') (default 'forward')
padding: Padding mode. Must be one of the following: Extrapolation, Tensor or number for constant extrapolation, name of extrapolation as str.
stack_dim: name of the new vector dimension listing the spatial_gradient w.r.t. the various axes
pad: How many cells to extend the result compared to grid. This value is added to the internal padding. For non-trivial extrapolations, this gives the correct result while manual padding before or after this operation would not respect the boundary locations.
padding_kwargs: Additional keyword arguments to be passed to pad().

Returns

Tensor

def sqrt(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes sqrt(x) of the Tensor or PhiTreeNode x.

def stack(values: Union[Sequence[~PhiTreeNodeType], Dict[str, ~PhiTreeNodeType]], dim: Union[str, phiml.math._shape.Shape], expand_values=False, simplify=False, layout_non_matching=False, **kwargs) ‑> ~PhiTreeNodeType

Stacks values along the new dimension dim. All values must have the same spatial, instance and channel dimensions. If the dimension sizes vary, the resulting tensor will be non-uniform. Batch dimensions will be added as needed.

Stacking tensors is performed lazily, i.e. the memory is allocated only when needed. This makes repeated stacking and slicing along the same dimension very efficient, i.e. jit-compiled functions will not perform these operations.

Args

values: Collection of Shapable, such as Tensor If a dict, keys must be of type str and are used as item names along dim.
dim: Shape with a least one dimension. None of these dimensions can be present with any of the values. If dim is a single-dimension shape, its size is determined from len(values) and can be left undefined (None). If dim is a multi-dimension shape, its volume must be equal to len(values).
expand_values: If True, will first add missing dimensions to all values, not just batch dimensions. This allows tensors with different dimensions to be stacked. The resulting tensor will have all dimensions that are present in values. If False, this may return a non-numeric object instead.
simplify: If True and all values are equal, returns one value without adding the dimension.
layout_non_matching: If non-matching values should be stacked using a Layout object, i.e. should be put into a named list instead.
**kwargs: Additional keyword arguments required by specific implementations. Adding spatial dimensions to fields requires the bounds: Box argument specifying the physical extent of the new dimensions. Adding batch dimensions must always work without keyword arguments.

Returns

Tensor containing values stacked along dim.

Examples

>>> stack({'x': 0, 'y': 1}, channel('vector'))
(x=0, y=1)

>>> stack([math.zeros(batch(b=2)), math.ones(batch(b=2))], channel(c='x,y'))
(x=0.000, y=1.000); (x=0.000, y=1.000) (bᵇ=2, cᶜ=x,y)

>>> stack([vec(x=1, y=0), vec(x=2, y=3.)], batch('b'))
(x=1.000, y=0.000); (x=2.000, y=3.000) (bᵇ=2, vectorᶜ=x,y)

def std(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> phiml.math._tensors.Tensor

Computes the standard deviation over values along the specified dimensions.

Warning: The standard deviation of non-uniform tensors along the stack dimension is undefined.

Args

value

Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

Returns

Tensor without the reduced dimensions.

def stop_gradient(x)

Disables gradients for the given tensor. This may switch off the gradients for x itself or create a copy of x with disabled gradients.

Implementations:

Args

x: Tensor or PhiTreeNode for which gradients should be disabled.

Returns

Copy of x.

def stored_indices(x: phiml.math._tensors.Tensor, list_dim=(entriesⁱ=None), index_dim=(indexᶜ=None), invalid='discard') ‑> phiml.math._tensors.Tensor

Returns the indices of the stored values for a given `Tensor``. For sparse tensors, this will return the stored indices tensor. For collapsed tensors, only the stored dimensions will be returned.

Args

x: Tensor
list_dim: Dimension along which stored indices should be laid out.
invalid: One of 'discard', 'clamp', 'keep' Filter result by valid indices. Internally, invalid indices may be stored for performance reasons.

Returns

Tensor representing all indices of stored values.

def stored_values(x: phiml.math._tensors.Tensor, list_dim=(entriesⁱ=None), invalid='discard') ‑> phiml.math._tensors.Tensor

Returns the stored values for a given `Tensor``.

For sparse tensors, this will return only the stored entries.

Dense tensors are reshaped so that all non-batch dimensions are packed into list_dim. Batch dimensions are preserved.

Args

x: Tensor
list_dim: Dimension along which stored values should be laid out.
invalid: One of 'discard', 'clamp', 'keep' Filter result by valid indices. Internally, invalid indices may be stored for performance reasons.

Returns

Tensor representing all values stored to represent x.

def sum(value: ~TensorOrTree, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function non_batch>) ‑> ~TensorOrTree

Sums values along the specified dimensions.

Args

value

(Sparse) Tensor or list / tuple of Tensors.

dim

Dimension or dimensions to be reduced. One of

None to reduce all non-batch dimensions
str containing single dimension or comma-separated list of dimensions
Tuple[str] or List[str]
Shape
batch(), instance(), spatial(), channel() to select dimensions by type
'0' when isinstance(value, (tuple, list)) to add up the sequence of Tensors

Returns

Tensor without the reduced dimensions.

def svd(x: phiml.math._tensors.Tensor, feature_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>, list_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = None, latent_dim=(singularᶜ=None), full_matrices=False)

Singular value decomposition.

The original matrix is approximated by (latent_to_value * singular.T) @ latents or latent_to_value @ (singular * latents).

Warning: Even for well-defined SVDs, different backend use different sign conventions, causing results to differ.

Args

x: Matrix containing feature_dim and list_dim.
feature_dim: Dimensions that list the features (columns).
list_dim: Dimensions that list the data points (rows).
latent_dim: Latent dimension. If a size is specified, truncates the SVD to this size.
full_matrices: If True, return full-sized (square) matrices for latent_by_example and latent_to_value. These may not match the singular values.

Returns

latents: Latent vectors of each item listed. Tensor with list_dim and latent_dim.
singular: List of singular values. Tensor with latent_dim.
features: Stacked normalized features / trends. This matrix can be used to compute the original value from a latent vector. Tensor with latent_dim and feature_dim.

def swap_axes(x, axes)

Swap the dimension order of x. This operation is generally not necessary for Tensors because tensors will be reshaped under the hood or when getting the native/numpy representations. It can be used to transpose native tensors.

Implementations:

Args

x: Tensor or native tensor or Shapable.
axes: tuple or list

Returns

Tensor or native tensor, depending on x.

def tan(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes tan(x) of the Tensor or PhiTreeNode x.

def tanh(x: ~TensorOrTree) ‑> ~TensorOrTree

Computes tanh(x) of the Tensor or PhiTreeNode x.

def tcat(values: Sequence[~PhiTreeNodeType], dim_type: Callable, expand_values=False, default_name='tcat') ‑> ~PhiTreeNodeType

Concatenate values by dim type. This function first packs all dimensions of dim_type into one dim, then concatenates all values. Values that do not have a dim of dim_type are considered a size-1 slice.

The name of the first matching dim of dim_type is used as the concatenated output dim name. If no value has a matching dim, default_name is used instead.

Args

values: Values to be concatenated.
dim_type: Dimension type along which to concatenate.
expand_values: Whether to add missing other non-batch dims to values as needed.
default_name: Concatenation dim name if none of the values have a matching dim.

Returns

Same type as any value.

def tensor(data, *shape: Union[str, phiml.math._shape.Shape], convert: bool = True, default_list_dim=(vectorᶜ=None)) ‑> phiml.math._tensors.Tensor

Create a Tensor from the specified data. If convert=True, converts data to the preferred format of the default backend.

data must be one of the following:

Number: returns a dimensionless Tensor.
Native tensor such as NumPy array, TensorFlow tensor or PyTorch tensor.
tuple or list of numbers: backs the Tensor with native tensor.
tuple or list of non-numbers: creates tensors for the items and stacks them.
Tensor: renames dimensions and dimension types if names is specified. Converts all internal native values of the tensor if convert=True.
Shape: creates a 1D tensor listing the dimension sizes.

While specifying names is optional in some cases, it is recommended to always specify them.

Dimension types are always inferred from the dimension names if specified.

Implementations:

NumPy: numpy.array
PyTorch: torch.tensor, torch.from_numpy
TensorFlow: tf.convert_to_tensor
Jax: jax.numpy.array

See Also: wrap() which uses convert=False, layout().

Args

data: native tensor, sparse COO / CSR / CSC matrix, scalar, sequence, Shape or Tensor
shape: Ordered dimensions and types. If sizes are defined, they will be checked against data.`
convert: If True, converts the data to the native format of the current default backend. If False, wraps the data in a Tensor but keeps the given data reference if possible.

Raises

AssertionError: if dimension names are not provided and cannot automatically be inferred
ValueError: if data is not tensor-like

Returns

Tensor containing same values as data

Examples

>>> tensor([1, 2, 3], channel(vector='x,y,z'))
(x=1, y=2, z=3)

>>> tensor([1., 2, 3], channel(vector='x,y,z'))
(x=1.000, y=2.000, z=3.000) float64

>>> tensor(numpy.zeros([10, 8, 6, 2]), batch('batch'), spatial('x,y'), channel(vector='x,y'))
(batchᵇ=10, xˢ=8, yˢ=6, vectorᶜ=x,y) float64 const 0.0

>>> tensor([(0, 1), (0, 2), (1, 3)], instance('particles'), channel(vector='x,y'))
(x=0, y=1); (x=0, y=2); (x=1, y=3) (particlesⁱ=3, vectorᶜ=x,y)

>>> tensor(numpy.random.randn(10))
(vectorᶜ=10) float64 -0.128 ± 1.197 (-2e+00...2e+00)

def tensor_like(existing_tensor: phiml.math._tensors.Tensor, values: Union[numbers.Number, phiml.math._tensors.Tensor, bool], value_order: str = None)

Creates a tensor with the same format and shape as existing_tensor.

Args

existing_tensor: Any Tensor, sparse or dense.
values: New values to replace the existing values by. If existing_tensor is sparse, values must broadcast to the instance dimension listing the stored indices.
value_order: Order of values compared to existing_tensor, only relevant if existing_tensor is sparse. If 'original', the values are ordered like the values that was used to create the first tensor with this sparsity pattern. If 'as existing', the values match the current order of existing_tensor. Note that the order of values may be changed upon creating a sparse tensor.

Returns

Tensor

def to_complex(x: ~TensorOrTree) ‑> ~TensorOrTree

Converts the given tensor to complex floating point format with the currently specified precision.

The precision can be set globally using math.set_global_precision() and locally using with math.precision().

See the documentation at https://tum-pbs.github.io/PhiML/Data_Types.html

Args

x: values to convert

Returns

Tensor of same shape as x

def to_device(value, device: phiml.backend._backend.ComputeDevice, convert=True, use_dlpack=True)

Allocates the tensors of value on device. If the value already exists on that device, this function may either create a copy of value or return value directly.

Args

value: Tensor or PhiTreeNode or native tensor.
device: Device to allocate value on. Either ComputeDevice or category str, such as 'CPU' or 'GPU'.
convert: Whether to convert tensors that do not belong to the corresponding backend to compatible native tensors. If False, this function has no effect on numpy tensors.
use_dlpack: Only if convert==True. Whether to use the DLPack library to convert from one GPU-enabled backend to another.

Returns

Same type as value.

def to_dict(value: Union[phiml.math._tensors.Tensor, phiml.math._shape.Shape])

Returns a serializable form of a Tensor or Shape. The result can be written to a JSON file, for example.

Args

value: Tensor or Shape

Returns

Serializable Python tree of primitives

def to_float(x: ~TensorOrTree) ‑> ~TensorOrTree

Converts the given tensor to floating point format with the currently specified precision.

The precision can be set globally using math.set_global_precision() and locally using with math.precision().

See the documentation at https://tum-pbs.github.io/PhiML/Data_Types.html

Args

x: Tensor or PhiTreeNode to convert

Returns

Tensor or PhiTreeNode matching x.

def to_format(x: phiml.math._tensors.Tensor, format: str)

Converts a Tensor to the specified sparse format or to a dense tensor.

Args

x: Sparse or dense Tensor
format: Target format. One of 'dense', 'coo', 'csr', or 'csc'. Additionally, 'sparse' can be passed to convert dense matrices to a sparse format, decided based on the backend for x.

Returns

Tensor of the specified format.

def to_int32(x: ~TensorOrTree) ‑> ~TensorOrTree

Converts the Tensor or PhiTreeNode x to 32-bit integer.

def to_int64(x: ~TensorOrTree) ‑> ~TensorOrTree

Converts the Tensor or PhiTreeNode x to 64-bit integer.

def trace_check(traced_function, *args, **kwargs) ‑> Tuple[bool, str]

Tests if f(*args, **kwargs) has already been traced for arguments compatible with args and kwargs. If true, jit-compiled functions are very fast since the Python function is not actually called anymore.

Args

traced_function: Transformed Function, e.g. jit-compiled or linear function.
*args: Hypothetical arguments to be passed to f
**kwargs: Hypothetical keyword arguments to be passed to f

Returns

result: True if there is an existing trace that can be used, False if f would have to be re-traced.
message: A str that, if result == False, gives hints as to why f needs to be re-traced given args and kwargs.

def unpack_dim(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None], *unpacked_dims: Union[phiml.math._shape.Shape, Sequence[phiml.math._shape.Shape]], **kwargs)

Decompresses a dimension by unstacking the elements along it. This function replaces the traditional reshape for these cases. The compressed dimension dim is assumed to contain elements laid out according to the order of unpacked_dims.

If dim does not exist on value, this function will return value as-is. This includes primitive types.

Args

value: Shapable, such as Tensor, for which one dimension should be split.
dim: Single dimension to be decompressed.
*unpacked_dims: Either vararg Shape, ordered dimensions to replace dim, fulfilling unpacked_dims.volume == shape(self)[dim].rank. This results in a single tensor output. Alternatively, pass a tuple or list of shapes to unpack a dim into multiple tensors whose combined volumes match dim.size.
**kwargs: Additional keyword arguments required by specific implementations. Adding spatial dimensions to fields requires the bounds: Box argument specifying the physical extent of the new dimensions. Adding batch dimensions must always work without keyword arguments.

Returns

Same type as value.

Examples

>>> unpack_dim(math.zeros(instance(points=12)), 'points', spatial(x=4, y=3))
(xˢ=4, yˢ=3) const 0.0

def unstack(value, dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None]) ‑> tuple

Un-stacks a Sliceable along one or multiple dimensions.

If multiple dimensions are given, the order of elements will be according to the dimension order in dim, i.e. elements along the last dimension will be neighbors in the returned tuple. If no dimension is given or none of the given dimensions exists on value, returns a list containing only value.

Args

value: Shapable, such as Tensor
dim: Dimensions as Shape or comma-separated str or dimension type, i.e. channel(), spatial(), instance(), batch().

Returns

tuple of objects matching the type of value.

Examples

>>> unstack(expand(0, spatial(x=5)), 'x')
(0.0, 0.0, 0.0, 0.0, 0.0)

def upsample2x(grid: phiml.math._tensors.Tensor, padding: Extrapolation = zero-gradient, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function spatial>, padding_kwargs: dict = None) ‑> phiml.math._tensors.Tensor

Resamples a regular grid to double the number of spatial sample points per dimension. The grid values at the new points are determined via linear interpolation.

Args

grid: half-size grid
padding: grid extrapolation
dims: dims along which up-sampling is applied. If None, up-sample along all spatial dims.
grid: Tensor:
padding: Extrapolation: (Default value = extrapolation.BOUNDARY)
dims: tuple or None: (Default value = None)
padding_kwargs: Additional keyword arguments to be passed to pad().

Returns

double-size grid

def use(backend: Union[str, phiml.backend._backend.Backend]) ‑> phiml.backend._backend.Backend

Sets the given backend as default. This setting can be overridden using with backend:.

See default_backend(), choose_backend_t().

Args

backend: Backend or backend name to set as default. Possible names are 'torch', 'tensorflow', 'jax', 'numpy'.

Returns

The chosen backend as a `Backend´ instance.

def vec(name: Union[str, phiml.math._shape.Shape] = 'vector', *sequence, tuple_dim=(sequenceˢ=None), list_dim=(sequenceⁱ=None), **components) ‑> phiml.math._tensors.Tensor

Lay out the given values along a channel dimension without converting them to the current backend.

Args

name: Dimension name.
*sequence: Component values that will also be used as item names. If specified, components must be empty.
**components: Values by component name. If specified, no additional positional arguments must be given.
tuple_dim: Dimension for tuple values passed as components, e.g. vec(x=(0, 1), ...)
list_dim: Dimension for list values passed as components, e.g. vec(x=[0, 1], ...)

Returns

Tensor

Examples

>>> vec(x=1, y=0, z=-1)
(x=1, y=0, z=-1)

>>> vec(x=1., z=0)
(x=1.000, z=0.000)

>>> vec(x=tensor([1, 2, 3], instance('particles')), y=0)
(x=1, y=0); (x=2, y=0); (x=3, y=0) (particlesⁱ=3, vectorᶜ=x,y)

>>> vec(x=0, y=[0, 1])
(x=0, y=0); (x=0, y=1) (vectorᶜ=x,y, sequenceⁱ=2)

>>> vec(x=0, y=(0, 1))
(x=0, y=0); (x=0, y=1) (sequenceˢ=2, vectorᶜ=x,y)

def vec_length(vec: phiml.math._tensors.Tensor, vec_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>, eps: Union[float, phiml.math._tensors.Tensor] = None)

Computes the vector length of vec().

Args

eps: Minimum valid vector length. Use to avoid inf gradients for zero-length vectors. Lengths shorter than eps are set to 0.

def vec_normalize(vec: phiml.math._tensors.Tensor, vec_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>, epsilon=None, allow_infinite=False, allow_zero=False)

Normalizes the vectors in vec(). If vec_dim is None, the combined channel dimensions of vec() are interpreted as a vector.

Args

vec: Tensor to normalize.
vec_dim: Dimensions to normalize over. By default, all channel dimensions are used to compute the vector length.
epsilon: (Optional) Zero-length threshold. Vectors shorter than this length yield the unit vector (1, 0, 0, …). If not specified, the zero-vector yields NaN as it cannot be normalized.
allow_infinite: Allow infinite components in vectors. These vectors will then only points towards the infinite components.
allow_zero: Whether to return zero vectors for inputs smaller epsilon instead of a unit vector.

def vec_squared(vec: phiml.math._tensors.Tensor, vec_dim: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = <function channel>)

Computes the squared length of vec(). If vec_dim is None, the combined channel dimensions of vec() are interpreted as a vector.

def when_available(runnable: Callable, *tensor_args: phiml.math._tensors.Tensor)

Calls runnable(*tensor_args) once the concrete values of all tensors are available. In eager mode, runnable is called immediately. When jit-compiled, runnable is called after the jit-compiled function has returned.

Args

runnable: Function to call as runnable(*tensor_args). This can be a lambda function.
*tensor_args: Tensor values to pass to runnable with concrete values.

def where(condition: Union[phiml.math._tensors.Tensor, float, int], value_true: Union[phiml.math._tensors.Tensor, float, int, Any] = None, value_false: Union[phiml.math._tensors.Tensor, float, int, Any] = None)

Builds a tensor by choosing either values from value_true or value_false depending on condition. If condition is not of type boolean, non-zero values are interpreted as True.

This function requires non-None values for value_true and value_false. To get the indices of True / non-zero values, use :func:nonzero().

Args

condition: determines where to choose values from value_true or from value_false
value_true: Values to pick where condition != 0 / True
value_false: Values to pick where condition == 0 / False

Returns

Tensor containing dimensions of all inputs.

def with_diagonal(matrix: phiml.math._tensors.Tensor, values: Union[float, phiml.math._tensors.Tensor], check_square=True)

Create a copy of matrix, replacing the diagonal elements. If matrix is sparse, diagonal zeros (and possibly other explicitly stored zeros) will be dropped from the sparse matrix.

This function currently only supports sparse COO,CSR,CSC SciPy matrices.

Args

matrix: Tensor with at least one dual dim.
values: Diagonal values
check_square: If True allow this function only for square matrices.

