Module phi.math.extrapolation

"""
Extrapolations are used for padding tensors and sampling coordinates lying outside the tensor bounds.
Standard extrapolations are listed as global variables in this module.

Extrapolations are an important part of sampled fields such as grids.
See the documentation at https://tum-pbs.github.io/PhiFlow/Fields.html#extrapolations .
"""
from typing import Union, Dict

from phi.math.backend._backend import get_spatial_derivative_order
from .backend import choose_backend
from ._shape import Shape, channel
from ._tensors import Tensor, NativeTensor, CollapsedTensor, TensorStack, wrap
from . import _ops as math  # TODO this executes _ops.py, can we avoid this?


class Extrapolation:
    """
    Extrapolations are used to determine values of grids or other structures outside the sampled bounds.
    They play a vital role in padding and sampling.
    """

    def __init__(self, pad_rank):
        """
        Args:
            pad_rank: low-ranking extrapolations are handled first during mixed-extrapolation padding.
                The typical order is periodic=1, boundary=2, symmetric=3, reflect=4, constant=5.
        """
        self.pad_rank = pad_rank

    def to_dict(self) -> dict:
        """
        Serialize this extrapolation to a dictionary that is serializable (JSON-writable).
        
        Use `from_dict()` to restore the Extrapolation object.
        """
        raise NotImplementedError()

    def spatial_gradient(self) -> 'Extrapolation':
        """Returns the extrapolation for the spatial spatial_gradient of a tensor/field with this extrapolation."""
        raise NotImplementedError()

    def valid_outer_faces(self, dim) -> tuple:
        """ `(lower: bool, upper: bool)` indicating whether the values sampled at the outer-most faces of a staggered grid with this extrapolation are valid, i.e. need to be stored and are not redundant. """
        raise NotImplementedError()

    @property
    def connects_to_outside(self) -> bool:
        """
        True if at any place, this extrapolation allows for dynamic flow out of the valid domain.
        `BOUNDARY` faces allow this, but constant extrapolations and `PERIODIC` do not.
        """
        raise NotImplementedError()

    def pad(self, value: Tensor, widths: dict) -> Tensor:
        """
        Pads a tensor using values from self.pad_values()

        Args:
          value: tensor to be padded
          widths: name: str -> (lower: int, upper: int)}
          value: Tensor: 
          widths: dict: 

        Returns:

        """
        for dim in widths:
            assert (w > 0 for w in widths[dim]), "Negative widths not allowed in Extrapolation.pad(). Use math.pad() instead."
            values = []
            if widths[dim][False] > 0:
                values.append(self.pad_values(value, widths[dim][False], dim, False))
            values.append(value)
            if widths[dim][True] > 0:
                values.append(self.pad_values(value, widths[dim][True], dim, True))
            value = math.concat(values, value.shape[dim])
        return value

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        """
        Determines the values with which the given tensor would be padded at the specified using this extrapolation.

        Args:
          value: tensor to be padded
          width: number of cells to pad perpendicular to the face. Must be larger than zero.
          dimension: axis in which to pad
          upper_edge: True for upper edge, False for lower edge
          value: Tensor: 
          width: int: 
          dimension: str: 
          upper_edge: bool: 

        Returns:
          tensor that can be concatenated to value for padding

        """
        raise NotImplementedError()

    def transform_coordinates(self, coordinates: Tensor, shape: Shape) -> Tensor:
        """
        If is_copy_pad, transforms outsider coordinates to point to the index from which the value should be copied.
        
        Otherwise, the grid tensor is assumed to hold the correct boundary values for this extrapolation at the edge.
        Coordinates are then snapped to the valid index range.
        This is the default implementation.

        Args:
          coordinates: integer coordinates in index space
          shape: tensor shape
          coordinates: Tensor: 
          shape: Shape: 

        Returns:
          transformed coordinates

        """
        return math.clip(coordinates, 0, math.wrap(shape.spatial - 1, channel('vector')))

    @property
    def is_copy_pad(self):
        """:return: True if all pad values are copies of existing values in the tensor to be padded"""
        return False

    @property
    def native_grid_sample_mode(self) -> Union[str, None]:
        return None

    def shortest_distance(self, start: Tensor, end: Tensor, domain_size: Tensor):
        """
        Computes the shortest distance between two points.
        Both points are assumed to lie within the domain

        Args:
            start: Start position.
            end: End position.
            domain_size: Domain side lengths as vector.