Returns

Tensor

def wrap(data, *shape: Union[str, phiml.math._shape.Shape], default_list_dim=(vectorᶜ=None)) ‑> phiml.math._tensors.Tensor

Short for tensor() with convert=False.

def zeros(*shape: phiml.math._shape.Shape, dtype: Union[phiml.backend._dtype.DType, tuple, type] = None) ‑> phiml.math._tensors.Tensor

Define a tensor with specified shape with value 0.0 / 0 / False everywhere.

This method may not immediately allocate the memory to store the values.

Args

*shape: This (possibly empty) sequence of Shapes is concatenated, preserving the order.
dtype: Data type as DType object. Defaults to float matching the current precision setting.

Returns

Tensor

def zeros_like(obj: Union[phiml.math._tensors.Tensor, PhiTreeNode]) ‑> Union[phiml.math._tensors.Tensor, PhiTreeNode]

Create a Tensor containing only 0.0 / 0 / False with the same shape and dtype as obj.

Classes

class ConvergenceException

Base class for exceptions raised when a solve does not converge.

Ancestors

builtins.RuntimeError
builtins.Exception
builtins.BaseException

Subclasses

phiml.math._optimize.Diverged
phiml.math._optimize.NotConverged

Instance variables

var result: SolveInfo holding information about the solve.

class DType (kind: type, bits: int = None, precision: int = None)

Instances of DType represent the kind and size of data elements. The data type of tensors can be obtained via Tensor.dtype.

The following kinds of data types are supported:

float with 32 / 64 bits
complex with 64 / 128 bits
int with 8 / 16 / 32 / 64 bits
bool with 8 bits
str with 8n bits

Unlike with many computing libraries, there are no global variables corresponding to the available types. Instead, data types can simply be instantiated as needed.

Args

kind: Python type, one of (bool, int, float, complex, str)
bits: number of bits per element, a multiple of 8.

Expand source code

class DType:
    """
    Instances of `DType` represent the kind and size of data elements.
    The data type of tensors can be obtained via `Tensor.dtype`.

    The following kinds of data types are supported:

    * `float` with 32 / 64 bits
    * `complex` with 64 / 128 bits
    * `int` with 8 / 16 / 32 / 64 bits
    * `bool` with 8 bits
    * `str` with 8*n* bits

    Unlike with many computing libraries, there are no global variables corresponding to the available types.
    Instead, data types can simply be instantiated as needed.
    """

    def __init__(self, kind: type, bits: int = None, precision: int = None):
        """
        Args:
            kind: Python type, one of `(bool, int, float, complex, str)`
            bits: number of bits per element, a multiple of 8.
        """
        assert kind in (bool, int, float, complex, str, object)
        if kind is bool:
            assert bits is None, "Bits may not be set for bool or object"
            assert precision is None, f"Precision may only be specified for float or complex but got {kind}, precision={precision}"
            bits = 8
        elif kind == object:
            assert bits is None, "bits may not be set for bool or object"
            assert precision is None, f"Precision may only be specified for float or complex but got {kind}, precision={precision}"
            bits = int(np.round(np.log2(sys.maxsize))) + 1
        elif precision is not None:
            assert bits is None, "Specify either bits or precision when creating a DType but not both."
            assert kind in [float, complex], f"Precision may only be specified for float or complex but got {kind}, precision={precision}"
            if kind == float:
                bits = precision
            else:
                bits = precision * 2
        else:
            assert isinstance(bits, int), f"bits must be an int but got {type(bits)}"
        self.kind = kind
        """ Python class corresponding to the type of data, ignoring precision. One of (bool, int, float, complex, str) """
        self.bits = bits
        """ Number of bits used to store a single value of this type. See `DType.itemsize`. """

    @property
    def precision(self):
        """ Floating point precision. Only defined if `kind in (float, complex)`. For complex values, returns half of `DType.bits`. """
        if self.kind == float:
            return self.bits
        if self.kind == complex:
            return self.bits // 2
        else:
            return None

    @property
    def itemsize(self):
        """ Number of bytes used to storea single value of this type. See `DType.bits`. """
        assert self.bits % 8 == 0
        return self.bits // 8

    def __eq__(self, other):
        return isinstance(other, DType) and self.kind == other.kind and self.bits == other.bits

    def __ne__(self, other):
        return not self == other

    def __hash__(self):
        return hash(self.kind) + hash(self.bits)

    def __repr__(self):
        return f"{self.kind.__name__}{self.bits}"

    @staticmethod
    def as_dtype(value: Union['DType', tuple, type, None]) -> Union['DType', None]:
        if isinstance(value, DType):
            return value
        elif value is int:
            return DType(int, 32)
        elif value is float:
            from . import get_precision
            return DType(float, get_precision())
        elif value is complex:
            from . import get_precision
            return DType(complex, 2 * get_precision())
        elif value is None:
            return None
        elif isinstance(value, tuple):
            return DType(*value)
        elif value is str:
            raise ValueError("str DTypes must specify precision")
        else:
            return DType(value)  # bool, object

Static methods

def as_dtype(value: Union[ForwardRef('DType'), tuple, type, None]) ‑> Optional[phiml.backend._dtype.DType]

Instance variables

var bits

Number of bits used to store a single value of this type. See DType.itemsize.

prop itemsize

Number of bytes used to storea single value of this type. See DType.bits.

Expand source code

@property
def itemsize(self):
    """ Number of bytes used to storea single value of this type. See `DType.bits`. """
    assert self.bits % 8 == 0
    return self.bits // 8

var kind

Python class corresponding to the type of data, ignoring precision. One of (bool, int, float, complex, str)

prop precision

Floating point precision. Only defined if kind in (float, complex). For complex values, returns half of DType.bits.

Expand source code

@property
def precision(self):
    """ Floating point precision. Only defined if `kind in (float, complex)`. For complex values, returns half of `DType.bits`. """
    if self.kind == float:
        return self.bits
    if self.kind == complex:
        return self.bits // 2
    else:
        return None

class Dict (*args, **kwargs)

Dictionary of Tensor or PhiTreeNode values. Dicts are not themselves tensors and do not have a shape. Use layout() to treat dict instances like tensors.

In addition to dictionary functions, supports mathematical operators with other Dicts and lookup via .key syntax. Dict implements PhiTreeNode so instances can be passed to math operations like sin().

Expand source code

class Dict(dict):
    """
    Dictionary of `Tensor` or `phiml.math.magic.PhiTreeNode` values.
    Dicts are not themselves tensors and do not have a shape.
    Use `layout()` to treat `dict` instances like tensors.

    In addition to dictionary functions, supports mathematical operators with other `Dict`s and lookup via `.key` syntax.
    `Dict` implements `phiml.math.magic.PhiTreeNode` so instances can be passed to math operations like `sin`.
    """

    def __value_attrs__(self):
        return tuple(self.keys())
    
    # --- Dict[key] ---

    def __getattr__(self, key):
        try:
            return self[key]
        except KeyError as k:
            raise AttributeError(k)

    def __setattr__(self, key, value):
        self[key] = value

    def __delattr__(self, key):
        try:
            del self[key]
        except KeyError as k:
            raise AttributeError(k)
        
    # --- operators ---
    
    def __neg__(self):
        return Dict({k: -v for k, v in self.items()})
    
    def __invert__(self):
        return Dict({k: ~v for k, v in self.items()})
    
    def __abs__(self):
        return Dict({k: abs(v) for k, v in self.items()})
    
    def __round__(self, n=None):
        return Dict({k: round(v) for k, v in self.items()})

    def __add__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val + other[key] for key, val in self.items()})
        else:
            return Dict({key: val + other for key, val in self.items()})

    def __radd__(self, other):
        if isinstance(other, Dict):
            return Dict({key: other[key] + val for key, val in self.items()})
        else:
            return Dict({key: other + val for key, val in self.items()})

    def __sub__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val - other[key] for key, val in self.items()})
        else:
            return Dict({key: val - other for key, val in self.items()})

    def __rsub__(self, other):
        if isinstance(other, Dict):
            return Dict({key: other[key] - val for key, val in self.items()})
        else:
            return Dict({key: other - val for key, val in self.items()})

    def __mul__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val * other[key] for key, val in self.items()})
        else:
            return Dict({key: val * other for key, val in self.items()})

    def __rmul__(self, other):
        if isinstance(other, Dict):
            return Dict({key: other[key] * val for key, val in self.items()})
        else:
            return Dict({key: other * val for key, val in self.items()})

    def __truediv__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val / other[key] for key, val in self.items()})
        else:
            return Dict({key: val / other for key, val in self.items()})

    def __rtruediv__(self, other):
        if isinstance(other, Dict):
            return Dict({key: other[key] / val for key, val in self.items()})
        else:
            return Dict({key: other / val for key, val in self.items()})

    def __floordiv__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val // other[key] for key, val in self.items()})
        else:
            return Dict({key: val // other for key, val in self.items()})

    def __rfloordiv__(self, other):
        if isinstance(other, Dict):
            return Dict({key: other[key] // val for key, val in self.items()})
        else:
            return Dict({key: other // val for key, val in self.items()})

    def __pow__(self, power, modulo=None):
        assert modulo is None
        if isinstance(power, Dict):
            return Dict({key: val ** power[key] for key, val in self.items()})
        else:
            return Dict({key: val ** power for key, val in self.items()})

    def __rpow__(self, other):
        if isinstance(other, Dict):
            return Dict({key: other[key] ** val for key, val in self.items()})
        else:
            return Dict({key: other ** val for key, val in self.items()})

    def __mod__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val % other[key] for key, val in self.items()})
        else:
            return Dict({key: val % other for key, val in self.items()})

    def __rmod__(self, other):
        if isinstance(other, Dict):
            return Dict({key: other[key] % val for key, val in self.items()})
        else:
            return Dict({key: other % val for key, val in self.items()})

    def __eq__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val == other[key] for key, val in self.items()})
        else:
            return Dict({key: val == other for key, val in self.items()})

    def __ne__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val != other[key] for key, val in self.items()})
        else:
            return Dict({key: val != other for key, val in self.items()})

    def __lt__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val < other[key] for key, val in self.items()})
        else:
            return Dict({key: val < other for key, val in self.items()})

    def __le__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val <= other[key] for key, val in self.items()})
        else:
            return Dict({key: val <= other for key, val in self.items()})

    def __gt__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val > other[key] for key, val in self.items()})
        else:
            return Dict({key: val > other for key, val in self.items()})

    def __ge__(self, other):
        if isinstance(other, Dict):
            return Dict({key: val >= other[key] for key, val in self.items()})
        else:
            return Dict({key: val >= other for key, val in self.items()})

    # --- overridden methods ---

    def copy(self):
        return Dict(self)

Ancestors

builtins.dict

Methods

def copy(self): D.copy() -> a shallow copy of D

class Diverged

Raised if the optimization was stopped prematurely and cannot continue. This may indicate that no solution exists.

The values of the last estimate x may or may not be finite.

This exception inherits from ConvergenceException.

Ancestors

phiml.math._optimize.ConvergenceException
builtins.RuntimeError
builtins.Exception
builtins.BaseException

class IncompatibleShapes (message, *shapes: phiml.math._shape.Shape)

Raised when the shape of a tensor does not match the other arguments.

Expand source code

class IncompatibleShapes(Exception):
    """
    Raised when the shape of a tensor does not match the other arguments.
    """
    def __init__(self, message, *shapes: Shape):
        Exception.__init__(self, message)
        self.shapes = shapes

Ancestors

builtins.Exception
builtins.BaseException

class LinearFunction

Just-in-time compiled linear function of Tensor arguments and return values.

Use jit_compile_linear() to create a linear function representation.

Expand source code

class LinearFunction(Generic[X, Y], Callable[[X], Y]):
    """
    Just-in-time compiled linear function of `Tensor` arguments and return values.

    Use `jit_compile_linear()` to create a linear function representation.
    """

    def __init__(self, f, auxiliary_args: Set[str], forget_traces: bool):
        self.f = f
        self.f_params = function_parameters(f)
        self.auxiliary_args = auxiliary_args
        self.forget_traces = forget_traces
        self.matrices_and_biases: Dict[SignatureKey, Tuple[SparseCoordinateTensor, Tensor, Tuple]] = {}
        self.nl_jit = JitFunction(f, self.auxiliary_args, forget_traces)  # for backends that do not support sparse matrices

    def _get_or_trace(self, key: SignatureKey, args: tuple, f_kwargs: dict):
        if not key.tracing and key in self.matrices_and_biases:
            return self.matrices_and_biases[key]
        else:
            if self.forget_traces:
                self.matrices_and_biases.clear()
            _TRACING_LINEAR.append(self)
            try:
                matrix, bias, raw_out = matrix_from_function(self.f, *args, **f_kwargs, auto_compress=True, _return_raw_output=True)
            finally:
                assert _TRACING_LINEAR.pop(-1) is self
            if not key.tracing:
                self.matrices_and_biases[key] = matrix, bias, raw_out
                if len(self.matrices_and_biases) >= 4:
                    warnings.warn(f"""Φ-ML-lin: The compiled linear function '{f_name(self.f)}' was traced {len(self.matrices_and_biases)} times.
Performing many traces may be slow and cause memory leaks.
Tensors in auxiliary arguments (all except the first parameter unless specified otherwise) are compared by reference, not by tensor values.
Auxiliary arguments: {key.auxiliary_kwargs}
Multiple linear traces can be avoided by jit-compiling the code that calls the linear function or setting forget_traces=True.""", RuntimeWarning, stacklevel=3)
            return matrix, bias, raw_out

    def __call__(self, *args: X, **kwargs) -> Y:
        try:
            key, tensors, natives, x, aux_kwargs = key_from_args(args, kwargs, self.f_params, cache=False, aux=self.auxiliary_args)
        except LinearTraceInProgress:
            return self.f(*args, **kwargs)
        assert tensors, "Linear function requires at least one argument"
        if any(isinstance(t, ShiftLinTracer) for t in tensors):
            # TODO: if t is identity, use cached ShiftLinTracer, otherwise multiply two ShiftLinTracers
            return self.f(*args, **kwargs)
        if not key.backend.supports(Backend.sparse_coo_tensor):  # This might be called inside a Jax linear solve
            # warnings.warn(f"Sparse matrices are not supported by {backend}. Falling back to regular jit compilation.", RuntimeWarning)
            if not math.all_available(*tensors):  # avoid nested tracing, Typical case jax.scipy.sparse.cg(LinearFunction). Nested traces cannot be reused which results in lots of traces per cg.
                ML_LOGGER.debug(f"Φ-ML-lin: Running '{f_name(self.f)}' as-is with {key.backend} because it is being traced.")
                return self.f(*args, **kwargs)
            else:
                return self.nl_jit(*args, **kwargs)
        matrix, bias, (out_tree, out_tensors) = self._get_or_trace(key, args, aux_kwargs)
        result = matrix @ tensors[0] + bias
        out_tensors = list(out_tensors)
        out_tensors[0] = result
        return assemble_tree(out_tree, out_tensors)

    def sparse_matrix(self, *args, **kwargs):
        """
        Create an explicit representation of this linear function as a sparse matrix.

        See Also:
            `sparse_matrix_and_bias()`.

        Args:
            *args: Function arguments. This determines the size of the matrix.
            **kwargs: Additional keyword arguments for the linear function.

        Returns:
            Sparse matrix representation with `values` property and `native()` method.
        """
        key, *_, aux_kwargs = key_from_args(args, kwargs, self.f_params, cache=False, aux=self.auxiliary_args)
        matrix, bias, *_ = self._get_or_trace(key, args, aux_kwargs)
        assert math.close(bias, 0), "This is an affine function and cannot be represented by a single matrix. Use sparse_matrix_and_bias() instead."
        return matrix

    def sparse_matrix_and_bias(self, *args, **kwargs):
        """
        Create an explicit representation of this affine function as a sparse matrix and a bias vector.

        Args:
            *args: Positional arguments to the linear function.
                This determines the size of the matrix.
            **kwargs: Additional keyword arguments for the linear function.

        Returns:
            matrix: Sparse matrix representation with `values` property and `native()` method.
            bias: `Tensor`
        """
        key, *_, aux_kwargs = key_from_args(args, kwargs, self.f_params, cache=False, aux=self.auxiliary_args)
        return self._get_or_trace(key, args, aux_kwargs)[:2]

    def __repr__(self):
        return f"lin({f_name(self.f)})"

Ancestors

collections.abc.Callable
typing.Generic

Methods

def sparse_matrix(self, *args, **kwargs)

Create an explicit representation of this linear function as a sparse matrix.

Args

*args: Function arguments. This determines the size of the matrix.
**kwargs: Additional keyword arguments for the linear function.

Returns

Sparse matrix representation with values property and native() method.

def sparse_matrix_and_bias(self, *args, **kwargs)

Create an explicit representation of this affine function as a sparse matrix and a bias vector.

Args

*args: Positional arguments to the linear function. This determines the size of the matrix.
**kwargs: Additional keyword arguments for the linear function.

Returns

matrix: Sparse matrix representation with values property and native() method.
bias: Tensor

class NotConverged

Raised during optimization if the desired accuracy was not reached within the maximum number of iterations.

This exception inherits from ConvergenceException.

Ancestors

phiml.math._optimize.ConvergenceException
builtins.RuntimeError
builtins.Exception
builtins.BaseException

class Shape

Shapes enumerate dimensions, each consisting of a name, size and type.

There are five types of dimensions: batch(), dual(), spatial(), channel(), and instance().

To construct a Shape, use batch(), dual(), spatial(), channel() or instance(), depending on the desired dimension type. To create a shape with multiple types, use merge_shapes(), concat_shapes() or the syntax shape1 & shape2.

The __init__ constructor is for internal use only.

Expand source code

class Shape:
    """
    Shapes enumerate dimensions, each consisting of a name, size and type.

    There are five types of dimensions: `batch`, `dual`, `spatial`, `channel`, and `instance`.
    """

    def __init__(self, sizes: tuple, names: tuple, types: tuple, item_names: tuple):
        """
        To construct a `Shape`, use `batch`, `dual`, `spatial`, `channel` or `instance`, depending on the desired dimension type.
        To create a shape with multiple types, use `merge_shapes()`, `concat_shapes()` or the syntax `shape1 & shape2`.

        The `__init__` constructor is for internal use only.
        """
        if len(sizes) > 0 and any(s is not None and not isinstance(s, int) for s in sizes):
            from ._tensors import Tensor
            sizes = tuple([s if isinstance(s, Tensor) or s is None else int(s) for s in sizes])  # TODO replace this by an assert
        self.sizes: tuple = sizes
        """
        Ordered dimension sizes as `tuple`.
        The size of a dimension can be an `int` or a `Tensor` for [non-uniform shapes](https://tum-pbs.github.io/PhiML/Non_Uniform.html).
        
        See Also:
            `Shape.get_size()`, `Shape.size`, `Shape.shape`.
        """
        self.names: Tuple[str] = names
        """
        Ordered dimension names as `tuple[str]`.
        
        See Also:
            `Shape.name`.
        """
        self.types: Tuple[str] = types  # undocumented, may be private
        self.item_names: Tuple[Optional[Tuple[str, ...]]] = (None,) * len(sizes) if item_names is None else item_names  # undocumented
        if DEBUG_CHECKS:
            assert len(sizes) == len(names) == len(types) == len(item_names), f"sizes={sizes}, names={names}, types={types}, item_names={item_names}"
            assert len(set(names)) == len(names), f"Duplicate dimension names: {names}"
            assert all(isinstance(n, str) for n in names), f"All names must be of type string but got {names}"
            assert isinstance(self.item_names, tuple)
            assert all([items is None or isinstance(items, tuple) for items in self.item_names])
            assert all([items is None or all([isinstance(n, str) for n in items]) for items in self.item_names])
            from ._tensors import Tensor
            for name, size in zip(names, sizes):
                if isinstance(size, Tensor):
                    assert size.rank > 0
            for name, size, item_names in zip(self.names, self.sizes, self.item_names):
                if item_names is not None:
                    try:
                        int(size)
                    except Exception:
                        raise AssertionError(f"When item names are present, the size must be an integer type")
                    assert len(item_names) == size, f"Number of item names ({len(item_names)}) does not match size {size}"
                    for item_name in item_names:
                        assert item_name, f"Empty item name"
                    assert len(set(item_names)) == len(item_names), f"Duplicate item names in shape {self} at dim '{name}': {item_names}"
            for name, type in zip(names, types):
                if type == DUAL_DIM:
                    assert name.startswith('~'), f"Dual dimensions must start with '~' but got '{name}' in {self}"

    def _check_is_valid_tensor_shape(self):
        if DEBUG_CHECKS:
            from ._tensors import Tensor
            for name, size in zip(self.names, self.sizes):
                if size is not None and isinstance(size, Tensor):
                    assert size.rank > 0
                    for dim in size.shape.names:
                        assert dim in self.names, f"Dimension {name} varies along {dim} but {dim} is not part of the Shape {self}"

    def _to_dict(self, include_sizes=True):
        result = dict(names=self.names, types=self.types, item_names=self.item_names)
        if include_sizes:
            if not all([isinstance(s, int)] for s in self.sizes):
                raise NotImplementedError()
            result['sizes'] = self.sizes
        return result

    @staticmethod
    def _from_dict(dict_: dict):
        names = tuple(dict_['names'])
        sizes = tuple(dict_['sizes']) if 'sizes' in dict_ else (None,) * len(names)
        item_names = tuple([None if n is None else tuple(n) for n in dict_['item_names']])
        return Shape(sizes, names, tuple(dict_['types']), item_names)

    @property
    def name_list(self):
        return list(self.names)

    @property
    def _named_sizes(self):
        return zip(self.names, self.sizes)

    @property
    def _dimensions(self):
        return zip(self.sizes, self.names, self.types, self.item_names)

    @property
    def untyped_dict(self):
        """
        Returns:
            `dict` containing dimension names as keys.
                The values are either the item names as `tuple` if available, otherwise the size.
        """
        return {name: self.get_item_names(i) or self.get_size(i) for i, name in enumerate(self.names)}

    def __len__(self):
        return len(self.sizes)

    def __contains__(self, item):
        if isinstance(item, (str, tuple, list)):
            dims = parse_dim_order(item)
            return all(dim in self.names for dim in dims)
        elif isinstance(item, Shape):
            return all([d in self.names for d in item.names])
        else:
            raise ValueError(item)

    def isdisjoint(self, other: Union['Shape', tuple, list, str]):
        """ Shapes are disjoint if all dimension names of one shape do not occur in the other shape. """
        other = parse_dim_order(other)
        return not any(dim in self.names for dim in other)

    def __iter__(self):
        return iter(self[i] for i in range(self.rank))

    def index(self, dim: Union[str, 'Shape', None]) -> int:
        """
        Finds the index of the dimension within this `Shape`.

        See Also:
            `Shape.indices()`.

        Args:
            dim: Dimension name or single-dimension `Shape`.

        Returns:
            Index as `int`.
        """
        if dim is None:
            return None
        elif isinstance(dim, str):
            if dim not in self.names:
                raise ValueError(f"Shape {self} has no dimension '{dim}'")
            return self.names.index(dim)
        elif isinstance(dim, Shape):
            assert dim.rank == 1, f"index() requires a single dimension as input but got {dim}. Use indices() for multiple dimensions."
            return self.names.index(dim.name)
        else:
            raise ValueError(f"index() requires a single dimension as input but got {dim}")

    def indices(self, dims: Union[tuple, list, 'Shape']) -> Tuple[int]:
        """
        Finds the indices of the given dimensions within this `Shape`.

        See Also:
            `Shape.index()`.

        Args:
            dims: Sequence of dimensions as `tuple`, `list` or `Shape`.

        Returns:
            Indices as `tuple[int]`.
        """
        if isinstance(dims, (list, tuple, set)):
            return tuple([self.index(n) for n in dims if n in self.names])
        elif isinstance(dims, Shape):
            return tuple([self.index(n) for n in dims.names if n in self.names])
        else:
            raise ValueError(f"indices() requires a sequence of dimensions but got {dims}")

    def get_size(self, dim: Union[str, 'Shape', int], default=None):
        """
        See Also:
            `Shape.get_sizes()`, `Shape.size`

        Args:
            dim: Dimension, either as name `str` or single-dimension `Shape` or index `int`.
            default: (Optional) If the dim does not exist, return this value instead of raising an error.

        Returns:
            Size associated with `dim` as `int` or `Tensor`.
        """
        if isinstance(dim, int):
            assert default is None, "Cannot use a default value when passing an int for dim"
            return self.sizes[dim]
        if isinstance(dim, Shape):
            assert dim.rank == 1, f"get_size() requires a single dimension but got {dim}. Use indices() to get multiple sizes."
            dim = dim.name
        if isinstance(dim, str):
            if dim not in self.names:
                if default is None:
                    raise KeyError(f"get_size() failed because '{dim}' is not part of Shape {self} and no default value was provided")
                else:
                    return default
            return self.sizes[self.names.index(dim)]
        else:
            raise ValueError(f"get_size() requires a single dim name but got {dim}. Use indices() to get multiple sizes.")

    def get_sizes(self, dims: Union[tuple, list, 'Shape']) -> tuple:
        """
        See Also:
            `Shape.get_size()`

        Args:
            dims: Dimensions as `tuple`, `list` or `Shape`.

        Returns:
            `tuple`
        """
        assert isinstance(dims, (tuple, list, Shape)), f"get_sizes() requires a sequence of dimensions but got {dims}"
        return tuple([self.get_size(dim) for dim in dims])

    def get_type(self, dim: Union[str, 'Shape']) -> str:
        # undocumented, use get_dim_type() instead.
        if isinstance(dim, str):
            return self.types[self.names.index(dim)]
        elif isinstance(dim, Shape):
            assert dim.rank == 1, f"Shape.get_type() only accepts single-dimension Shapes but got {dim}"
            return self.types[self.names.index(dim.name)]
        else:
            raise ValueError(dim)

    def get_dim_type(self, dim: Union[str, 'Shape']) -> Callable:
        """
        Args:
            dim: Dimension, either as name `str` or single-dimension `Shape`.

        Returns:
            Dimension type, one of `batch`, `spatial`, `instance`, `channel`.
        """
        return DIM_FUNCTIONS[self.get_type(dim)]

    def get_types(self, dims: Union[tuple, list, 'Shape']) -> tuple:
        # undocumented, do not use
        if isinstance(dims, (tuple, list)):
            return tuple(self.get_type(n) for n in dims)
        elif isinstance(dims, Shape):
            return tuple(self.get_type(n) for n in dims.names)
        else:
            raise ValueError(dims)

    def get_item_names(self, dim: Union[str, 'Shape', int], fallback_spatial=False) -> Union[tuple, None]:
        """
        Args:
            fallback_spatial: If `True` and no item names are defined for `dim` and `dim` is a channel dimension, the spatial dimension names are interpreted as item names along `dim` in the order they are listed in this `Shape`.
            dim: Dimension, either as `int` index, `str` name or single-dimension `Shape`.

        Returns:
            Item names as `tuple` or `None` if not defined.
        """
        if isinstance(dim, int):
            result = self.item_names[dim]
        elif isinstance(dim, str):
            result = self.item_names[self.index(dim)]
        elif isinstance(dim, Shape):
            assert dim.rank == 1, f"Shape.get_type() only accepts single-dimension Shapes but got {dim}"
            result = self.item_names[self.names.index(dim.name)]
        else:
            raise ValueError(dim)
        if result is not None:
            return result
        elif fallback_spatial and self.spatial_rank == self.get_size(dim) and self.get_type(dim) == CHANNEL_DIM:
            return self.spatial.names
        else:
            return None

    def flipped(self, dims: Union[List[str], Tuple[str]]):
        item_names = list(self.item_names)
        for dim in dims:
            if dim in self.names:
                dim_i_n = self.get_item_names(dim)
                if dim_i_n is not None:
                    item_names[self.index(dim)] = tuple(reversed(dim_i_n))
        return Shape(self.sizes, self.names, self.types, tuple(item_names))

    def __getitem__(self, selection):
        if isinstance(selection, int):
            return Shape((self.sizes[selection],), (self.names[selection],), (self.types[selection],), (self.item_names[selection],))
        elif isinstance(selection, slice):
            return Shape(self.sizes[selection], self.names[selection], self.types[selection], self.item_names[selection])
        elif isinstance(selection, str):
            if ',' in selection:
                selection = [self.index(s.strip()) for s in selection.split(',')]
            else:
                selection = self.index(selection)
            return self[selection]
        elif isinstance(selection, Shape):
            selection = selection.names
        if isinstance(selection, (tuple, list)):
            selection = [self.index(s) if isinstance(s, str) else s for s in selection]
            return Shape(tuple([self.sizes[i] for i in selection]), tuple([self.names[i] for i in selection]), tuple([self.types[i] for i in selection]), tuple([self.item_names[i] for i in selection]))
        raise AssertionError("Can only access shape elements as shape[int], shape[str], shape[slice], shape[Sequence] or shape[Shape]")

    @property
    def reversed(self):
        return Shape(tuple(reversed(self.sizes)), tuple(reversed(self.names)), tuple(reversed(self.types)), tuple(reversed(self.item_names)))

    @property
    def batch(self) -> 'Shape':
        """
        Filters this shape, returning only the batch dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t == BATCH_DIM]]

    @property
    def non_batch(self) -> 'Shape':
        """
        Filters this shape, returning only the non-batch dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t != BATCH_DIM]]

    @property
    def spatial(self) -> 'Shape':
        """
        Filters this shape, returning only the spatial dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t == SPATIAL_DIM]]

    @property
    def non_spatial(self) -> 'Shape':
        """
        Filters this shape, returning only the non-spatial dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t != SPATIAL_DIM]]

    @property
    def instance(self) -> 'Shape':
        """
        Filters this shape, returning only the instance dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t == INSTANCE_DIM]]

    @property
    def non_instance(self) -> 'Shape':
        """
        Filters this shape, returning only the non-instance dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t != INSTANCE_DIM]]

    @property
    def channel(self) -> 'Shape':
        """
        Filters this shape, returning only the channel dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t == CHANNEL_DIM]]

    @property
    def non_channel(self) -> 'Shape':
        """
        Filters this shape, returning only the non-channel dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t != CHANNEL_DIM]]

    @property
    def dual(self) -> 'Shape':
        """
        Filters this shape, returning only the dual dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t == DUAL_DIM]]

    @property
    def non_dual(self) -> 'Shape':
        """
        Filters this shape, returning only the non-dual dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t != DUAL_DIM]]

    @property
    def primal(self) -> 'Shape':
        """
        Filters this shape, returning only the dual dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t not in [DUAL_DIM, BATCH_DIM]]]

    @property
    def non_primal(self) -> 'Shape':
        """
        Filters this shape, returning only batch and dual dimensions as a new `Shape` object.

        See also:
            `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, t in enumerate(self.types) if t in [DUAL_DIM, BATCH_DIM]]]

    @property
    def transposed(self):
        if self.channel_rank > 0:
            replacement = {DUAL_DIM: CHANNEL_DIM, CHANNEL_DIM: DUAL_DIM}
        elif self.instance_rank > 0:
            replacement = {DUAL_DIM: INSTANCE_DIM, INSTANCE_DIM: DUAL_DIM}
        elif self.spatial_rank > 0:
            replacement = {DUAL_DIM: SPATIAL_DIM, SPATIAL_DIM: DUAL_DIM}
        elif self.dual_rank > 0:
            warnings.warn(f"Transposing {self} is ill-defined because there are not primal dims. Replacing dual dims by channel dims.", SyntaxWarning)
            replacement = {DUAL_DIM: CHANNEL_DIM}
        else:
            raise ValueError(f"Cannot transpose shape {self} as it has no channel or instance or spatial dims.")
        return self._with_types(tuple([replacement.get(t, t) for t in self.types]))

    def transpose(self, dims: DimFilter):
        if callable(dims) and dims in TYPE_BY_FUNCTION:
            dims = TYPE_BY_FUNCTION[dims]
            replacement = {DUAL_DIM: dims, dims: DUAL_DIM}
            return self._with_types(tuple([replacement.get(t, t) for t in self.types]))
        dims = self.only(dims)
        return self.replace(dims, dims.transposed)

    @property
    def non_singleton(self) -> 'Shape':
        """
        Filters this shape, returning only non-singleton dimensions as a new `Shape` object.
        Dimensions are singleton if their size is exactly `1`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, s in enumerate(self.sizes) if not _size_equal(s, 1)]]

    @property
    def singleton(self) -> 'Shape':
        """
        Filters this shape, returning only singleton dimensions as a new `Shape` object.
        Dimensions are singleton if their size is exactly `1`.

        Returns:
            New `Shape` object
        """
        return self[[i for i, s in enumerate(self.sizes) if _size_equal(s, 1)]]

    def assert_all_sizes_defined(self):
        """
        Filters this shape, returning only singleton dimensions as a new `Shape` object.
        Dimensions are singleton if their size is exactly `1`.

        Returns:
            New `Shape` object
        """
        for n, s in zip(self.names, self.sizes):
            assert s is not None, f"All sizes must be defined but dim '{n}' is undefined in shape {self}"

    def as_channel(self):
        """Returns a copy of this `Shape` with all dimensions of type *channel*."""
        return channel(**self.untyped_dict)

    def as_batch(self):
        """Returns a copy of this `Shape` with all dimensions of type *batch*."""
        return batch(**self.untyped_dict)

    def as_spatial(self):
        """Returns a copy of this `Shape` with all dimensions of type *spatial*."""
        return spatial(**self.untyped_dict)

    def as_instance(self):
        """Returns a copy of this `Shape` with all dimensions of type *instance*."""
        return instance(**self.untyped_dict)

    def as_dual(self):
        """Returns a copy of this `Shape` with all dimensions of type *dual*."""
        return dual(**self.untyped_dict)

    def as_type(self, new_type: Callable):
        """Returns a copy of this `Shape` with all dimensions of the given type, either `batch`, `dual`, `spatial`, `instance`, or `channel` ."""
        return new_type(**self.untyped_dict)

    def _more_dual(self):
        return Shape(self.sizes, tuple('~' + n for n in self.names), (DUAL_DIM,) * len(self.names), self.item_names)

    def _less_dual(self, default_type='unknown_primal'):
        names = tuple(n[1:] if n.startswith('~') else n for n in self.names)
        types = [t if t != DUAL_DIM else (DUAL_DIM if n.startswith('~~') else default_type) for n, t in zip(self.names, self.types)]
        return Shape(self.sizes, names, tuple(types), self.item_names)

    def unstack(self, dim='dims') -> Tuple['Shape']:
        """
        Slices this `Shape` along a dimension.
        The dimension listing the sizes of the shape is referred to as `'dims'`.

        Non-uniform tensor shapes may be unstacked along other dimensions as well, see
        https://tum-pbs.github.io/PhiML/Non_Uniform.html

        Args:
            dim: dimension to unstack

        Returns:
            slices of this shape
        """
        if dim == 'dims':
            return tuple(Shape((self.sizes[i],), (self.names[i],), (self.types[i],), (self.item_names[i],)) for i in range(self.rank))
        if dim not in self and self.is_uniform:
            return tuple([self])
        from ._tensors import Tensor
        if dim in self:
            inner = self.without(dim)
            dim_size = self.get_size(dim)
        else:
            inner = self
            dim_size = self.shape.get_size(dim)
        sizes = []
        for size in inner.sizes:
            if isinstance(size, Tensor) and dim in size.shape:
                sizes.append(size._unstack(dim))
                dim_size = size.shape.get_size(dim)
            else:
                sizes.append(size)
        assert isinstance(dim_size, int)
        shapes = tuple(Shape(tuple([int(size[i]) if isinstance(size, tuple) else size for size in sizes]), inner.names, inner.types, inner.item_names) for i in range(dim_size))
        return shapes

    @property
    def name(self) -> str:
        """
        Only for Shapes containing exactly one single dimension.
        Returns the name of the dimension.

        See Also:
            `Shape.names`.
        """
        assert self.rank == 1, f"Shape.name is only defined for shapes of rank 1. shape={self}"
        return self.names[0]

    @property
    def size(self):
        """
        Only for Shapes containing exactly one single dimension.
        Returns the size of the dimension.

        See Also:
            `Shape.sizes`, `Shape.get_size()`.
        """
        assert self.rank == 1, f"Shape.size is only defined for shapes of rank 1 but has dims {self}"
        return self.sizes[0]

    @property
    def type(self) -> str:
        """
        Only for Shapes containing exactly one single dimension.
        Returns the type of the dimension.

        See Also:
            `Shape.get_type()`.
        """
        assert self.rank == 1, "Shape.type is only defined for shapes of rank 1."
        return self.types[0]

    @property
    def dim_type(self):
        types = set(self.types)
        assert len(types) == 1, f"Shape contains multiple types: {self}"
        return DIM_FUNCTIONS[next(iter(types))]

    def __int__(self):
        assert self.rank == 1, "int(Shape) is only defined for shapes of rank 1."
        return self.sizes[0]

    def mask(self, names: Union[tuple, list, set, 'Shape']):
        """
        Returns a binary sequence corresponding to the names of this Shape.
        A value of 1 means that a dimension of this Shape is contained in `names`.

        Args:
          names: instance of dimension
          names: tuple or list or set: 

        Returns:
          binary sequence

        """
        if isinstance(names, str):
            names = [names]
        elif isinstance(names, Shape):
            names = names.names
        mask = [1 if name in names else 0 for name in self.names]
        return tuple(mask)

    def __repr__(self):
        def size_repr(size, items):
            if items is not None:
                items_str = ",".join(items)
                return items_str if len(items_str) <= 12 else f"{size}:{items[0][:5]}..."
            return size

        strings = [f"{name}{SUPERSCRIPT.get(dim_type, '?')}={size_repr(size, items)}" for size, name, dim_type, items in self._dimensions]
        return '(' + ', '.join(strings) + ')'

    def __eq__(self, other):
        if not isinstance(other, Shape):
            return False
        if self.names != other.names or self.types != other.types:
            return False
        for size1, size2 in zip(self.sizes, other.sizes):
            equal = size1 == size2
            assert isinstance(equal, (bool, math.Tensor))
            if isinstance(equal, math.Tensor):
                equal = equal.all
            if not equal:
                return False
        for names1, names2 in zip(self.item_names, other.item_names):
            if names1 != names2:
                return False
        return True

    def __ne__(self, other):
        return not self == other

    def __bool__(self):
        return self.rank > 0

    def _reorder(self, names: Union[tuple, list, 'Shape']) -> 'Shape':
        assert len(names) == self.rank
        if isinstance(names, Shape):
            names = names.names
        order = [self.index(n) for n in names]
        return self[order]

    def _order_group(self, names: Union[tuple, list, 'Shape']) -> list:
        """ Reorders the dimensions of this `Shape` so that `names` are clustered together and occur in the specified order. """
        if isinstance(names, Shape):
            names = names.names
        result = []
        for dim in self.names:
            if dim not in result:
                if dim in names:
                    result.extend(names)
                else:
                    result.append(dim)
        return result

    def __and__(self, other):
        if other is dual:
            return concat_shapes(self, self.primal.as_dual())
        return merge_shapes(self, other)

    def __rand__(self, other):
        if other is dual:
            return concat_shapes(self.primal.as_dual(), self)
        return merge_shapes(self, other)

    def _expand(self, dim: 'Shape', pos=None) -> 'Shape':
        """**Deprecated.** Use `phiml.math.merge_shapes()` or `phiml.math.concat_shapes()` instead. """
        if not dim:
            return self
        assert dim.name not in self, f"Cannot expand shape {self} by {dim} because dimension already exists."
        assert isinstance(dim, Shape) and dim.rank == 1, f"Shape.expand() requires a single dimension as a Shape but got {dim}"
        if pos is None:
            same_type_dims = self[[i for i, t in enumerate(self.types) if t == dim.type]]
            if len(same_type_dims) > 0:
                pos = self.index(same_type_dims.names[0])
            else:
                pos = {
                    BATCH_DIM: 0,
                    DUAL_DIM: self.batch_rank,
                    INSTANCE_DIM: self.batch_rank + self.dual_rank,
                    SPATIAL_DIM: self.batch.rank + self.dual_rank + self.instance_rank,
                    CHANNEL_DIM: self.rank + 1
                }[dim.type]
        elif pos < 0:
            pos += self.rank + 1
        sizes = list(self.sizes)
        names = list(self.names)
        types = list(self.types)
        item_names = list(self.item_names)
        sizes.insert(pos, dim.size)
        names.insert(pos, dim.name)
        types.insert(pos, dim.type)
        item_names.insert(pos, dim.item_names[0])
        return Shape(tuple(sizes), tuple(names), tuple(types), tuple(item_names))

    def without(self, dims: 'DimFilter') -> 'Shape':
        """
        Builds a new shape from this one that is missing all given dimensions.
        Dimensions in `dims` that are not part of this Shape are ignored.
        