        Returns:
            Shortest distance from `start` to `end`.
        """
        return end - start

    def __getitem__(self, item):
        return self


class ConstantExtrapolation(Extrapolation):
    """
    Extrapolate with a constant value.
    """

    def __init__(self, value: Tensor or float):
        Extrapolation.__init__(self, 5)
        self.value = wrap(value)
        """ Extrapolation value """

    def __repr__(self):
        return repr(self.value)

    def to_dict(self) -> dict:
        return {'type': 'constant', 'value': self.value.numpy()}

    def __value_attrs__(self):
        return 'value',

    def spatial_gradient(self):
        return ZERO

    def valid_outer_faces(self, dim) -> tuple:
        return False, False

    @property
    def connects_to_outside(self) -> bool:
        return False

    def pad(self, value: Tensor, widths: dict):
        """
        Pads a tensor using CONSTANT values

        Args:
          value: tensor to be padded
          widths: name: str -> (lower: int, upper: int)}
          value: Tensor: 
          widths: dict: 

        Returns:

        """
        derivative = get_spatial_derivative_order()
        pad_value = self.value if derivative == 0 else math.zeros()
        value = value._simplify()
        from phi.math._functional import is_tracer
        if isinstance(value, NativeTensor):
            native = value._native
            ordered_pad_widths = order_by_shape(value.shape, widths, default=(0, 0))
            backend = choose_backend(native)
            result_tensor = backend.pad(native, ordered_pad_widths, 'constant', pad_value.native())
            new_shape = value.shape.with_sizes(backend.staticshape(result_tensor))
            return NativeTensor(result_tensor, new_shape)
        elif isinstance(value, CollapsedTensor):
            if value._inner.shape.volume > 1 or not math.all_available(pad_value, value) or not math.close(pad_value, value._inner):  # .inner should be safe after _simplify
                return self.pad(value._cache(), widths)
            else:  # Stays constant value, only extend shape
                new_sizes = []
                for size, dim, *_ in value.shape._dimensions:
                    if dim not in widths:
                        new_sizes.append(size)
                    else:
                        delta = sum(widths[dim]) if isinstance(widths[dim], (tuple, list)) else 2 * widths[dim]
                        new_sizes.append(size + int(delta))
                new_shape = value.shape.with_sizes(new_sizes)
                return CollapsedTensor(value._inner, new_shape)
        # elif isinstance(value, SparseLinearOperation):
        #     return pad_operator(value, pad_width, mode)
        elif isinstance(value, TensorStack):
            if not value.requires_broadcast:
                return self.pad(value._cache(), widths)
            inner_widths = {dim: w for dim, w in widths.items() if dim != value.stack_dim_name}
            tensors = [self.pad(t, inner_widths) for t in value.tensors]
            return TensorStack(tensors, value.stack_dim)
        elif is_tracer(value):
            lower = {dim: -lo for dim, (lo, _) in widths.items()}
            return value.shift(lower, value.shape.after_pad(widths), lambda v: ZERO.pad(v, widths), lambda b: self.pad(b, widths))
        else:
            raise NotImplementedError()

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        shape = value.shape.after_gather({dimension: slice(0, width)})
        return math.expand(self.value, shape)

    def __eq__(self, other):
        return isinstance(other, ConstantExtrapolation) and math.close(self.value, other.value)

    def __hash__(self):
        return hash(self.__class__)

    def is_zero(self):
        return self == ZERO

    def is_one(self):
        return self == ONE

    @property
    def native_grid_sample_mode(self) -> Union[str, None]:
        return 'zeros' if self.is_zero() else None

    def __add__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(self.value + other.value)
        elif self.is_zero():
            return other
        else:
            return NotImplemented

    def __sub__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(self.value - other.value)
        else:
            return NotImplemented

    def __rsub__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(other.value - self.value)
        elif self.is_zero():
            return other
        else:
            return NotImplemented

    def __mul__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(self.value * other.value)
        elif self.is_one():
            return other
        elif self.is_zero():
            return self
        else:
            return NotImplemented

    def __truediv__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(self.value / other.value)
        elif self.is_zero():
            return self
        else:
            return NotImplemented

    def __rtruediv__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(other.value / self.value)
        elif self.is_one():
            return other
        else:
            return NotImplemented

    def __lt__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(self.value < other.value)
        else:
            return NotImplemented

    def __gt__(self, other):
        if isinstance(other, ConstantExtrapolation):
            return ConstantExtrapolation(self.value > other.value)
        else:
            return NotImplemented

    def __abs__(self):
        return ConstantExtrapolation(abs(self.value))

    def _op1(self, operator):
        return ConstantExtrapolation(self.value._op1(operator))


class _CopyExtrapolation(Extrapolation):