        The complementary operation is `Shape.only()`.

        Args:
          dims: Single dimension (str) or instance of dimensions (tuple, list, Shape)
          dims: Dimensions to exclude as `str` or `tuple` or `list` or `Shape`. Dimensions that are not included in this shape are ignored.

        Returns:
          Shape without specified dimensions
        """
        if dims is None:  # subtract none
            return self
        elif callable(dims):
            dims = dims(self)
        if isinstance(dims, str):
            return self[[i for i in range(self.rank) if self.names[i] not in parse_dim_order(dims)]]
        elif isinstance(dims, Shape):
            return self[[i for i in range(self.rank) if self.names[i] not in dims.names]]
        if isinstance(dims, (tuple, list, set)) and all([isinstance(d, str) for d in dims]):
            return self[[i for i in range(self.rank) if self.names[i] not in dims]]
        elif isinstance(dims, (tuple, list, set)):
            result = self
            for wo in dims:
                result = result.without(wo)
            return result
        else:
            raise ValueError(dims)

    def only(self, dims: 'DimFilter', reorder=False):
        """
        Builds a new shape from this one that only contains the given dimensions.
        Dimensions in `dims` that are not part of this Shape are ignored.
        
        The complementary operation is :func:`Shape.without`.

        Args:
          dims: comma-separated dimension names (str) or instance of dimensions (tuple, list, Shape) or filter function.
          reorder: If `False`, keeps the dimension order as defined in this shape.
            If `True`, reorders the dimensions of this shape to match the order of `dims`.

        Returns:
          Shape containing only specified dimensions

        """
        if dims is None:  # keep none
            return EMPTY_SHAPE
        if callable(dims):
            dims = dims(self)
        if isinstance(dims, str):
            dims = parse_dim_order(dims)
        if isinstance(dims, Shape):
            dims = dims.names
        if isinstance(dims, (tuple, list, set)):
            if all(isinstance(d, int) for d in dims):
                if not reorder:
                    dims = tuple(sorted(dims))
                return self[dims]
            dim_names = []
            for d in dims:
                if callable(d):
                    d = d(self)
                if isinstance(d, str):
                    dim_names.append(d)
                elif isinstance(d, Shape):
                    dim_names.extend(d.names)
                else:
                    raise ValueError(f"Format not understood for Shape.only(): {dims}")
            if reorder:
                dim_names = [d.name if isinstance(d, Shape) else d for d in dim_names]
                assert all(isinstance(d, str) for d in dim_names)
                return self[[self.names.index(d) for d in dim_names if d in self.names]]
            else:
                dim_names = [d.name if isinstance(d, Shape) else d for d in dim_names]
                assert all(isinstance(d, str) for d in dim_names)
                return self[[i for i in range(self.rank) if self.names[i] in dim_names]]
        raise ValueError(dims)

    def is_compatible(self, *others: 'Shape'):
        """
        Checks if this shape and the others can be broadcast.

        Args:
            others: Other shapes.

        Returns:
            `True` only if all shapes are compatible.
        """
        try:
            merge_shapes(self, *others)
            return True
        except IncompatibleShapes:
            return False

    @property
    def rank(self) -> int:
        """
        Returns the number of dimensions.
        Equal to `len(shape)`.

        See `Shape.is_empty`, `Shape.batch_rank`, `Shape.spatial_rank`, `Shape.channel_rank`.
        """
        return len(self.sizes)

    @property
    def batch_rank(self) -> int:
        """ Number of batch dimensions """
        return sum([1 for ty in self.types if ty == BATCH_DIM])

    @property
    def instance_rank(self) -> int:
        return sum([1 for ty in self.types if ty == INSTANCE_DIM])

    @property
    def spatial_rank(self) -> int:
        """ Number of spatial dimensions """
        return sum([1 for ty in self.types if ty == SPATIAL_DIM])

    @property
    def dual_rank(self) -> int:
        """ Number of spatial dimensions """
        return sum([1 for ty in self.types if ty == DUAL_DIM])

    @property
    def channel_rank(self) -> int:
        """ Number of channel dimensions """
        return sum([1 for ty in self.types if ty == CHANNEL_DIM])

    @property
    def well_defined(self):
        """
        Returns `True` if no dimension size is `None`.

        Shapes with undefined sizes may be used in `phiml.math.tensor()`, `phiml.math.wrap()`, `phiml.math.stack()` or `phiml.math.concat()`.

        To create an undefined size, call a constructor function (`batch()`, `spatial()`, `channel()`, `instance()`)
        with positional `str` arguments, e.g. `spatial('x')`.
        """
        for size in self.sizes:
            if size is None:
                return False
        return True

    @property
    def defined(self):
        return self[[i for i, size in enumerate(self.sizes) if size is not None]]

    @property
    def undefined(self):
        return self[[i for i, size in enumerate(self.sizes) if size is None]]

    @property
    def shape(self) -> 'Shape':
        """
        Higher-order `Shape`.
        The returned shape will always contain the channel dimension `dims` with a size equal to the `Shape.rank` of this shape.

        For uniform shapes, `Shape.shape` will only contain the dimension `dims` but the shapes of [non-uniform shapes](https://tum-pbs.github.io/PhiML/Non_Uniform.html)
        may contain additional dimensions.

        See Also:
            `Shape.is_uniform`.

        Returns:
            `Shape`.
        """
        from . import Tensor
        shape = Shape((self.rank,), ('dims',), (CHANNEL_DIM,), (self.names,))
        for size in self.sizes:
            if isinstance(size, Tensor):
                shape = shape & size.shape
        return shape

    @property
    def is_uniform(self) -> bool:
        """
        A shape is uniform if it all sizes have a single integer value.

        See Also:
            `Shape.is_non_uniform`, `Shape.shape`.
        """
        from ._tensors import Tensor
        return all(not isinstance(s, Tensor) for s in self.sizes)

    @property
    def is_non_uniform(self) -> bool:
        """
        A shape is non-uniform if the size of any dimension varies along another dimension.

        See Also:
            `Shape.is_uniform`, `Shape.shape`.
        """
        return not self.is_uniform

    @property
    def non_uniform(self) -> 'Shape':
        """
        Returns only the non-uniform dimensions of this shape, i.e. the dimensions whose size varies along another dimension.

        See Also
            `Shape.non_uniform_shape`
        """
        from . import Tensor
        indices = [i for i, size in enumerate(self.sizes) if isinstance(size, Tensor) and size.rank > 0]
        return self[indices]

    @property
    def non_uniform_shape(self):
        """
        Returns the stack dimensions of non-uniform shapes.
        This is equal to `Shape.shape` excluding the `dims` dimension.

        For example, when stacking `(x=3)` and `(x=2)` along `vector`, the resulting shape is non_uniform.
        Its `non_uniform_shape` is `vector` and its `non_uniform` dimension is `x`.

        See Also
            `Shape.non_uniform`.
        """
        from . import Tensor
        shape = EMPTY_SHAPE
        for size in self.sizes:
            if isinstance(size, Tensor):
                shape = shape & size.shape
        return shape

    def with_size(self, size: Union[int, Sequence[str]]):
        """
        Only for single-dimension shapes.
        Returns a `Shape` representing this dimension but with a different size.

        See Also:
            `Shape.with_sizes()`.

        Args:
            size: Replacement size for this dimension.

        Returns:
            `Shape`
        """
        assert self.rank == 1, "Shape.with_size() is only defined for shapes of rank 1."
        return self.with_sizes([size])

    def with_sizes(self, sizes: Union[Sequence[int], Sequence[Tuple[str, ...]], 'Shape', int], keep_item_names=True):
        """
        Returns a new `Shape` matching the dimension names and types of `self` but with different sizes.

        See Also:
            `Shape.with_size()`.

        Args:
            sizes: One of

                * `tuple` / `list` of same length as `self` containing replacement sizes or replacement item names.
                * `Shape` of any rank. Replaces sizes for dimensions shared by `sizes` and `self`.
                * `int`: new size for all dimensions

            keep_item_names: If `False`, forgets all item names.
                If `True`, keeps item names where the size does not change.

        Returns:
            `Shape` with same names and types as `self`.
        """
        if isinstance(sizes, int):
            sizes = [sizes] * len(self.sizes)
        if isinstance(sizes, Shape):
            item_names = [sizes.get_item_names(dim) if dim in sizes else self.get_item_names(dim) for dim in self.names]
            sizes = [sizes.get_size(dim) if dim in sizes else s for dim, s in self._named_sizes]
            return Shape(tuple(sizes), self.names, self.types, tuple(item_names))
        else:
            assert len(sizes) == len(self.sizes), f"Failed to set sizes of Shape {self} to {sizes} because rank does not match."
            sizes_ = []
            item_names = []
            for i, obj in enumerate(sizes):
                new_size, new_item_names = Shape._size_and_item_names_from_obj(obj, self.sizes[i], self.item_names[i], keep_item_names)
                sizes_.append(new_size)
                item_names.append(new_item_names)
            return Shape(tuple(sizes_), self.names, self.types, tuple(item_names))

    @staticmethod
    def _size_and_item_names_from_obj(obj, prev_size, prev_item_names, keep_item_names=True):
        if isinstance(obj, str):
            obj = [s.strip() for s in obj.split(',')]
        if isinstance(obj, (tuple, list)):
            return len(obj), tuple(obj)
        elif isinstance(obj, Number):
            return obj, prev_item_names if keep_item_names and (prev_size is None or _size_equal(obj, prev_size)) else None
        elif isinstance(obj, math.Tensor) or obj is None:
            return obj, None
        elif isinstance(obj, Shape):
            return obj.rank, obj.names
        else:
            raise ValueError(f"sizes can only contain int, str or Tensor but got {type(obj)}")

    def without_sizes(self):
        """
        Returns:
            `Shape` with all sizes undefined (`None`)
        """
        return Shape((None,) * self.rank, self.names, self.types, (None,) * self.rank)

    def _replace_single_size(self, dim: str, size: int, keep_item_names: bool = False):
        new_sizes = list(self.sizes)
        new_sizes[self.index(dim)] = size
        return self.with_sizes(new_sizes, keep_item_names=keep_item_names)

    def with_dim_size(self, dim: Union[str, 'Shape'], size: Union[int, 'math.Tensor', str, tuple, list], keep_item_names=True):
        """
        Returns a new `Shape` that has a different size for `dim`.

        Args:
            dim: Dimension for which to replace the size, `Shape` or `str`.
            size: New size, `int` or `Tensor`

        Returns:
            `Shape` with same names and types as `self`.
        """
        if isinstance(dim, Shape):
            dim = dim.name
        assert isinstance(dim, str)
        new_size, new_item_names = Shape._size_and_item_names_from_obj(size, self.get_size(dim), self.get_item_names(dim), keep_item_names)
        return self.replace(dim, Shape((new_size,), (dim,), (self.get_type(dim),), (new_item_names,)), keep_item_names=keep_item_names)

    def _with_names(self, names: Union[str, tuple, list]):
        if isinstance(names, str):
            names = parse_dim_names(names, self.rank)
            names = [n if n is not None else o for n, o in zip(names, self.names)]
        return Shape(self.sizes, tuple(names), self.types, self.item_names)

    def _replace_names_and_types(self,
                                 dims: Union['Shape', str, tuple, list],
                                 new: Union['Shape', str, tuple, list]) -> 'Shape':
        """
        Returns a copy of `self` with `dims` replaced by `new`.
        Dimensions that are not present in `self` are ignored.

        The dimension order is preserved.

        Args:
            dims: Dimensions to replace.
            new: New dimensions, must have same length as `dims`.
                If a `Shape` is given, replaces the dimension types and item names as well.

        Returns:
            `Shape` with same rank and dimension order as `self`.
        """
        dims = parse_dim_order(dims)
        sizes = [math.rename_dims(s, dims, new) if isinstance(s, math.Tensor) else s for s in self.sizes]
        new = parse_dim_order(new) if isinstance(new, str) else new
        names = list(self.names)
        types = list(self.types)
        item_names = list(self.item_names)
        for old_name, new_dim in zip(dims, new):
            if old_name in self:
                if isinstance(new_dim, Shape):
                    names[self.index(old_name)] = new_dim.name
                    types[self.index(old_name)] = new_dim.type
                    item_names[self.index(old_name)] = new_dim.item_names[0]
                else:
                    names[self.index(old_name)] = _apply_prefix(new_dim, types[self.index(old_name)])
        return Shape(tuple(sizes), tuple(names), tuple(types), tuple(item_names))

    def replace(self, dims: Union['Shape', str, tuple, list], new: 'Shape', keep_item_names=True, replace_item_names: DimFilter = None) -> 'Shape':
        """
        Returns a copy of `self` with `dims` replaced by `new`.
        Dimensions that are not present in `self` are ignored.

        The dimension order is preserved.

        Args:
            dims: Dimensions to replace.
            new: New dimensions, must have same length as `dims`.
                If a `Shape` is given, replaces the dimension types and item names as well.
            keep_item_names: Keeps existing item names for dimensions where `new` does not specify item names if the new dimension has the same size.
            replace_item_names: For which dims the item names should be replaced as well.

        Returns:
            `Shape` with same rank and dimension order as `self`.
        """
        dims = parse_dim_order(dims)
        assert isinstance(new, Shape), f"new must be a Shape but got {new}"
        names = list(self.names)
        sizes = list(self.sizes)
        types = list(self.types)
        item_names = list(self.item_names)
        for i in self.indices(self.only(replace_item_names)):
            if item_names[i]:
                if len(new) > len(dims):
                    raise NotImplementedError
                else:
                    name_map = {d: n for d, n in zip(dims, new.names)}
                    item_names[i] = tuple([name_map.get(n, n) for n in item_names[i]])
        if len(new) > len(dims):  # Put all in one spot
            assert len(dims) == 1, "Cannot replace 2+ dims by more replacements"
            index = self.index(dims[0])
            return concat_shapes(self[:index], new, self[index+1:])
        for old_name, new_dim in zip(dims, new):
            if old_name in self:
                names[self.index(old_name)] = new_dim.name
                types[self.index(old_name)] = new_dim.type
                if new_dim.item_names[0]:
                    item_names[self.index(old_name)] = new_dim.item_names[0]
                elif not _size_equal(new_dim.size, self.get_size(old_name)) or not keep_item_names:
                    item_names[self.index(old_name)] = None  # forget previous item names
                sizes[self.index(old_name)] = new_dim.size
        replaced = Shape(tuple(sizes), tuple(names), tuple(types), tuple(item_names))
        if len(new) == len(dims):
            return replaced
        to_remove = dims[-(len(dims) - len(new)):]
        return replaced.without(to_remove)

    def _with_types(self, types: Union['Shape', str, Tuple[str, ...], List[str]]):
        """
        Only for internal use.
        Note: This method does not rename dimensions to comply with type requirements (e.g. ~ for dual dims).
        """
        if isinstance(types, Shape):
            types = tuple([types.get_type(name) if name in types else self_type for name, self_type in zip(self.names, self.types)])
        elif isinstance(types, str):
            types = (types,) * self.rank
        elif isinstance(types, (tuple, list)):
            types = tuple(types)
        else:
            raise ValueError(types)
        names = tuple([_apply_prefix(name, t) for name, t in zip(self.names, types)])
        return Shape(self.sizes, names, types, self.item_names)

    def _with_item_names(self, item_names: tuple):
        return Shape(self.sizes, self.names, self.types, item_names)

    def _with_item_name(self, dim: str, item_name: tuple):
        if dim not in self:
            return self
        item_names = list(self.item_names)
        item_names[self.index(dim)] = item_name
        return Shape(self.sizes, self.names, self.types, tuple(item_names))

    def _perm(self, names: Tuple[str]) -> List[int]:
        assert len(set(names)) == len(names), f"No duplicates allowed but got {names}"
        assert len(names) >= len(self.names), f"Cannot find permutation for {self} given {names} because names {set(self.names) - set(names)} are missing"
        assert len(names) <= len(self.names), f"Cannot find permutation for {self} given {names} because too many names were passed: {names}"
        perm = [self.names.index(name) for name in names]
        return perm

    @property
    def volume(self) -> Union[int, None]:
        """
        Returns the total number of values contained in a tensor of this shape.
        This is the product of all dimension sizes.

        Returns:
            volume as `int` or `Tensor` or `None` if the shape is not `Shape.well_defined`
        """
        result = 1
        for size in self.sizes:
            if size is None:
                return None
            result *= size
        from ._tensors import Tensor
        if not isinstance(result, Tensor):
            return result
        result /= self.non_uniform_shape.volume  # We summed up the items -> undo multiplication
        return int(result.sum)

    @property
    def is_empty(self) -> bool:
        """ True if this shape has no dimensions. Equivalent to `Shape.rank` `== 0`. """
        return len(self.sizes) == 0

    def after_pad(self, widths: dict) -> 'Shape':
        sizes = list(self.sizes)
        item_names = list(self.item_names)
        for dim, (lo, up) in widths.items():
            if dim in self.names:
                sizes[self.index(dim)] += lo + up
                item_names[self.index(dim)] = None
        return Shape(tuple(sizes), self.names, self.types, tuple(item_names))

    def prepare_gather(self, dim: str, selection: Union[slice, int, 'Shape', str, tuple, list]):
        """
        Parse a slice object for a specific dimension.

        Args:
            dim: Name of dimension to slice.
            selection: Slice object.

        Returns:

        """
        if isinstance(selection, Shape):
            selection = selection.name if selection.rank == 1 else selection.names
        if isinstance(selection, str) and ',' in selection:
            selection = parse_dim_order(selection)
        if isinstance(selection, str):  # single item name
            item_names = self.get_item_names(dim, fallback_spatial=True)
            assert item_names is not None, f"No item names defined for dim '{dim}' in tensor {self.shape} and dimension size does not match spatial rank."
            assert selection in item_names, f"Accessing tensor.{dim}['{selection}'] failed. Item names are {item_names}."
            selection = item_names.index(selection)
        if isinstance(selection, (tuple, list)):
            selection = list(selection)
            if any([isinstance(s, str) for s in selection]):
                item_names = self.get_item_names(dim, fallback_spatial=True)
                for i, s in enumerate(selection):
                    if isinstance(s, str):
                        assert item_names is not None, f"Accessing tensor.{dim}['{s}'] failed because no item names are present on tensor {self.shape}"
                        assert s in item_names, f"Accessing tensor.{dim}['{s}'] failed. Item names are {item_names}."
                        selection[i] = item_names.index(s)
            if not selection:  # empty
                selection = slice(0, 0)
        return selection

    def resolve_index(self, index: Dict[str, Union[slice, int, 'Shape', str, tuple, list]]) -> Dict[str, Union[slice, int, tuple, list]]:
        """
        Replaces item names by the corresponding indices.

        Args:
            index: n-dimensional index or slice.