    @property
    def is_copy_pad(self):
        return True

    def to_dict(self) -> dict:
        return {'type': repr(self)}

    def __value_attrs__(self):
        return ()

    def valid_outer_faces(self, dim):
        return True, True

    def pad(self, value: Tensor, widths: dict) -> Tensor:
        value = value._simplify()
        from phi.math._functional import is_tracer
        if isinstance(value, NativeTensor):
            native = value._native
            ordered_pad_widths = order_by_shape(value.shape, widths, default=(0, 0))
            result_tensor = choose_backend(native).pad(native, ordered_pad_widths, repr(self))
            if result_tensor is NotImplemented:
                return Extrapolation.pad(self, value, widths)
            new_shape = value.shape.with_sizes(result_tensor.shape)
            return NativeTensor(result_tensor, new_shape)
        elif isinstance(value, CollapsedTensor):
            inner = value._inner  # should be fine after _simplify
            inner_widths = {dim: w for dim, w in widths.items() if dim in inner.shape}
            if len(inner_widths) > 0:
                inner = self.pad(inner, widths)
            new_sizes = []
            for size, dim, *_ in value.shape._dimensions:
                if dim not in widths:
                    new_sizes.append(size)
                else:
                    delta = sum(widths[dim]) if isinstance(widths[dim], (tuple, list)) else 2 * widths[dim]
                    new_sizes.append(size + int(delta))
            new_shape = value.shape.with_sizes(new_sizes)
            return CollapsedTensor(inner, new_shape)
        # elif isinstance(value, SparseLinearOperation):
        #     return pad_operator(value, widths, mode)
        elif isinstance(value, TensorStack):
            if not value.requires_broadcast:
                return self.pad(value._cache(), widths)
            inner_widths = {dim: w for dim, w in widths.items() if dim != value.stack_dim_name}
            tensors = [self.pad(t, inner_widths) for t in value.tensors]
            return TensorStack(tensors, value.stack_dim)
        elif is_tracer(value):
            return self._pad_linear_tracer(value, widths)
        else:
            raise NotImplementedError(f'{type(value)} not supported')

    def _pad_linear_tracer(self, value, widths: dict):
        raise NotImplementedError()

    @property
    def native_grid_sample_mode(self) -> Union[str, None]:
        return str(self)

    def __eq__(self, other):
        return type(other) == type(self)

    def __hash__(self):
        return hash(self.__class__)

    def _op(self, other, op):
        if type(other) == type(self):
            return self
        elif isinstance(other, Extrapolation) and not isinstance(other, _CopyExtrapolation):
            op = getattr(other, op.__name__)
            return op(self)
        else:
            return NotImplemented

    def __add__(self, other):
        return self._op(other, ConstantExtrapolation.__add__)

    def __mul__(self, other):
        return self._op(other, ConstantExtrapolation.__mul__)

    def __sub__(self, other):
        return self._op(other, ConstantExtrapolation.__rsub__)

    def __truediv__(self, other):
        return self._op(other, ConstantExtrapolation.__rtruediv__)

    def __lt__(self, other):
        return self._op(other, ConstantExtrapolation.__gt__)

    def __gt__(self, other):
        return self._op(other, ConstantExtrapolation.__lt__)

    def __neg__(self):
        return self  # assume also applied to values

    def __abs__(self):
        return self  # assume also applied to values

    def _op1(self, operator):
        return self  # assume also applied to values


class _BoundaryExtrapolation(_CopyExtrapolation):
    """Uses the closest defined value for points lying outside the defined region."""