        Returns:
            Same index but without any reference to item names.
        """
        return {dim: self.prepare_gather(dim, s) for dim, s in index.items()}

    def after_gather(self, selection: dict) -> 'Shape':
        from . import Tensor
        if self.is_non_uniform:
            sizes = [(s[selection] if isinstance(s, Tensor) else s) for s in self.sizes]
            sizes = [(int(s) if isinstance(s, Tensor) and s.rank == 0 else s) for s in sizes]
            result = self.with_sizes(sizes)
        else:
            result = self
        for sel_dim, sel in selection.items():
            if sel_dim not in self.names:
                continue
            sel = self.prepare_gather(sel_dim, sel)
            if isinstance(sel, int):
                result = result.without(sel_dim)
            elif isinstance(sel, slice):
                step = int(sel.step) if sel.step is not None else 1
                start = int(sel.start) if sel.start is not None else (0 if step > 0 else self.get_size(sel_dim)-1)
                stop = int(sel.stop) if sel.stop is not None else (self.get_size(sel_dim) if step > 0 else -1)
                if stop < 0 and step > 0:
                    stop += self.get_size(sel_dim)
                    assert stop >= 0
                if start < 0 and step > 0:
                    start += self.get_size(sel_dim)
                    assert start >= 0
                stop = min(stop, self.get_size(sel_dim))
                new_size = math.to_int64(math.ceil(math.wrap((stop - start) / step)))
                if new_size.rank == 0:
                    new_size = int(new_size)  # NumPy array not allowed because not hashable
                result = result._replace_single_size(sel_dim, new_size, keep_item_names=True)
                if step < 0:
                    result = result.flipped([sel_dim])
                if self.get_item_names(sel_dim) is not None:
                    result = result._with_item_name(sel_dim, tuple(self.get_item_names(sel_dim)[sel]))
            elif isinstance(sel, (tuple, list)):
                result = result._replace_single_size(sel_dim, len(sel))
                if self.get_item_names(sel_dim) is not None:
                    result = result._with_item_name(sel_dim, tuple([self.get_item_names(sel_dim)[i] for i in sel]))
            elif isinstance(sel, Tensor):
                if sel.dtype.kind == bool:
                    raise NotImplementedError("Shape.after_gather(Tensor[bool]) not yet implemented")
                    # from ._ops import nonzero
                    # sel = nonzero(sel)
                if sel.dtype.kind == int:
                    assert len(selection) == 1, f"When slicing a Shape with Tensor[int], only one sel item is allowed but got {sel}"
                    sel_shape = shape(sel)
                    assert sel_shape.channel_rank == 1 and sel_shape.channel.item_names[0], f"Shape.after_gather(Tensor[int]) requires indices to have a single channel dim with item names but got {sel}"
                    indexed = sel_shape.channel.item_names[0]
                    assert indexed in self, f"All indexed dims {indexed} must be part of sliced Shape {self}"
                    from ._ops import slice_
                    sizes = [slice_(s, sel) for s in self.sizes]
                    return self.with_sizes(sizes).without(indexed) & sel_shape.non_channel
            else:
                raise NotImplementedError(f"{type(sel)} not supported. Only (int, slice) allowed.")
        return result

    def meshgrid(self, names=False):
        """
        Builds a sequence containing all multi-indices within a tensor of this shape.
        All indices are returned as `dict` mapping dimension names to `int` indices.

        The corresponding values can be retrieved from Tensors and other Sliceables using `tensor[index]`.

        This function currently only supports uniform tensors.

        Args:
            names: If `True`, replace indices by their item names if available.

        Returns:
            `dict` iterator.
        """
        assert self.is_uniform, f"Shape.meshgrid() is currently not supported for non-uniform tensors, {self}"
        indices = [0] * self.rank
        while True:
            if names:
                yield {dim: (names[index] if names is not None else index) for dim, index, names in zip(self.names, indices, self.item_names)}
            else:
                yield {dim: index for dim, index in zip(self.names, indices)}
            for i in range(self.rank-1, -1, -1):
                indices[i] = (indices[i] + 1) % self.sizes[i]
                if indices[i] != 0:
                    break
            else:
                return

    def first_index(self, names=False):
        return next(iter(self.meshgrid(names=names)))

    def are_adjacent(self, dims: Union[str, tuple, list, set, 'Shape']):
        indices = self.indices(dims)
        return (max(indices) - min(indices)) == len(dims) - 1

    def __add__(self, other):
        if isinstance(other, Shape) and self.isdisjoint(other):
            return concat_shapes(self, other)
        return self._op2(other, lambda s, o: s + o, 0)

    def __radd__(self, other):
        return self._op2(other, lambda s, o: o + s, 0)

    def __sub__(self, other):
        if isinstance(other, (str, Shape, tuple, list, set)) or callable(other):
            return self.without(other)
        return self._op2(other, lambda s, o: s - o, 0)

    def __rsub__(self, other):
        return self._op2(other, lambda s, o: o - s, 0)

    def __mul__(self, other):
        return self._op2(other, lambda s, o: s * o, 1)

    def __rmul__(self, other):
        return self._op2(other, lambda s, o: o * s, 1)

    def _op2(self, other, fun, default: int):
        if isinstance(other, int):
            return Shape(tuple([fun(s, other) for s in self.sizes]), self.names, self.types, (None,) * self.rank)
        else:
            return NotImplemented

    def __hash__(self):
        return hash(self.names)

    @staticmethod
    def __stack__(values, dim: 'Shape', **kwargs):
        return shape_stack(dim, *values)

Instance variables

prop batch : Shape

Filters this shape, returning only the batch dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def batch(self) -> 'Shape':
    """
    Filters this shape, returning only the batch dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t == BATCH_DIM]]

prop batch_rank : int

Number of batch dimensions

Expand source code

@property
def batch_rank(self) -> int:
    """ Number of batch dimensions """
    return sum([1 for ty in self.types if ty == BATCH_DIM])

prop channel : Shape

Filters this shape, returning only the channel dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def channel(self) -> 'Shape':
    """
    Filters this shape, returning only the channel dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t == CHANNEL_DIM]]

prop channel_rank : int

Number of channel dimensions

Expand source code

@property
def channel_rank(self) -> int:
    """ Number of channel dimensions """
    return sum([1 for ty in self.types if ty == CHANNEL_DIM])

prop defined

Expand source code

@property
def defined(self):
    return self[[i for i, size in enumerate(self.sizes) if size is not None]]

prop dim_type

Expand source code

@property
def dim_type(self):
    types = set(self.types)
    assert len(types) == 1, f"Shape contains multiple types: {self}"
    return DIM_FUNCTIONS[next(iter(types))]

prop dual : Shape

Filters this shape, returning only the dual dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def dual(self) -> 'Shape':
    """
    Filters this shape, returning only the dual dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t == DUAL_DIM]]

prop dual_rank : int

Number of spatial dimensions

Expand source code

@property
def dual_rank(self) -> int:
    """ Number of spatial dimensions """
    return sum([1 for ty in self.types if ty == DUAL_DIM])

prop instance : Shape

Filters this shape, returning only the instance dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def instance(self) -> 'Shape':
    """
    Filters this shape, returning only the instance dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t == INSTANCE_DIM]]

prop instance_rank : int

Expand source code

@property
def instance_rank(self) -> int:
    return sum([1 for ty in self.types if ty == INSTANCE_DIM])

prop is_empty : bool

True if this shape has no dimensions. Equivalent to Shape.rank == 0.

Expand source code

@property
def is_empty(self) -> bool:
    """ True if this shape has no dimensions. Equivalent to `Shape.rank` `== 0`. """
    return len(self.sizes) == 0

prop is_non_uniform : bool

A shape is non-uniform if the size of any dimension varies along another dimension.

See Also: Shape.is_uniform, Shape.shape.

Expand source code

@property
def is_non_uniform(self) -> bool:
    """
    A shape is non-uniform if the size of any dimension varies along another dimension.

    See Also:
        `Shape.is_uniform`, `Shape.shape`.
    """
    return not self.is_uniform

prop is_uniform : bool

A shape is uniform if it all sizes have a single integer value.

See Also: Shape.is_non_uniform, Shape.shape.

Expand source code

@property
def is_uniform(self) -> bool:
    """
    A shape is uniform if it all sizes have a single integer value.

    See Also:
        `Shape.is_non_uniform`, `Shape.shape`.
    """
    from ._tensors import Tensor
    return all(not isinstance(s, Tensor) for s in self.sizes)

prop name : str

Only for Shapes containing exactly one single dimension. Returns the name of the dimension.

Returns

New Shape object

Expand source code

@property
def non_batch(self) -> 'Shape':
    """
    Filters this shape, returning only the non-batch dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t != BATCH_DIM]]

prop non_channel : Shape

Filters this shape, returning only the non-channel dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def non_channel(self) -> 'Shape':
    """
    Filters this shape, returning only the non-channel dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t != CHANNEL_DIM]]

prop non_dual : Shape

Filters this shape, returning only the non-dual dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def non_dual(self) -> 'Shape':
    """
    Filters this shape, returning only the non-dual dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t != DUAL_DIM]]

prop non_instance : Shape

Filters this shape, returning only the non-instance dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def non_instance(self) -> 'Shape':
    """
    Filters this shape, returning only the non-instance dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t != INSTANCE_DIM]]

prop non_primal : Shape

Filters this shape, returning only batch and dual dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def non_primal(self) -> 'Shape':
    """
    Filters this shape, returning only batch and dual dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t in [DUAL_DIM, BATCH_DIM]]]

prop non_singleton : Shape

Filters this shape, returning only non-singleton dimensions as a new Shape object. Dimensions are singleton if their size is exactly 1.

Returns

New Shape object

Expand source code

@property
def non_singleton(self) -> 'Shape':
    """
    Filters this shape, returning only non-singleton dimensions as a new `Shape` object.
    Dimensions are singleton if their size is exactly `1`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, s in enumerate(self.sizes) if not _size_equal(s, 1)]]

prop non_spatial : Shape

Filters this shape, returning only the non-spatial dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def non_spatial(self) -> 'Shape':
    """
    Filters this shape, returning only the non-spatial dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t != SPATIAL_DIM]]

prop non_uniform : Shape

Returns only the non-uniform dimensions of this shape, i.e. the dimensions whose size varies along another dimension.

Returns

New Shape object

Expand source code

@property
def primal(self) -> 'Shape':
    """
    Filters this shape, returning only the dual dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t not in [DUAL_DIM, BATCH_DIM]]]

prop rank : int

Returns the number of dimensions. Equal to len(shape()).

See Shape.is_empty, Shape.batch_rank, Shape.spatial_rank, Shape.channel_rank.

Expand source code

@property
def rank(self) -> int:
    """
    Returns the number of dimensions.
    Equal to `len(shape)`.

    See `Shape.is_empty`, `Shape.batch_rank`, `Shape.spatial_rank`, `Shape.channel_rank`.
    """
    return len(self.sizes)

prop reversed

Expand source code

@property
def reversed(self):
    return Shape(tuple(reversed(self.sizes)), tuple(reversed(self.names)), tuple(reversed(self.types)), tuple(reversed(self.item_names)))

prop shape : Shape

Higher-order Shape. The returned shape will always contain the channel dimension dims with a size equal to the Shape.rank of this shape.

For uniform shapes, Shape.shape will only contain the dimension dims but the shapes of non-uniform shapes may contain additional dimensions.

Returns

Shape.

Expand source code

@property
def shape(self) -> 'Shape':
    """
    Higher-order `Shape`.
    The returned shape will always contain the channel dimension `dims` with a size equal to the `Shape.rank` of this shape.

    For uniform shapes, `Shape.shape` will only contain the dimension `dims` but the shapes of [non-uniform shapes](https://tum-pbs.github.io/PhiML/Non_Uniform.html)
    may contain additional dimensions.

    See Also:
        `Shape.is_uniform`.

    Returns:
        `Shape`.
    """
    from . import Tensor
    shape = Shape((self.rank,), ('dims',), (CHANNEL_DIM,), (self.names,))
    for size in self.sizes:
        if isinstance(size, Tensor):
            shape = shape & size.shape
    return shape

prop singleton : Shape

Filters this shape, returning only singleton dimensions as a new Shape object. Dimensions are singleton if their size is exactly 1.

Returns

New Shape object

Expand source code

@property
def singleton(self) -> 'Shape':
    """
    Filters this shape, returning only singleton dimensions as a new `Shape` object.
    Dimensions are singleton if their size is exactly `1`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, s in enumerate(self.sizes) if _size_equal(s, 1)]]

prop size

Only for Shapes containing exactly one single dimension. Returns the size of the dimension.

See Also: Shape.sizes, Shape.get_size().

Expand source code

@property
def size(self):
    """
    Only for Shapes containing exactly one single dimension.
    Returns the size of the dimension.

    See Also:
        `Shape.sizes`, `Shape.get_size()`.
    """
    assert self.rank == 1, f"Shape.size is only defined for shapes of rank 1 but has dims {self}"
    return self.sizes[0]

var sizes

Ordered dimension sizes as tuple. The size of a dimension can be an int or a Tensor for non-uniform shapes.

See Also: Shape.get_size(), Shape.size, Shape.shape.

prop spatial : Shape

Filters this shape, returning only the spatial dimensions as a new Shape object.

Returns

New Shape object

Expand source code

@property
def spatial(self) -> 'Shape':
    """
    Filters this shape, returning only the spatial dimensions as a new `Shape` object.

    See also:
        `Shape.batch`, `Shape.spatial`, `Shape.instance`, `Shape.channel`, `Shape.dual`, `Shape.non_batch`, `Shape.non_spatial`, `Shape.non_instance`, `Shape.non_channel`, `Shape.non_dual`.

    Returns:
        New `Shape` object
    """
    return self[[i for i, t in enumerate(self.types) if t == SPATIAL_DIM]]

prop spatial_rank : int

Number of spatial dimensions

Expand source code

@property
def spatial_rank(self) -> int:
    """ Number of spatial dimensions """
    return sum([1 for ty in self.types if ty == SPATIAL_DIM])

prop transposed

Expand source code

@property
def transposed(self):
    if self.channel_rank > 0:
        replacement = {DUAL_DIM: CHANNEL_DIM, CHANNEL_DIM: DUAL_DIM}
    elif self.instance_rank > 0:
        replacement = {DUAL_DIM: INSTANCE_DIM, INSTANCE_DIM: DUAL_DIM}
    elif self.spatial_rank > 0:
        replacement = {DUAL_DIM: SPATIAL_DIM, SPATIAL_DIM: DUAL_DIM}
    elif self.dual_rank > 0:
        warnings.warn(f"Transposing {self} is ill-defined because there are not primal dims. Replacing dual dims by channel dims.", SyntaxWarning)
        replacement = {DUAL_DIM: CHANNEL_DIM}
    else:
        raise ValueError(f"Cannot transpose shape {self} as it has no channel or instance or spatial dims.")
    return self._with_types(tuple([replacement.get(t, t) for t in self.types]))

prop type : str

Only for Shapes containing exactly one single dimension. Returns the type of the dimension.

Returns

dict containing dimension names as keys. The values are either the item names as tuple if available, otherwise the size.

Expand source code

@property
def untyped_dict(self):
    """
    Returns:
        `dict` containing dimension names as keys.
            The values are either the item names as `tuple` if available, otherwise the size.
    """
    return {name: self.get_item_names(i) or self.get_size(i) for i, name in enumerate(self.names)}

prop volume : Optional[int]

Returns the total number of values contained in a tensor of this shape. This is the product of all dimension sizes.

Returns

volume as int or Tensor or None if the shape is not Shape.well_defined

Expand source code

@property
def volume(self) -> Union[int, None]:
    """
    Returns the total number of values contained in a tensor of this shape.
    This is the product of all dimension sizes.

    Returns:
        volume as `int` or `Tensor` or `None` if the shape is not `Shape.well_defined`
    """
    result = 1
    for size in self.sizes:
        if size is None:
            return None
        result *= size
    from ._tensors import Tensor
    if not isinstance(result, Tensor):
        return result
    result /= self.non_uniform_shape.volume  # We summed up the items -> undo multiplication
    return int(result.sum)

prop well_defined

Returns True if no dimension size is None.

Shapes with undefined sizes may be used in tensor(), wrap(), stack() or concat().

To create an undefined size, call a constructor function (batch(), spatial(), channel(), instance()) with positional str arguments, e.g. spatial('x').

Expand source code

@property
def well_defined(self):
    """
    Returns `True` if no dimension size is `None`.

    Shapes with undefined sizes may be used in `phiml.math.tensor()`, `phiml.math.wrap()`, `phiml.math.stack()` or `phiml.math.concat()`.

    To create an undefined size, call a constructor function (`batch()`, `spatial()`, `channel()`, `instance()`)
    with positional `str` arguments, e.g. `spatial('x')`.
    """
    for size in self.sizes:
        if size is None:
            return False
    return True

Methods

def after_gather(self, selection: dict) ‑> phiml.math._shape.Shape

def after_pad(self, widths: dict) ‑> phiml.math._shape.Shape

def are_adjacent(self, dims: Union[str, tuple, list, set, ForwardRef('Shape')])

def as_batch(self)

Returns a copy of this Shape with all dimensions of type batch.

def as_channel(self)

Returns a copy of this Shape with all dimensions of type channel.

def as_dual(self)

Returns a copy of this Shape with all dimensions of type dual.

def as_instance(self)

Returns a copy of this Shape with all dimensions of type instance.

def as_spatial(self)

Returns a copy of this Shape with all dimensions of type spatial.

def as_type(self, new_type: Callable)

Returns a copy of this Shape with all dimensions of the given type, either batch(), dual(), spatial(), instance(), or channel() .

def assert_all_sizes_defined(self)

Filters this shape, returning only singleton dimensions as a new Shape object. Dimensions are singleton if their size is exactly 1.

Returns

New Shape object

def first_index(self, names=False)

def flipped(self, dims: Union[List[str], Tuple[str]])

def get_dim_type(self, dim: Union[str, ForwardRef('Shape')]) ‑> Callable

Args

dim: Dimension, either as name str or single-dimension Shape.

Returns

Dimension type, one of batch(), spatial(), instance(), channel().

def get_item_names(self, dim: Union[str, ForwardRef('Shape'), int], fallback_spatial=False) ‑> Optional[tuple]

Args

fallback_spatial: If True and no item names are defined for dim and dim is a channel dimension, the spatial dimension names are interpreted as item names along dim in the order they are listed in this Shape.
dim: Dimension, either as int index, str name or single-dimension Shape.

Returns

Item names as tuple or None if not defined.

def get_size(self, dim: Union[str, ForwardRef('Shape'), int], default=None)

See Also: Shape.get_sizes(), Shape.size

Args

dim: Dimension, either as name str or single-dimension Shape or index int.
default: (Optional) If the dim does not exist, return this value instead of raising an error.

Returns

Size associated with dim as int or Tensor.

def get_sizes(self, dims: Union[tuple, list, ForwardRef('Shape')]) ‑> tuple

Args

dims: Dimensions as tuple, list or Shape.

Returns

tuple

def get_type(self, dim: Union[str, ForwardRef('Shape')]) ‑> str

def get_types(self, dims: Union[tuple, list, ForwardRef('Shape')]) ‑> tuple

def index(self, dim: Union[str, ForwardRef('Shape'), None]) ‑> int

Finds the index of the dimension within this Shape.

Args

dim: Dimension name or single-dimension Shape.

Returns

Index as int.

def indices(self, dims: Union[tuple, list, ForwardRef('Shape')]) ‑> Tuple[int]

Finds the indices of the given dimensions within this Shape.

Args

dims: Sequence of dimensions as tuple, list or Shape.

Returns

Indices as tuple[int].

def is_compatible(self, *others: Shape)

Checks if this shape and the others can be broadcast.

Args

others: Other shapes.

Returns

True only if all shapes are compatible.

def isdisjoint(self, other: Union[ForwardRef('Shape'), tuple, list, str])

Shapes are disjoint if all dimension names of one shape do not occur in the other shape.

def mask(self, names: Union[tuple, list, set, ForwardRef('Shape')])

Returns a binary sequence corresponding to the names of this Shape. A value of 1 means that a dimension of this Shape is contained in names.