    _CACHED_LOWER_MASKS = {}
    _CACHED_UPPER_MASKS = {}

    def __repr__(self):
        return 'boundary'

    def spatial_gradient(self):
        return ZERO

    @property
    def connects_to_outside(self) -> bool:
        return True

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        if upper_edge:
            edge = value[{dimension: slice(-1, None)}]
        else:
            edge = value[{dimension: slice(1)}]
        return math.concat([edge] * width, value.shape[dimension])

    def _pad_linear_tracer(self, value: '_trace.ShiftLinTracer', widths: dict) -> '_trace.ShiftLinTracer':
        """
        *Warning*:
        This implementation discards corners, i.e. values that lie outside the original tensor in more than one dimension.
        These are typically sliced off in differential operators. Corners are instead assigned the value 0.
        To take corners into account, call pad() for each axis individually. This is inefficient with ShiftLinTracer.

        Args:
          value: ShiftLinTracer:
          widths: dict: 

        Returns:

        """
        lower = {dim: -lo for dim, (lo, _) in widths.items()}
        result = value.shift(lower, value.shape.after_pad(widths), lambda v: ZERO.pad(v, widths), lambda b: ZERO.pad(b, widths))  # inner values  ~half the computation time
        for bound_dim, (bound_lo, bound_hi) in widths.items():
            for i in range(bound_lo):  # i=0 means outer
                # this sets corners to 0
                lower = {dim: -i if dim == bound_dim else -lo for dim, (lo, _) in widths.items()}
                mask = self._lower_mask(value.shape.only(result.dependent_dims), widths, bound_dim, bound_lo, bound_hi, i)
                boundary = value.shift(lower, result.shape, lambda v: self.pad(v, widths) * mask, lambda b: ZERO.pad(b, widths))
                result += boundary
            for i in range(bound_hi):
                lower = {dim: i - lo - hi if dim == bound_dim else -lo for dim, (lo, hi) in widths.items()}
                mask = self._upper_mask(value.shape.only(result.dependent_dims), widths, bound_dim, bound_lo, bound_hi, i)
                boundary = value.shift(lower, result.shape, lambda v: self.pad(v, widths) * mask, lambda b: ZERO.pad(b, widths))  # ~ half the computation time
                result += boundary  # this does basically nothing if value is the identity
        return result

    def _lower_mask(self, shape, widths, bound_dim, bound_lo, bound_hi, i):
        # key = (shape, tuple(widths.keys()), tuple(widths.values()), bound_dim, bound_lo, bound_hi, i)
        # if key in _BoundaryExtrapolation._CACHED_LOWER_MASKS:
        #     result = math.tensor(_BoundaryExtrapolation._CACHED_LOWER_MASKS[key])
        #     _BoundaryExtrapolation._CACHED_LOWER_MASKS[key] = result
        #     return result
        # else:
            mask = ZERO.pad(math.zeros(shape), {bound_dim: (bound_lo - i - 1, 0)})
            mask = ONE.pad(mask, {bound_dim: (1, 0)})
            mask = ZERO.pad(mask, {dim: (i, bound_hi) if dim == bound_dim else (lo, hi) for dim, (lo, hi) in widths.items()})
            # _BoundaryExtrapolation._CACHED_LOWER_MASKS[key] = mask
            return mask

    def _upper_mask(self, shape, widths, bound_dim, bound_lo, bound_hi, i):
        # key = (shape, tuple(widths.keys()), tuple(widths.values()), bound_dim, bound_lo, bound_hi, i)
        # if key in _BoundaryExtrapolation._CACHED_UPPER_MASKS:
        #     result = math.tensor(_BoundaryExtrapolation._CACHED_UPPER_MASKS[key])
        #     _BoundaryExtrapolation._CACHED_UPPER_MASKS[key] = result
        #     return result
        # else:
            mask = ZERO.pad(math.zeros(shape), {bound_dim: (0, bound_hi - i - 1)})
            mask = ONE.pad(mask, {bound_dim: (0, 1)})
            mask = ZERO.pad(mask, {dim: (bound_lo, i) if dim == bound_dim else (lo, hi) for dim, (lo, hi) in widths.items()})
            # _BoundaryExtrapolation._CACHED_UPPER_MASKS[key] = mask
            return mask


class _PeriodicExtrapolation(_CopyExtrapolation):
    def __repr__(self):
        return 'periodic'

    def spatial_gradient(self):
        return self

    def valid_outer_faces(self, dim):
        return True, False

    @property
    def connects_to_outside(self) -> bool:
        return False

    def transform_coordinates(self, coordinates: Tensor, shape: Shape) -> Tensor:
        return coordinates % shape.spatial