Args

names: instance of dimension
names: tuple or list or set:

Returns

binary sequence

def meshgrid(self, names=False)

Builds a sequence containing all multi-indices within a tensor of this shape. All indices are returned as dict mapping dimension names to int indices.

The corresponding values can be retrieved from Tensors and other Sliceables using tensor()[index].

This function currently only supports uniform tensors.

Args

names: If True, replace indices by their item names if available.

Returns

dict iterator.

def only(self, dims: DimFilter, reorder=False)

Builds a new shape from this one that only contains the given dimensions. Dimensions in dims that are not part of this Shape are ignored.

The complementary operation is :func:Shape.without().

Args

dims: comma-separated dimension names (str) or instance of dimensions (tuple, list, Shape) or filter function.
reorder: If False, keeps the dimension order as defined in this shape. If True, reorders the dimensions of this shape to match the order of dims.

Returns

Shape containing only specified dimensions

def prepare_gather(self, dim: str, selection: Union[slice_(), int, ForwardRef('Shape'), str, tuple, list])

Parse a slice object for a specific dimension.

Args

dim: Name of dimension to slice.
selection: Slice object.

Returns:

def replace(self, dims: Union[ForwardRef('Shape'), tuple, list, str], new: Shape, keep_item_names=True, replace_item_names: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None] = None) ‑> phiml.math._shape.Shape

Returns a copy of self with dims replaced by new. Dimensions that are not present in self are ignored.

The dimension order is preserved.

Args

dims: Dimensions to replace.
new: New dimensions, must have same length as dims. If a Shape is given, replaces the dimension types and item names as well.
keep_item_names: Keeps existing item names for dimensions where new does not specify item names if the new dimension has the same size.
replace_item_names: For which dims the item names should be replaced as well.

Returns

Shape with same rank and dimension order as self.

def resolve_index(self, index: Dict[str, Union[slice_(), int, ForwardRef('Shape'), str, tuple, list]]) ‑> Dict[str, Union[slice_(), int, tuple, list]]

Replaces item names by the corresponding indices.

Args

index: n-dimensional index or slice.

Returns

Same index but without any reference to item names.

def transpose(self, dims: Union[str, Sequence[+T_co], set, ForwardRef('Shape'), Callable, None])

def unstack(self, dim='dims') ‑> Tuple[phiml.math._shape.Shape]

Slices this Shape along a dimension. The dimension listing the sizes of the shape is referred to as 'dims'.

Non-uniform tensor shapes may be unstacked along other dimensions as well, see https://tum-pbs.github.io/PhiML/Non_Uniform.html

Args

dim: dimension to unstack

Returns

slices of this shape

def with_dim_size(self, dim: Union[str, ForwardRef('Shape')], size: Union[int, ForwardRef('math.Tensor'), str, tuple, list], keep_item_names=True)

Returns a new Shape that has a different size for dim.

Args

dim: Dimension for which to replace the size, Shape or str.
size: New size, int or Tensor

Returns

Shape with same names and types as self.

def with_size(self, size: Union[int, Sequence[str]])

Only for single-dimension shapes. Returns a Shape representing this dimension but with a different size.

Args

size: Replacement size for this dimension.

Returns

Shape

def with_sizes(self, sizes: Union[Sequence[int], Sequence[Tuple[str, ...]], ForwardRef('Shape'), int], keep_item_names=True)

Returns a new Shape matching the dimension names and types of self but with different sizes.

Args

sizes

One of

tuple / list of same length as self containing replacement sizes or replacement item names.
Shape of any rank. Replaces sizes for dimensions shared by sizes and self.
int: new size for all dimensions

keep_item_names

If False, forgets all item names. If True, keeps item names where the size does not change.

Returns

Shape with same names and types as self.

def without(self, dims: DimFilter) ‑> phiml.math._shape.Shape

Builds a new shape from this one that is missing all given dimensions. Dimensions in dims that are not part of this Shape are ignored.

The complementary operation is Shape.only().

Args

dims: Single dimension (str) or instance of dimensions (tuple, list, Shape)
dims: Dimensions to exclude as str or tuple or list or Shape. Dimensions that are not included in this shape are ignored.

Returns

Shape without specified dimensions

def without_sizes(self)

Returns

Shape with all sizes undefined (None)

class Solve (method: Optional[str] = 'auto', rel_tol: Union[float, phiml.math._tensors.Tensor] = None, abs_tol: Union[float, phiml.math._tensors.Tensor] = None, x0: Union[~X, Any] = None, max_iterations: Union[int, phiml.math._tensors.Tensor] = 1000, suppress: Union[tuple, list] = (), preprocess_y: Callable = None, preprocess_y_args: tuple = (), preconditioner: Optional[str] = None, rank_deficiency: int = None, gradient_solve: Optional[ForwardRef('Solve[Y, X]')] = None)

Specifies parameters and stopping criteria for solving a minimization problem or system of equations.

Expand source code

class Solve(Generic[X, Y]):
    """
    Specifies parameters and stopping criteria for solving a minimization problem or system of equations.
    """

    def __init__(self,
                 method: Union[str, None] = 'auto',
                 rel_tol: Union[float, Tensor] = None,
                 abs_tol: Union[float, Tensor] = None,
                 x0: Union[X, Any] = None,
                 max_iterations: Union[int, Tensor] = 1000,
                 suppress: Union[tuple, list] = (),
                 preprocess_y: Callable = None,
                 preprocess_y_args: tuple = (),
                 preconditioner: Optional[str] = None,
                 rank_deficiency: int = None,
                 gradient_solve: Union['Solve[Y, X]', None] = None):
        method = method or 'auto'
        assert isinstance(method, str)
        self.method: str = method
        """ Optimization method to use. Available solvers depend on the solve function that is used to perform the solve. """
        self.rel_tol: Tensor = math.to_float(wrap(rel_tol)) if rel_tol is not None else None
        """Relative tolerance for linear solves only, defaults to 1e-5 for singe precision solves and 1e-12 for double precision solves.
        This must be unset or `0` for minimization problems.
        For systems of equations *f(x)=y*, the final tolerance is `max(rel_tol * norm(y), abs_tol)`. """
        self.abs_tol: Tensor = math.to_float(wrap(abs_tol)) if abs_tol is not None else None
        """ Absolut tolerance for optimization problems and linear solves.
        Defaults to 1e-5 for singe precision solves and 1e-12 for double precision solves.
        For systems of equations *f(x)=y*, the final tolerance is `max(rel_tol * norm(y), abs_tol)`. """
        self.max_iterations: Tensor = math.to_int32(wrap(max_iterations))
        """ Maximum number of iterations to perform before raising a `NotConverged` error is raised. """
        self.x0 = x0
        """ Initial guess for the method, of same type and dimensionality as the solve result.
         This property must be set to a value compatible with the solution `x` before running a method. """
        self.preprocess_y: Callable = preprocess_y
        """ Function to be applied to the right-hand-side vector of an equation system before solving the system.
        This property is propagated to gradient solves by default. """
        self.preprocess_y_args: tuple = preprocess_y_args
        assert all(issubclass(err, ConvergenceException) for err in suppress)
        self.suppress: tuple = tuple(suppress)
        """ Error types to suppress; `tuple` of `ConvergenceException` types. For these errors, the solve function will instead return the partial result without raising the error. """
        self.preconditioner = preconditioner
        self.rank_deficiency: int = rank_deficiency
        """Rank deficiency of matrix or linear function. If not specified, will be determined for (implicit or explicit) matrix solves and assumed 0 for function-based solves."""
        self._gradient_solve: Solve[Y, X] = gradient_solve
        self.id = str(uuid.uuid4())  # not altered by copy_with(), so that the lookup SolveTape[Solve] works after solve has been copied

    @property
    def gradient_solve(self) -> 'Solve[Y, X]':
        """
        Parameters to use for the gradient pass when an implicit gradient is computed.
        If `None`, a duplicate of this `Solve` is created for the gradient solve.

        In any case, the gradient solve information will be stored in `gradient_solve.result`.
        """
        if self._gradient_solve is None:
            self._gradient_solve = Solve(self.method, self.rel_tol, self.abs_tol, None, self.max_iterations, self.suppress, self.preprocess_y, self.preprocess_y_args)
        return self._gradient_solve

    def __repr__(self):
        return f"{self.method} with tolerance {self.rel_tol} (rel), {self.abs_tol} (abs), max_iterations={self.max_iterations}" + (" including preprocessing" if self.preprocess_y else "")

    def __eq__(self, other):
        if not isinstance(other, Solve):
            return False
        if self.method != other.method \
                or not math.equal(self.abs_tol, other.abs_tol) \
                or not math.equal(self.rel_tol, other.rel_tol) \
                or (self.max_iterations != other.max_iterations).any \
                or self.preprocess_y is not other.preprocess_y \
                or self.suppress != other.suppress:
            return False
        return self.x0 == other.x0

    def __variable_attrs__(self):
        return 'x0', 'rel_tol', 'abs_tol', 'max_iterations'

    def __value_attrs__(self):
        return self.__variable_attrs__()

    def with_defaults(self, mode: str):
        assert mode in ('solve', 'optimization')
        result = self
        if result.rel_tol is None:
            result = copy_with(result, rel_tol=_default_tolerance() if mode == 'solve' else wrap(0.))
        if result.abs_tol is None:
            result = copy_with(result, abs_tol=_default_tolerance())
        return result

    def with_preprocessing(self, preprocess_y: Callable, *args) -> 'Solve':
        """
        Adds preprocessing to this `Solve` and all corresponding gradient solves.

        Args:
            preprocess_y: Preprocessing function.
            *args: Arguments for the preprocessing function.

        Returns:
            Copy of this `Solve` with given preprocessing.
        """
        assert self.preprocess_y is None, f"preprocessing for linear solve '{self}' already set"
        gradient_solve = self._gradient_solve.with_preprocessing(preprocess_y, *args) if self._gradient_solve is not None else None
        return copy_with(self, preprocess_y=preprocess_y, preprocess_y_args=args, _gradient_solve=gradient_solve)

Ancestors

typing.Generic

Instance variables

var abs_tol

Absolut tolerance for optimization problems and linear solves. Defaults to 1e-5 for singe precision solves and 1e-12 for double precision solves. For systems of equations f(x)=y, the final tolerance is max(rel_tol * norm(y), abs_tol).

prop gradient_solve : Solve[Y, X]

Parameters to use for the gradient pass when an implicit gradient is computed. If None, a duplicate of this Solve is created for the gradient solve.

In any case, the gradient solve information will be stored in gradient_solve.result.

Expand source code

@property
def gradient_solve(self) -> 'Solve[Y, X]':
    """
    Parameters to use for the gradient pass when an implicit gradient is computed.
    If `None`, a duplicate of this `Solve` is created for the gradient solve.

    In any case, the gradient solve information will be stored in `gradient_solve.result`.
    """
    if self._gradient_solve is None:
        self._gradient_solve = Solve(self.method, self.rel_tol, self.abs_tol, None, self.max_iterations, self.suppress, self.preprocess_y, self.preprocess_y_args)
    return self._gradient_solve

var max_iterations

Maximum number of iterations to perform before raising a NotConverged error is raised.

var method

Optimization method to use. Available solvers depend on the solve function that is used to perform the solve.

var preprocess_y

Function to be applied to the right-hand-side vector of an equation system before solving the system. This property is propagated to gradient solves by default.

var rank_deficiency

Rank deficiency of matrix or linear function. If not specified, will be determined for (implicit or explicit) matrix solves and assumed 0 for function-based solves.

var rel_tol

Relative tolerance for linear solves only, defaults to 1e-5 for singe precision solves and 1e-12 for double precision solves. This must be unset or 0 for minimization problems. For systems of equations f(x)=y, the final tolerance is max(rel_tol * norm(y), abs_tol).

var suppress

Error types to suppress; tuple of ConvergenceException types. For these errors, the solve function will instead return the partial result without raising the error.

var x0

Initial guess for the method, of same type and dimensionality as the solve result. This property must be set to a value compatible with the solution x before running a method.

Methods

def with_defaults(self, mode: str)

def with_preprocessing(self, preprocess_y: Callable, *args) ‑> phiml.math._optimize.Solve

Adds preprocessing to this Solve and all corresponding gradient solves.

Args

preprocess_y: Preprocessing function.
*args: Arguments for the preprocessing function.

Returns

Copy of this Solve with given preprocessing.

class SolveInfo

Stores information about the solution or trajectory of a solve.

When representing the full optimization trajectory, all tracked quantities will have an additional trajectory batch dimension.

Expand source code

class SolveInfo(Generic[X, Y]):
    """
    Stores information about the solution or trajectory of a solve.

    When representing the full optimization trajectory, all tracked quantities will have an additional `trajectory` batch dimension.
    """

    def __init__(self,
                 solve: Solve,
                 x: X,
                 residual: Union[Y, None],
                 iterations: Union[Tensor, None],
                 function_evaluations: Union[Tensor, None],
                 converged: Tensor,
                 diverged: Tensor,
                 method: str,
                 msg: Tensor,
                 solve_time: float):
        # tuple.__new__(SolveInfo, (x, residual, iterations, function_evaluations, converged, diverged))
        self.solve: Solve[X, Y] = solve
        """ `Solve`, Parameters specified for the solve. """
        self.x: X = x
        """ `Tensor` or `phiml.math.magic.PhiTreeNode`, solution estimate. """
        self.residual: Y = residual
        """ `Tensor` or `phiml.math.magic.PhiTreeNode`, residual vector for systems of equations or function value for minimization problems. """
        self.iterations: Tensor = iterations
        """ `Tensor`, number of performed iterations to reach this state. """
        self.function_evaluations: Tensor = function_evaluations
        """ `Tensor`, how often the function (or its gradient function) was called. """
        self.converged: Tensor = converged
        """ `Tensor`, whether the residual is within the specified tolerance. """
        self.diverged: Tensor = diverged
        """ `Tensor`, whether the solve has diverged at this point. """
        self.method = method
        """ `str`, which method and implementation that was used. """
        if all_available(diverged, converged, iterations):
            _, res_tensors = disassemble_tree(residual, cache=False)
            msg_fun = partial(_default_solve_info_msg, solve=solve)
            msg = map_(msg_fun, msg, converged.trajectory[-1], diverged.trajectory[-1], iterations.trajectory[-1], method=method, residual=res_tensors[0], dims=converged.shape.without('trajectory'))
        self.msg = msg
        """ `str`, termination message """
        self.solve_time = solve_time
        """ Time spent in Backend solve function (in seconds) """

    def __repr__(self):
        return f"{self.method}: {self.converged.trajectory[-1].sum} converged, {self.diverged.trajectory[-1].sum} diverged"

    def snapshot(self, index):
        return SolveInfo(self.solve, self.x.trajectory[index], self.residual.trajectory[index], self.iterations.trajectory[index], self.function_evaluations.trajectory[index],
                         self.converged.trajectory[index], self.diverged.trajectory[index], self.method, self.msg, self.solve_time)

    def convergence_check(self, only_warn: bool):
        if not all_available(self.diverged, self.converged):
            return
        if self.diverged.any:
            if Diverged not in self.solve.suppress:
                if only_warn:
                    warnings.warn(self.msg, ConvergenceWarning)
                else:
                    raise Diverged(self)
        if not self.converged.trajectory[-1].all:
            if NotConverged not in self.solve.suppress:
                if only_warn:
                    warnings.warn(self.msg, ConvergenceWarning)
                else:
                    raise NotConverged(self)

Ancestors

typing.Generic

Instance variables

var converged: Tensor, whether the residual is within the specified tolerance.
var diverged: Tensor, whether the solve has diverged at this point.
var function_evaluations: Tensor, how often the function (or its gradient function) was called.
var iterations: Tensor, number of performed iterations to reach this state.
var method: str, which method and implementation that was used.
var msg: str, termination message
var residual: Tensor or PhiTreeNode, residual vector for systems of equations or function value for minimization problems.
var solve: Solve, Parameters specified for the solve.
var solve_time: Time spent in Backend solve function (in seconds)
var x: Tensor or PhiTreeNode, solution estimate.

Methods

def convergence_check(self, only_warn: bool)
def snapshot(self, index)

class SolveTape (*solves: phiml.math._optimize.Solve, record_trajectories=False)

Used to record additional information about solves invoked via solve_linear(), solve_nonlinear() or minimize(). While a SolveTape is active, certain performance optimizations and algorithm implementations may be disabled.

To access a SolveInfo of a recorded solve, use

>>> solve = Solve(method, ...)
>>> with SolveTape() as solves:
>>>     x = math.solve_linear(f, y, solve)
>>> result: SolveInfo = solves[solve]  # get by Solve
>>> result: SolveInfo = solves[0]  # get by index

Args

*solves: (Optional) Select specific solves to be recorded. If none is given, records all solves that occur within the scope of this SolveTape.
record_trajectories: When enabled, the entries of SolveInfo will contain an additional batch dimension named trajectory.

Expand source code

class SolveTape:
    """
    Used to record additional information about solves invoked via `solve_linear()`, `solve_nonlinear()` or `minimize()`.
    While a `SolveTape` is active, certain performance optimizations and algorithm implementations may be disabled.

    To access a `SolveInfo` of a recorded solve, use
    >>> solve = Solve(method, ...)
    >>> with SolveTape() as solves:
    >>>     x = math.solve_linear(f, y, solve)
    >>> result: SolveInfo = solves[solve]  # get by Solve
    >>> result: SolveInfo = solves[0]  # get by index
    """

    def __init__(self, *solves: Solve, record_trajectories=False):
        """
        Args:
            *solves: (Optional) Select specific `solves` to be recorded.
                If none is given, records all solves that occur within the scope of this `SolveTape`.
            record_trajectories: When enabled, the entries of `SolveInfo` will contain an additional batch dimension named `trajectory`.
        """
        self.record_only_ids = [s.id for s in solves]
        self.record_trajectories = record_trajectories
        self.solves: List[SolveInfo] = []

    def should_record_trajectory_for(self, solve: Solve):
        if not self.record_trajectories:
            return False
        if not self.record_only_ids:
            return True
        return solve.id in self.record_only_ids

    def __enter__(self):
        _SOLVE_TAPES.append(self)
        return self

    def __exit__(self, exc_type, exc_val, exc_tb):
        _SOLVE_TAPES.remove(self)

    def _add(self, solve: Solve, trj: bool, result: SolveInfo):
        if any(s.solve.id == solve.id for s in self.solves):
            warnings.warn("SolveTape contains two results for the same solve settings. SolveTape[solve] will return the first solve result.", RuntimeWarning)
        if self.record_only_ids and solve.id not in self.record_only_ids:
            return  # this solve should not be recorded
        if self.record_trajectories:
            assert trj, "Solve did not record a trajectory."
            self.solves.append(result)
        elif trj:
            self.solves.append(result.snapshot(-1))
        else:
            self.solves.append(result)

    def __getitem__(self, item) -> SolveInfo:
        if isinstance(item, int):
            return self.solves[item]
        else:
            assert isinstance(item, Solve)
            solves = [s for s in self.solves if s.solve.id == item.id]
            if len(solves) == 0:
                raise KeyError(f"No solve recorded with key '{item}'.")
            assert len(solves) == 1
            return solves[0]

    def __iter__(self):
        return iter(self.solves)

    def __len__(self):
        return len(self.solves)

Methods

def should_record_trajectory_for(self, solve: phiml.math._optimize.Solve)

class Tensor

Abstract base class to represent structured data of one data type. This class replaces the native tensor classes numpy.ndarray, torch.Tensor, tensorflow.Tensor or jax.numpy.ndarray as the main data container in Φ-ML.

Tensor instances are different from native tensors in two important ways:

The dimensions of Tensors have names and types.
Tensors can have non-uniform shapes, meaning that the size of dimensions can vary along other dimensions.

To check whether a value is a tensor, use isinstance(value, Tensor).

To construct a Tensor, use tensor(), wrap() or one of the basic tensor creation functions, see https://tum-pbs.github.io/PhiML/Tensors.html .

Tensors are not editable. When backed by an editable native tensor, e.g. a numpy.ndarray, do not edit the underlying data structure.

Expand source code

class Tensor:
    """
    Abstract base class to represent structured data of one data type.
    This class replaces the native tensor classes `numpy.ndarray`, `torch.Tensor`, `tensorflow.Tensor` or `jax.numpy.ndarray` as the main data container in Φ-ML.

    `Tensor` instances are different from native tensors in two important ways:

    * The dimensions of Tensors have *names* and *types*.
    * Tensors can have non-uniform shapes, meaning that the size of dimensions can vary along other dimensions.

    To check whether a value is a tensor, use `isinstance(value, Tensor)`.

    To construct a Tensor, use `phiml.math.tensor()`, `phiml.math.wrap()` or one of the basic tensor creation functions,
    see https://tum-pbs.github.io/PhiML/Tensors.html .

    Tensors are not editable.
    When backed by an editable native tensor, e.g. a `numpy.ndarray`, do not edit the underlying data structure.
    """

    def __init__(self):
        if DEBUG_CHECKS:
            self._init_stack = traceback.extract_stack()

    def native(self, order: Union[str, tuple, list, Shape] = None, force_expand=True, to_numpy=False):
        """
        Returns a native tensor object with the dimensions ordered according to `order`.
        
        Transposes the underlying tensor to match the name order and adds singleton dimensions for new dimension names.
        If a dimension of the tensor is not listed in `order`, a `ValueError` is raised.

        Additionally, groups of dimensions can be specified to pack dims, see `phiml.math.reshaped_native()`.

        Args:
            order: (Optional) Order of dimension names as comma-separated string, list or `Shape`.
            force_expand: If `False`, dimensions along which values are guaranteed to be constant will not be expanded to their true size but returned as singleton dimensions.
            to_numpy: Whether to convert the tensor to a NumPy `ndarray`.

        Returns:
            Native tensor representation, such as PyTorch tensor or NumPy array.

        Raises:
            ValueError if the tensor cannot be transposed to match target_shape
        """
        if isinstance(order, (tuple, list)):
            return reshaped_native(self, order, force_expand=force_expand, to_numpy=to_numpy)
        elif order is None:
            assert self.rank <= 1, f"When calling Tensor.native() or Tensor.numpy(), the dimension order must be specified for Tensors with more than one dimension, e.g. '{','.join(self._shape.names)}'. The listed default dimension order can vary depending on the chosen backend. Consider using math.reshaped_native(Tensor) instead."
            order = self._shape.names
        else:
            order = parse_dim_order(order)
        native = self._transposed_native(order, force_expand)
        return choose_backend(native).numpy(native) if to_numpy else native

    def _transposed_native(self, order: Sequence[str], force_expand: bool):
        raise NotImplementedError(self.__class__)

    def numpy(self, order: Union[str, tuple, list, Shape] = None, force_expand=True) -> np.ndarray:
        """
        Converts this tensor to a `numpy.ndarray` with dimensions ordered according to `order`.
        
        *Note*: Using this function breaks the autograd chain. The returned tensor is not differentiable.
        To get a differentiable tensor, use `Tensor.native()` instead.
        
        Transposes the underlying tensor to match the name order and adds singleton dimensions for new dimension names.
        If a dimension of the tensor is not listed in `order`, a `ValueError` is raised.

        If this `Tensor` is backed by a NumPy array, a reference to this array may be returned.

        See Also:
            `phiml.math.numpy()`

        Args:
            order: (Optional) Order of dimension names as comma-separated string, list or `Shape`.

        Returns:
            NumPy representation

        Raises:
            ValueError if the tensor cannot be transposed to match target_shape
        """
        return self.native(order, force_expand, to_numpy=True)

    def __array__(self, dtype=None):  # NumPy conversion
        if self.rank > 1:
            warnings.warn("Automatic conversion of Φ-ML tensors to NumPy can cause problems because the dimension order is not guaranteed.", SyntaxWarning, stacklevel=3)
        return self.numpy(self._shape)

    def __array_ufunc__(self, ufunc, method, *inputs, **kwargs):  # NumPy interface
        if len(inputs) != 2:
            return NotImplemented
        if ufunc.__name__ == 'multiply':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x * y, lambda x, y: choose_backend(x, y).mul(x, y), 'mul', '*')
            else:
                return self._op2(inputs[0], lambda x, y: y * x, lambda x, y: choose_backend(x, y).mul(y, x), 'rmul', '*')
        if ufunc.__name__ == 'add':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x + y, lambda x, y: choose_backend(x, y).add(x, y), 'add', '+')
            else:
                return self._op2(inputs[0], lambda x, y: y + x, lambda x, y: choose_backend(x, y).add(y, x), 'radd', '+')
        if ufunc.__name__ == 'subtract':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x - y, lambda x, y: choose_backend(x, y).sub(x, y), 'add', '-')
            else:
                return self._op2(inputs[0], lambda x, y: y - x, lambda x, y: choose_backend(x, y).sub(y, x), 'rsub', '-')
        if ufunc.__name__ in ['divide', 'true_divide']:
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x / y, lambda x, y: choose_backend(x, y).div(x, y), 'true_divide', '/')
            else:
                return self._op2(inputs[0], lambda x, y: y / x, lambda x, y: choose_backend(x, y).div(y, x), 'r_true_divide', '/')
        if ufunc.__name__ == 'floor_divide':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x // y, lambda x, y: choose_backend(x, y).floordiv(x, y), 'floor_divide', '//')
            else:
                return self._op2(inputs[0], lambda x, y: y // x, lambda x, y: choose_backend(x, y).floordiv(y, x), 'r_floor_divide', '//')
        if ufunc.__name__ == 'remainder':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x % y, lambda x, y: choose_backend(x, y).mod(x, y), 'remainder', '%')
            else:
                return self._op2(inputs[0], lambda x, y: y % x, lambda x, y: choose_backend(x, y).mod(y, x), 'r_remainder', '%')
        if ufunc.__name__ == 'power':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x ** y, lambda x, y: choose_backend(x, y).pow(x, y), 'power', '**')
            else:
                return self._op2(inputs[0], lambda x, y: y ** x, lambda x, y: choose_backend(x, y).pow(y, x), 'r_power', '**')
        if ufunc.__name__ == 'equal':
            if _EQUALITY_REDUCE[-1] == 'ref':
                return wrap(inputs[0] is inputs[1])
            elif _EQUALITY_REDUCE[-1] == 'shape_and_value':
                if set(inputs[0].shape) != set(inputs[1].shape):
                    return wrap(False)
                from ._ops import close
                return wrap(close(inputs[0], inputs[1], rel_tolerance=0, abs_tolerance=0))
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x == y, lambda x, y: choose_backend(x, y).equal(x, y), 'equal', '==')
            else:
                return self._op2(inputs[0], lambda x, y: y == x, lambda x, y: choose_backend(x, y).equal(y, x), 'r_equal', '==')
        if ufunc.__name__ == 'not_equal':
            if _EQUALITY_REDUCE[-1] == 'ref':
                return wrap(inputs[0] is not inputs[1])
            elif _EQUALITY_REDUCE[-1] == 'shape_and_value':
                if set(inputs[0].shape) != set(inputs[1].shape):
                    return wrap(True)
                from ._ops import close
                return wrap(not close(inputs[0], inputs[1], rel_tolerance=0, abs_tolerance=0))
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x != y, lambda x, y: choose_backend(x, y).not_equal(x, y), 'equal', '!=')
            else:
                return self._op2(inputs[0], lambda x, y: y != x, lambda x, y: choose_backend(x, y).not_equal(y, x), 'r_equal', '!=')
        if ufunc.__name__ == 'greater':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x > y, lambda x, y: choose_backend(x, y).greater_than(x, y), 'greater', '>')
            else:
                return self._op2(inputs[0], lambda x, y: y > x, lambda x, y: choose_backend(x, y).greater_than(y, x), 'r_greater', '>')
        if ufunc.__name__ == 'greater_equal':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x >= y, lambda x, y: choose_backend(x, y).greater_or_equal(x, y), 'greater_equal', '>=')
            else:
                return self._op2(inputs[0], lambda x, y: y >= x, lambda x, y: choose_backend(x, y).greater_or_equal(y, x), 'r_greater_equal', '>=')
        if ufunc.__name__ == 'less':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x < y, lambda x, y: choose_backend(x, y).greater_than(y, x), 'less', '<')
            else:
                return self._op2(inputs[0], lambda x, y: y < x, lambda x, y: choose_backend(x, y).greater_than(x, y), 'r_less', '<')
        if ufunc.__name__ == 'less_equal':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x <= y, lambda x, y: choose_backend(x, y).greater_or_equal(y, x), 'less_equal', '<=')
            else:
                return self._op2(inputs[0], lambda x, y: y <= x, lambda x, y: choose_backend(x, y).greater_or_equal(x, y), 'r_less_equal', '<=')
        if ufunc.__name__ == 'left_shift':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x << y, lambda x, y: choose_backend(x, y).shift_bits_left(x, y), 'left_shift', '<<')
            else:
                return self._op2(inputs[0], lambda x, y: y << x, lambda x, y: choose_backend(x, y).shift_bits_left(y, x), 'r_left_shift', '<<')
        if ufunc.__name__ == 'right_shift':
            if inputs[0] is self:
                return self._op2(inputs[1], lambda x, y: x >> y, lambda x, y: choose_backend(x, y).shift_bits_right(x, y), 'right_shift', '>>')
            else:
                return self._op2(inputs[0], lambda x, y: y >> x, lambda x, y: choose_backend(x, y).shift_bits_right(y, x), 'r_right_shift', '>>')
        raise NotImplementedError(f"NumPy function '{ufunc.__name__}' is not compatible with Φ-ML tensors.")

    @property
    def dtype(self) -> DType:
        """ Data type of the elements of this `Tensor`. """
        raise NotImplementedError(self.__class__)

    @property
    def shape(self) -> Shape:
        """ The `Shape` lists the dimensions with their sizes, names and types. """
        raise NotImplementedError(self.__class__)

    @property
    def backend(self) -> Backend:
        from ._ops import choose_backend_t
        return choose_backend_t(self)

    default_backend = backend

    def _with_shape_replaced(self, new_shape: Shape):
        raise NotImplementedError(self.__class__)

    def _with_natives_replaced(self, natives: list):
        """ Replaces all n _natives() of this Tensor with the first n elements of the list and removes them from the list. """
        raise NotImplementedError(self.__class__)

    @property
    def rank(self) -> int:
        """
        Number of explicit dimensions of this `Tensor`. Equal to `tensor.shape.rank`.
        This replaces [`numpy.ndarray.ndim`](https://numpy.org/doc/stable/reference/generated/numpy.ndarray.ndim.html) /
        [`torch.Tensor.dim`](https://pytorch.org/docs/master/generated/torch.Tensor.dim.html) /
        [`tf.rank()`](https://www.tensorflow.org/api_docs/python/tf/rank) /
        [`jax.numpy.ndim()`](https://jax.readthedocs.io/en/latest/_autosummary/jax.numpy.ndim.html).
        """
        return self.shape.rank

    @property
    def _is_tracer(self) -> bool:
        """
        Tracers store additional internal information.
        They should not be converted to `native()` in intermediate operations.
        
        TensorStack prevents performing the actual stack operation if one of its component tensors is special.
        """
        raise NotImplementedError(self.__class__)

    def _to_dict(self):
        return cached(self)._to_dict()

    def __len__(self):
        return self.shape.volume if self.rank == 1 else NotImplemented

    def __bool__(self):
        assert self.rank == 0, f"Cannot convert tensor with non-empty shape {self.shape} to bool. Use tensor.any or tensor.all instead."
        from ._ops import all_
        if not self.default_backend.supports(Backend.jit_compile):  # NumPy
            return bool(self.native()) if self.rank == 0 else bool(all_(self).native())
        else:
            # __bool__ does not work with TensorFlow tracing.
            # TensorFlow needs to see a tf.Tensor in loop conditions but won't allow bool() invocations.
            # However, this function must always return a Python bool.
            raise AssertionError("To evaluate the boolean value of a Tensor, use 'Tensor.all'.")

    @property
    def all(self):
        """ Whether all values of this `Tensor` are `True` as a native bool. """
        from ._ops import all_, cast
        if self.rank == 0:
            return cast(self, DType(bool)).native()
        else:
            return all_(self, dim=self.shape).native()

    @property
    def any(self):
        """ Whether this `Tensor` contains a `True` value as a native bool. """
        from ._ops import any_, cast
        if self.rank == 0:
            return cast(self, DType(bool)).native()
        else:
            return any_(self, dim=self.shape).native()

    @property
    def mean(self):
        """ Mean value of this `Tensor` as a native scalar. """
        from ._ops import mean
        return mean(self, dim=self.shape).native()

    @property
    def finite_mean(self):
        """ Mean value of all finite values in this `Tensor` as a native scalar. """
        from ._ops import finite_mean
        return finite_mean(self, dim=self.shape).native()

    @property
    def std(self):
        """ Standard deviation of this `Tensor` as a native scalar. """
        from ._ops import std
        return std(self, dim=self.shape).native()

    @property
    def sum(self):
        """ Sum of all values of this `Tensor` as a native scalar. """
        from ._ops import sum_
        return sum_(self, dim=self.shape).native()

    @property
    def finite_sum(self):
        """ Sum of all finite values of this `Tensor` as a native scalar. """
        from ._ops import finite_sum
        return finite_sum(self, dim=self.shape).native()

    @property
    def min(self):
        """ Minimum value of this `Tensor` as a native scalar. """
        from ._ops import min_
        return min_(self, dim=self.shape).native()

    @property
    def finite_min(self):
        """ Minimum finite value of this `Tensor` as a native scalar. """
        from ._ops import finite_min
        return finite_min(self, dim=self.shape).native()

    @property
    def max(self):
        """ Maximum value of this `Tensor` as a native scalar. """
        from ._ops import max_
        return max_(self, dim=self.shape).native()

    @property
    def finite_max(self):
        """ Maximum finite value of this `Tensor` as a native scalar. """
        from ._ops import finite_max
        return finite_max(self, dim=self.shape).native()

    @property
    def real(self) -> 'Tensor':
        """
        Returns the real part of this tensor.

        See Also:
            `phiml.math.real()`
        """
        from ._ops import real
        return real(self)

    @property
    def imag(self) -> 'Tensor':
        """
        Returns the imaginary part of this tensor.
        If this tensor does not store complex numbers, returns a zero tensor with the same shape and dtype as this tensor.

        See Also:
            `phiml.math.imag()`
        """
        from ._ops import imag
        return imag(self)

    @property
    def available(self) -> bool:
        """
        A tensor is available if it stores concrete values and these can currently be read.

        Tracers used inside jit compilation are typically not available.

        See Also:
            `phiml.math.jit_compile()`.
        """
        if self._is_tracer:
            return False
        natives = self._natives()
        natives_available = [choose_backend(native).is_available(native) for native in natives]
        return all(natives_available)

    @property
    def device(self) -> Union[ComputeDevice, None]:
        """
        Returns the `ComputeDevice` that this tensor is allocated on.
        The device belongs to this tensor's `default_backend`.

        See Also:
            `Tensor.default_backend`.
        """
        natives = self._natives()
        if not natives:
            return None
        return self.default_backend.get_device(natives[0])

    def __int__(self):
        return int(self.native()) if self.shape.volume == 1 else NotImplemented

    def __float__(self):
        return float(self.native()) if self.shape.volume == 1 else NotImplemented

    def __complex__(self):
        return complex(self.native()) if self.shape.volume == 1 else NotImplemented

    def __index__(self):
        assert self.shape.volume == 1, f"Only scalar tensors can be converted to index but has shape {self.shape}"
        assert self.dtype.kind == int, f"Only int tensors can be converted to index but dtype is {self.dtype}"
        return int(self.native())

    def __contains__(self, item):
        if isinstance(item, Shape):
            return item in self.shape
        elif isinstance(item, BoundDim):
            return item.name in self.shape
        elif isinstance(item, _BoundDims):
            return item.dims in self.shape
        elif isinstance(item, str):
            assert self.dtype.kind != object, "str in Tensor not allowed for object-type Tensors"
            return item in self.shape
        raise ValueError(f"'dim in Tensor' requires dim to be a Shape or str but got {item}")

    def __repr__(self):
        return format_tensor(self, PrintOptions())

    def _repr_pretty_(self, printer, cycle):
        printer.text(format_tensor(self, PrintOptions(colors=DEFAULT_COLORS)))

    def print(self, layout='full', float_format=None, threshold=8, include_shape=None, include_dtype=None):
        print(format_tensor(self, PrintOptions(layout=layout, float_format=float_format, threshold=threshold, colors=DEFAULT_COLORS, include_shape=include_shape, include_dtype=include_dtype)))

    def __format__(self, format_spec: str):
        if BROADCAST_FORMATTER.values is not None:
            return BROADCAST_FORMATTER.register_formatted(self, format_spec)
        specs = format_spec.split(':')
        layout_ = 'auto'
        for possible_layout in ['summary', 'full', 'row', 'numpy']:
            if possible_layout in specs:
                assert layout_ == 'auto', f"Two layout identifiers encountered in '{format_spec}'"
                layout_ = possible_layout
        include_shape = 'shape' in specs or (False if 'no-shape' in specs else None)
        include_dtype = 'dtype' in specs or (False if 'no-dtype' in specs else None)
        color = 'color' in specs or (False if 'no-color' in specs else None)
        threshold = 8
        float_format = None
        for spec in specs:
            if spec.startswith('threshold='):
                threshold = int(spec[len('threshold='):])
            elif '.' in spec:
                float_format = spec
        result = format_tensor(self, PrintOptions(layout_, float_format, threshold, color, include_shape, include_dtype))
        return result

    def __getitem__(self, item) -> 'Tensor':
        if isinstance(item, Tensor):
            if item.dtype.kind == bool:
                from ._ops import boolean_mask
                return boolean_mask(self, item.shape.non_batch or item.shape, item)
            elif item.dtype.kind == int:
                from ._ops import gather
                return gather(self, item)
            else:
                raise AssertionError(f"Index tensor must be of dtype int (gather) or bool (boolean_mask) but got {item}")
        item = slicing_dict(self, item)
        selections = {}
        sliced = self
        for dim, selection in item.items():
            if dim not in self.shape:
                continue
            selection = self.shape.prepare_gather(dim, selection)
            # Either handle slicing directly or add it to the dict
            if isinstance(selection, (tuple, list)):
                result = [sliced[{dim: i}] for i in selection]
                stack_dim = sliced.shape[dim].after_gather({dim: selection})
                sliced = stack(result, stack_dim)
            elif isinstance(selection, Tensor) and selection.dtype.kind == bool:
                from ._ops import boolean_mask
                sliced = boolean_mask(sliced, dim, selection)
            elif isinstance(selection, Tensor) and selection.dtype.kind == int:
                from ._ops import gather
                sliced = gather(sliced, selection, dims=dim)
            else:
                selections[dim] = selection
        return sliced._getitem(selections) if selections else sliced

    def _getitem(self, selection: dict) -> 'Tensor':
        """
        Slice the tensor along specified dimensions.

        Args:
          selection: dim_name: str -> Union[int, slice]
          selection: dict: 

        Returns:

        """
        raise NotImplementedError()

    def __setitem__(self, key, value):
        raise SyntaxError("Tensors are not editable to preserve the autodiff chain. This feature might be added in the future. To update part of a tensor, use math.where() or math.scatter()")

    def __unstack__(self, dims: Tuple[str, ...]) -> Tuple['Tensor', ...]:  # from phiml.math.magic.Sliceable
        if len(dims) == 1:
            return self._unstack(dims[0])
        else:
            return NotImplemented

    def _unstack(self, dim: str):
        """
        Splits this tensor along the specified dimension.
        The returned tensors have the same dimensions as this tensor save the unstacked dimension.

        Raises an error if the dimension is not part of the `Shape` of this `Tensor`.