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        if upper_edge:
            return value[{dimension: slice(width)}]
        else:
            return value[{dimension: slice(-width, None)}]

    def _pad_linear_tracer(self, value: '_trace.ShiftLinTracer', widths: dict) -> '_trace.ShiftLinTracer':
        if value.shape.get_sizes(tuple(widths.keys())) != value.source.shape.get_sizes(tuple(widths.keys())):
            raise NotImplementedError("Periodicity does not match input: %s but input has %s. This can happen when padding an already padded or sliced tensor." % (value.shape.only(tuple(widths.keys())), value.source.shape.only(tuple(widths.keys()))))
        lower = {dim: -lo for dim, (lo, _) in widths.items()}
        return value.shift(lower, value.shape.after_pad(widths), lambda v: self.pad(v, widths), lambda b: ZERO.pad(b, widths))

    def shortest_distance(self, start: Tensor, end: Tensor, domain_size: Tensor):
        dx = end - start
        return (dx + domain_size / 2) % domain_size - domain_size / 2


class _SymmetricExtrapolation(_CopyExtrapolation):
    """Mirror with the boundary value occurring twice."""

    def __repr__(self):
        return 'symmetric'

    def spatial_gradient(self):
        return -self

    @property
    def connects_to_outside(self) -> bool:
        return True

    def transform_coordinates(self, coordinates: Tensor, shape: Shape) -> Tensor:
        coordinates = coordinates % (2 * shape)
        return ((2 * shape - 1) - abs((2 * shape - 1) - 2 * coordinates)) // 2

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        raise NotImplementedError()
        # raise NotImplementedError()  # only used by PyTorch which does not support ::-1 axis flips
        # dims = range(math.ndims(value))
        # for dim in dims:
        #     pad_lower, pad_upper = pad_width[dim]
        #     if pad_lower == 0 and pad_upper == 0:
        #         continue  # Nothing to pad
        #     top_rows = value[
        #         tuple([slice(value.shape[dim] - pad_upper, None) if d == dim else slice(None) for d in dims])]
        #     bottom_rows = value[tuple([slice(None, pad_lower) if d == dim else slice(None) for d in dims])]
        #     top_rows = math.flip_axis(top_rows, dim)
        #     bottom_rows = math.flip_axis(bottom_rows, dim)
        #     value = math.concat([bottom_rows, value, top_rows], axis=dim)
        # return value


class _ReflectExtrapolation(_CopyExtrapolation):
    """Mirror of inner elements. The boundary value is not duplicated."""

    def __repr__(self):
        return 'reflect'

    def spatial_gradient(self):
        return -self

    @property
    def connects_to_outside(self) -> bool:
        return True

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        if upper_edge:
            return value[{dimension: slice(-1-width, -1)}].flip(dimension)
        else:
            return value[{dimension: slice(1, width+1)}].flip(dimension)

    def transform_coordinates(self, coordinates: Tensor, shape: Shape) -> Tensor:
        coordinates = coordinates % (2 * shape - 2)
        return (shape - 1) - math.abs_((shape - 1) - coordinates)


class _NoExtrapolation(Extrapolation):
    def to_dict(self) -> dict:
        return {}

    def pad(self, value: Tensor, widths: dict) -> Tensor:
        return value

    def spatial_gradient(self) -> 'Extrapolation':
        return self

    def valid_outer_faces(self, dim):
        return True, True

    @property
    def connects_to_outside(self) -> bool:
        raise AssertionError(f"connects_to_outside not defined by {self.__class__}")

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        raise AssertionError("Invalid extrapolation")

    def __repr__(self):
        return "none"