        See Also:
            `TensorDim.unstack()`

        Args:
            dim: name of dimension to unstack

        Returns:
            tuple of tensors

        """
        raise NotImplementedError()

    @staticmethod
    def __stack__(values: tuple, dim: Shape, **_kwargs) -> 'Tensor':
        if any(isinstance(v, Layout) for v in values):
            layout_ = [v for v in values if isinstance(v, Layout)][0]
            return layout_.__stack__(values, dim, **_kwargs)
        from ._ops import stack_tensors
        return stack_tensors(values, dim)

    def __expand__(self, dims: Shape, **kwargs) -> 'Tensor':
        return expand_tensor(self, dims)

    @staticmethod
    def __concat__(values: tuple, dim: str, **kwargs) -> 'Tensor':
        from ._ops import concat_tensor
        return concat_tensor(values, dim)

    def __replace_dims__(self, dims: Tuple[str, ...], new_dims: Shape, **kwargs) -> 'Tensor':
        return self._with_shape_replaced(rename_dims(self.shape, dims, new_dims))

    def __unpack_dim__(self, dim: str, unpacked_dims: Shape, **kwargs) -> 'Tensor':
        if self.shape.is_uniform:
            native = self._transposed_native(self.shape.names, True)
            new_shape = self.shape.replace(dim, unpacked_dims)
            if not new_shape.well_defined:
                assert new_shape.undefined.rank <= 1, f"At most one dim can have an undefined size to be inferred during un-packing but got {new_shape}"
                missing = self.shape.volume / new_shape.defined.volume
                sizes = [missing if s is None else s for s in new_shape.sizes]
                new_shape = new_shape.with_sizes(sizes)
            if new_shape.is_uniform:
                native_reshaped = choose_backend(native).reshape(native, new_shape.sizes)
                return NativeTensor(native_reshaped, new_shape)
            else:
                split_dim = new_shape.non_uniform_shape[-1]
                i = 0
                result = []
                for idx in split_dim.meshgrid():
                    s = new_shape.after_gather(idx).get_size(new_shape.non_uniform.name)
                    sliced = self[{dim: slice(i, i + s)}]
                    result.append(sliced._with_shape_replaced(sliced.shape.replace(dim, unpacked_dims - split_dim)))
                    i += s
                return stack(result, split_dim)
        else:
            tensors = self._tensors
            if dim == self._stack_dim.name:
                for udim in unpacked_dims:
                    tensors = [TensorStack(tensors[o::len(tensors)//udim.size], udim) for o in range(len(tensors)//udim.size)]
                assert len(tensors) == 1
                return tensors[0]
            raise NotImplementedError

    def __pack_dims__(self, dims: Tuple[str, ...], packed_dim: Shape, pos: Union[int, None], **kwargs) -> 'Tensor':
        order = self.shape._order_group(dims)
        if self.shape.is_uniform:
            native = self._transposed_native(order, force_expand=True)
            if pos is None:
                pos = min(self.shape.indices(dims))
            new_shape = self.shape.without(dims)._expand(packed_dim.with_sizes([self.shape.only(dims).volume]), pos)
            native = choose_backend(native).reshape(native, new_shape.sizes)
            return NativeTensor(native, new_shape)
        else:
            from ._ops import concat_tensor
            value = cached(self)
            assert isinstance(value, TensorStack)
            inner_packed = [pack_dims(t, dims, packed_dim) for t in value._tensors]
            return concat_tensor(inner_packed, packed_dim.name)

    def __cast__(self, dtype: DType):
        return self._op1(lambda native: choose_backend(native).cast(native, dtype=dtype))

    def dimension(self, name: Union[str, Shape]) -> 'TensorDim':
        """
        Returns a reference to a specific dimension of this tensor.
        This is equivalent to the syntax `tensor.<name>`.

        The dimension need not be part of the `Tensor.shape` in which case its size is 1.

        Args:
            name: dimension name

        Returns:
            `TensorDim` corresponding to a dimension of this tensor
        """
        if isinstance(name, str):
            return TensorDim(self, name)
        elif isinstance(name, Shape):
            return TensorDim(self, name.name)
        else:
            raise ValueError(name)

    def pack(self, dims, packed_dim):
        """ See `pack_dims()` """
        from ._ops import pack_dims
        return pack_dims(self, dims, packed_dim)

    def unpack(self, dim, unpacked_dims):
        """ See `unpack_dim()` """
        from ._ops import unpack_dim
        return unpack_dim(self, dim, unpacked_dims)

    @property
    def T(self):
        return self._with_shape_replaced(self.shape.transposed)

    @property
    def Ti(self):
        return self._with_shape_replaced(self.shape.transpose(instance))

    @property
    def Tc(self):
        return self._with_shape_replaced(self.shape.transpose(channel))

    @property
    def Ts(self):
        return self._with_shape_replaced(self.shape.transpose(channel))

    def map(self, function: Callable, dims=shape_, range=range, unwrap_scalars=True, **kwargs):
        from ._functional import map_
        return map_(function, self, dims=dims, range=range, unwrap_scalars=unwrap_scalars, **kwargs)

    def __getattr__(self, name):
        if name.startswith('__'):  # called by hasattr in magic ops
            raise AttributeError
        if name.startswith('_'):
            raise AttributeError(f"'{type(self)}' object has no attribute '{name}'")
        if name == 'is_tensor_like':  # TensorFlow replaces abs() while tracing and checks for this attribute
            raise AttributeError(f"'{type(self)}' object has no attribute '{name}'")
        assert name not in ('shape', '_shape', 'tensor'), name
        return TensorDim(self, name)

    def __add__(self, other):
        return self._op2(other, lambda x, y: x + y, lambda x, y: choose_backend(x, y).add(x, y), 'add', '+')

    def __radd__(self, other):
        return self._op2(other, lambda x, y: y + x, lambda x, y: choose_backend(x, y).add(y, x), 'radd', '+')

    def __sub__(self, other):
            return self._op2(other, lambda x, y: x - y, lambda x, y: choose_backend(x, y).sub(x, y), 'sub', '-')

    def __rsub__(self, other):
        return self._op2(other, lambda x, y: y - x, lambda x, y: choose_backend(x, y).sub(y, x), 'rsub', '-')

    def __and__(self, other):
        return self._op2(other, lambda x, y: x & y, lambda x, y: choose_backend(x, y).and_(x, y), 'and', '&')

    def __rand__(self, other):
        return self._op2(other, lambda x, y: y & x, lambda x, y: choose_backend(x, y).and_(y, x), 'rand', '&')

    def __or__(self, other):
        return self._op2(other, lambda x, y: x | y, lambda x, y: choose_backend(x, y).or_(x, y), 'or', '|')

    def __ror__(self, other):
        return self._op2(other, lambda x, y: y | x, lambda x, y: choose_backend(x, y).or_(y, x), 'ror', '|')

    def __xor__(self, other):
        return self._op2(other, lambda x, y: x ^ y, lambda x, y: choose_backend(x, y).xor(x, y), 'xor', '^')

    def __rxor__(self, other):
        return self._op2(other, lambda x, y: y ^ x, lambda x, y: choose_backend(x, y).xor(y, x), 'rxor', '^')

    def __mul__(self, other):
        return self._op2(other, lambda x, y: x * y, lambda x, y: choose_backend(x, y).mul(x, y), 'mul', '*')

    def __rmul__(self, other):
        return self._op2(other, lambda x, y: y * x, lambda x, y: choose_backend(x, y).mul(y, x), 'rmul', '*')

    def __truediv__(self, other):
        return self._op2(other, lambda x, y: x / y, lambda x, y: choose_backend(x, y).div(x, y), 'truediv', '/')

    def __rtruediv__(self, other):
        return self._op2(other, lambda x, y: y / x, lambda x, y: choose_backend(x, y).div(y, x), 'rtruediv', '/')

    def __divmod__(self, other):
        return self._op2(other, lambda x, y: divmod(x, y), lambda x, y: divmod(x, y), 'divmod', 'divmod')

    def __rdivmod__(self, other):
        return self._op2(other, lambda x, y: divmod(y, x), lambda x, y: divmod(y, x), 'rdivmod', 'divmod')

    def __floordiv__(self, other):
        return self._op2(other, lambda x, y: x // y, lambda x, y: choose_backend(x, y).floordiv(x, y), 'floordiv', '//')

    def __rfloordiv__(self, other):
        return self._op2(other, lambda x, y: y // x, lambda x, y: choose_backend(x, y).floordiv(y, x), 'rfloordiv', '//')

    def __pow__(self, power, modulo=None):
        assert modulo is None
        return self._op2(power, lambda x, y: x ** y, lambda x, y: choose_backend(x, y).pow(x, y), 'pow', '**')

    def __rpow__(self, other):
        return self._op2(other, lambda x, y: y ** x, lambda x, y: choose_backend(x, y).pow(y, x), 'rpow', '**')

    def __mod__(self, other):
        return self._op2(other, lambda x, y: x % y, lambda x, y: choose_backend(x, y).mod(x, y), 'mod', '%')

    def __rmod__(self, other):
        return self._op2(other, lambda x, y: y % x, lambda x, y: choose_backend(x, y).mod(y, x), 'rmod', '%')

    def __eq__(self, other) -> 'Tensor':
        if _EQUALITY_REDUCE[-1] == 'ref':
            return wrap(self is other)
        elif _EQUALITY_REDUCE[-1] == 'shape_and_value':
            if set(self.shape) != set(other.shape):
                return wrap(False)
            from ._ops import close
            return wrap(close(self, other, rel_tolerance=0, abs_tolerance=0))
        if other is None:
            other = float('nan')
        return self._op2(other, lambda x, y: x == y, lambda x, y: choose_backend(x, y).equal(x, y), 'eq', '==')

    def __ne__(self, other) -> 'Tensor':
        if _EQUALITY_REDUCE[-1] == 'ref':
            return wrap(self is not other)
        elif _EQUALITY_REDUCE[-1] == 'shape_and_value':
            if set(self.shape) != set(other.shape):
                return wrap(True)
            from ._ops import close
            return wrap(not close(self, other, rel_tolerance=0, abs_tolerance=0))
        if other is None:
            other = float('nan')
        return self._op2(other, lambda x, y: x != y, lambda x, y: choose_backend(x, y).not_equal(x, y), 'ne', '!=')

    def __lt__(self, other):
        return self._op2(other, lambda x, y: x < y, lambda x, y: choose_backend(x, y).greater_than(y, x), 'lt', '<')

    def __le__(self, other):
        return self._op2(other, lambda x, y: x <= y, lambda x, y: choose_backend(x, y).greater_or_equal(y, x), 'le', '<=')

    def __gt__(self, other):
        return self._op2(other, lambda x, y: x > y, lambda x, y: choose_backend(x, y).greater_than(x, y), 'gt', '>')

    def __ge__(self, other):
        return self._op2(other, lambda x, y: x >= y, lambda x, y: choose_backend(x, y).greater_or_equal(x, y), 'ge', '>=')

    def __lshift__(self, other):
        return self._op2(other, lambda x, y: x << y, lambda x, y: choose_backend(x, y).shift_bits_left(x, y), 'lshift', '<<')

    def __rlshift__(self, other):
        return self._op2(other, lambda y, x: x << y, lambda y, x: choose_backend(x, y).shift_bits_left(x, y), 'lshift', '<<')

    def __rshift__(self, other):
        return self._op2(other, lambda x, y: x >> y, lambda x, y: choose_backend(x, y).shift_bits_right(x, y), 'rshift', '>>')

    def __rrshift__(self, other):
        return self._op2(other, lambda y, x: x >> y, lambda y, x: choose_backend(x, y).shift_bits_right(x, y), 'rshift', '>>')

    def __abs__(self):
        return self._op1(lambda t: choose_backend(t).abs(t))

    def __round__(self, n=None):
        return self._op1(lambda t: choose_backend(t).round(t))

    def __copy__(self):
        return self._op1(lambda t: choose_backend(t).copy(t, only_mutable=True))

    def __deepcopy__(self, memodict={}):
        return self._op1(lambda t: choose_backend(t).copy(t, only_mutable=False))

    def __neg__(self) -> 'Tensor':
        return self._op1(lambda t: -t)

    def __invert__(self) -> 'Tensor':
        return self._op1(lambda t: choose_backend(t).invert(t))

    def __reversed__(self):
        assert self.shape.channel.rank == 1
        return self[::-1]

    def __iter__(self):
        if self.rank == 1:
            return iter(self.native())
        elif self.rank == 0:
            return iter([self.native()])
        else:
            native = reshaped_native(self, [self.shape])
            return iter(native)

    def __matmul__(self, other):
        from ._ops import dot
        assert isinstance(other, Tensor), f"Matmul '@' requires two Tensor arguments but got {type(other)}"
        if not self.shape.dual_rank and self.shape.channel_rank:
            match = self.shape.channel.only(other.shape.channel)
            if match:
                return dot(self, match, other, match)
        match_names = self.shape.dual.as_batch().names
        if not match_names:  # this is not a matrix
            assert self.shape.primal.only(other.shape).is_empty, f"Cannot compute matmul {self.shape} @ {other.shape}. First argument is not a matrix; it has no dual dimensions."
            return self * other
        match_primal = other.shape.only(match_names, reorder=True)
        if not match_primal:
            assert non_batch(other).non_dual.rank == 1, f"Cannot multiply {self.shape} @ {other.shape} because arg2 does not have appropriate non-dual dimensions"
            assert non_batch(other).non_dual.size == match_primal.volume, f"Cannot multiply {self.shape} @ {other.shape} because dual dims of arg1 have no match"
            match_primal = non_batch(other).non_dual
        match_dual = self.shape.dual.only(match_primal.as_dual(), reorder=True)
        left_arg = pack_dims(self, match_dual, dual('_reduce'))
        right_arg = pack_dims(other, match_primal, channel('_reduce'))
        return dot(left_arg, '~_reduce', right_arg, '_reduce')

    # def __rmatmul__(self, other):

    def _tensor(self, other) -> 'Tensor':
        if isinstance(other, Tensor):
            return other
        elif isinstance(other, (tuple, list)) and any(isinstance(v, Tensor) for v in other):
            if 'vector' in self.shape:
                outer_dim = self.shape['vector']
            elif self.shape.channel_rank == 1:
                outer_dim = self.shape.channel
            else:
                raise ValueError(f"Cannot combine tensor of shape {self.shape} with tuple {tuple([type(v).__name__ for v in other])}")
            remaining_shape = self.shape.without(outer_dim)
            other_items = [v if isinstance(v, Tensor) else compatible_tensor(v, compat_shape=remaining_shape, compat_natives=self._natives(), convert=False) for v in other]
            other_stacked = stack(other_items, outer_dim, expand_values=True)
            return other_stacked
        else:
            return compatible_tensor(other, compat_shape=self.shape, compat_natives=self._natives(), convert=False)

    def _op1(self, native_function) -> 'Tensor':
        """
        Transform the values of this tensor given a function that can be applied to any native tensor.

        Args:
          native_function:

        Returns:

        """
        raise NotImplementedError(self.__class__)

    def _op2(self, other, operator: Callable, native_function: Callable, op_name: str = 'unknown', op_symbol: str = '?') -> 'Tensor':
        """
        Apply a broadcast operation on two tensors.

        Args:
            other: second argument
            operator: function (Tensor, Tensor) -> Tensor, used to propagate the operation to children tensors to have Python choose the callee
            native_function: function (native tensor, native tensor) -> native tensor
            op_name: Name of the python function without leading and trailing `__`.
                Examples: 'add', 'radd', 'sub', 'mul', 'and', 'eq', 'ge'.
            op_symbol: Operation symbol, such as '+', '-', '&', '%', '>='

        Returns:
            `Tensor`
        """
        raise NotImplementedError(self.__class__)

    def _natives(self) -> tuple:
        raise NotImplementedError(self.__class__)

    def _spec_dict(self) -> dict:
        raise NotImplementedError(self.__class__)

    @classmethod
    def _from_spec_and_natives(cls, spec: dict, natives: list):
        raise NotImplementedError(cls)

    def _simplify(self):
        """ Does not cache this value but if it is already cached, returns the cached version. """
        return self

Subclasses

phiml.math._sparse.CompactSparseTensor
phiml.math._sparse.CompressedSparseMatrix
phiml.math._sparse.SparseCoordinateTensor
phiml.math._tensors.Layout
phiml.math._tensors.NativeTensor
phiml.math._tensors.TensorStack
phiml.math._trace.GatherLinTracer
phiml.math._trace.ShiftLinTracer
phiml.math._trace.SparseLinTracer

Instance variables

prop T

Expand source code

@property
def T(self):
    return self._with_shape_replaced(self.shape.transposed)

prop Tc

Expand source code

@property
def Tc(self):
    return self._with_shape_replaced(self.shape.transpose(channel))

prop Ti

Expand source code

@property
def Ti(self):
    return self._with_shape_replaced(self.shape.transpose(instance))

prop Ts

Expand source code

@property
def Ts(self):
    return self._with_shape_replaced(self.shape.transpose(channel))

prop all

Whether all values of this Tensor are True as a native bool.

Expand source code

@property
def all(self):
    """ Whether all values of this `Tensor` are `True` as a native bool. """
    from ._ops import all_, cast
    if self.rank == 0:
        return cast(self, DType(bool)).native()
    else:
        return all_(self, dim=self.shape).native()

prop any

Whether this Tensor contains a True value as a native bool.

Expand source code

@property
def any(self):
    """ Whether this `Tensor` contains a `True` value as a native bool. """
    from ._ops import any_, cast
    if self.rank == 0:
        return cast(self, DType(bool)).native()
    else:
        return any_(self, dim=self.shape).native()

prop available : bool

A tensor is available if it stores concrete values and these can currently be read.

Tracers used inside jit compilation are typically not available.

Methods

def dimension(self, name: Union[str, phiml.math._shape.Shape]) ‑> phiml.math._tensors.TensorDim

Returns a reference to a specific dimension of this tensor. This is equivalent to the syntax tensor.<name>.

The dimension need not be part of the Tensor.shape in which case its size is 1.

Args

name: dimension name

Returns

TensorDim corresponding to a dimension of this tensor

def map(self, function: Callable, dims=<function shape>, range=builtins.range, unwrap_scalars=True, **kwargs)

def native(self, order: Union[phiml.math._shape.Shape, tuple, list, str] = None, force_expand=True, to_numpy=False)

Returns a native tensor object with the dimensions ordered according to order.

Transposes the underlying tensor to match the name order and adds singleton dimensions for new dimension names. If a dimension of the tensor is not listed in order, a ValueError is raised.

Additionally, groups of dimensions can be specified to pack dims, see reshaped_native().

Args

order: (Optional) Order of dimension names as comma-separated string, list or Shape.
force_expand: If False, dimensions along which values are guaranteed to be constant will not be expanded to their true size but returned as singleton dimensions.
to_numpy: Whether to convert the tensor to a NumPy ndarray.

Returns

Native tensor representation, such as PyTorch tensor or NumPy array.

Raises

ValueError if the tensor cannot be transposed to match target_shape

def numpy(self, order: Union[phiml.math._shape.Shape, tuple, list, str] = None, force_expand=True) ‑> numpy.ndarray

Converts this tensor to a numpy.ndarray with dimensions ordered according to order.

Note: Using this function breaks the autograd chain. The returned tensor is not differentiable. To get a differentiable tensor, use Tensor.native() instead.

Transposes the underlying tensor to match the name order and adds singleton dimensions for new dimension names. If a dimension of the tensor is not listed in order, a ValueError is raised.

If this Tensor is backed by a NumPy array, a reference to this array may be returned.

Args

order: (Optional) Order of dimension names as comma-separated string, list or Shape.

Returns

NumPy representation

Raises

ValueError if the tensor cannot be transposed to match target_shape

def pack(self, dims, packed_dim)

See pack_dims()

def print(self, layout='full', float_format=None, threshold=8, include_shape=None, include_dtype=None)

def unpack(self, dim, unpacked_dims)

See unpack_dim()