    def __add__(self, other):
        return self

    def __radd__(self, other):
        return self

    def __sub__(self, other):
        return self

    def __rsub__(self, other):
        return self

    def __mul__(self, other):
        return self

    def __rmul__(self, other):
        return self

    def __truediv__(self, other):
        return self

    def __rtruediv__(self, other):
        return self


ZERO = ConstantExtrapolation(0)
""" Extrapolates with the constant value 0 (Dirichlet boundary condition). """
ONE = ConstantExtrapolation(1)
""" Extrapolates with the constant value 1 (Dirichlet boundary condition). """
PERIODIC = _PeriodicExtrapolation(1)
""" Extends a grid by tiling it (Periodic boundary condition). """
BOUNDARY = _BoundaryExtrapolation(2)
""" Extends a grid with its edge values (Neumann boundary condition). The value of a point lying outside the grid is determined by the closest grid value(s). """
SYMMETRIC = _SymmetricExtrapolation(3)
""" Extends a grid by tiling it. Every other copy of the grid is flipped. Edge values occur twice per seam. """
REFLECT = _ReflectExtrapolation(4)
""" Like SYMMETRIC but the edge values are not copied and only occur once per seam. """
NONE = _NoExtrapolation(-1)
""" Raises AssertionError when used to determine outside values. Padding operations will have no effect with this extrapolation. """


def combine_sides(**extrapolations: Extrapolation or tuple) -> Extrapolation:
    """
    Specify extrapolations for each side / face of a box.

    Args:
        **extrapolations: map from dim: str -> `Extrapolation` or `tuple` (lower, upper)

    Returns:
        `Extrapolation`
    """
    values = set()
    for ext in extrapolations.values():
        if isinstance(ext, Extrapolation):
            values.add(ext)
        else:
            values.add(ext[0])
            values.add(ext[1])
    if len(values) == 1:
        return next(iter(values))
    else:
        return _MixedExtrapolation(extrapolations)


class _MixedExtrapolation(Extrapolation):

    def __init__(self, extrapolations: dict):
        """
        A mixed extrapolation uses different extrapolations for different sides.

        Args:
          extrapolations: axis: str -> (lower: Extrapolation, upper: Extrapolation) or Extrapolation
        """
        Extrapolation.__init__(self, None, )
        self.ext = {dim: (e, e) if isinstance(e, Extrapolation) else tuple(e) for dim, e in extrapolations.items()}

    def to_dict(self) -> dict:
        return {
            'type': 'mixed',
            'dims': {ax: (es[0].to_dict(), es[1].to_dict()) for ax, es in self.ext.items()}
        }

    def __eq__(self, other):
        if isinstance(other, _MixedExtrapolation):
            return self.ext == other.ext
        else:
            simplified = combine_sides(**self.ext)
            if not isinstance(simplified, _MixedExtrapolation):
                return simplified == other
            else:
                return False

    def __hash__(self):
        simplified = combine_sides(**self.ext)
        if not isinstance(simplified, _MixedExtrapolation):
            return hash(simplified)
        else:
            return hash(frozenset(self.ext.items()))

    def __repr__(self):
        return repr(self.ext)

    def spatial_gradient(self) -> Extrapolation:
        return combine_sides(**{ax: (es[0].spatial_gradient(), es[1].spatial_gradient()) for ax, es in self.ext.items()})

    def valid_outer_faces(self, dim):
        e_lower, e_upper = self.ext[dim]
        return e_lower.valid_outer_faces(dim)[0], e_upper.valid_outer_faces(dim)[1]

    @property
    def connects_to_outside(self) -> bool:
        result_by_dim = [lo.connects_to_outside or up.connects_to_outside for lo, up in self.ext.values()]
        return any(result_by_dim)

    def pad(self, value: Tensor, widths: dict) -> Tensor:
        """
        Pads a tensor using mixed values

        Args:
          value: tensor to be padded
          widths: name: str -> (lower: int, upper: int)}
          value: Tensor: 
          widths: dict: 

        Returns:

        """
        extrapolations = set(sum(self.ext.values(), ()))
        extrapolations = tuple(sorted(extrapolations, key=lambda e: e.pad_rank))
        for ext in extrapolations:
            ext_widths = {ax: (l if self.ext[ax][0] == ext else 0, u if self.ext[ax][1] == ext else 0)
                          for ax, (l, u) in widths.items()}
            value = ext.pad(value, ext_widths)
        return value

    def pad_values(self, value: Tensor, width: int, dimension: str, upper_edge: bool) -> Tensor:
        extrap: Extrapolation = self.ext[dimension][upper_edge]
        return extrap.pad_values(value, width, dimension, upper_edge)

    def transform_coordinates(self, coordinates: Tensor, shape: Shape) -> Tensor:
        coordinates = coordinates.vector.unstack()
        assert len(self.ext) == len(shape.spatial) == len(coordinates)
        result = []
        for dim, dim_coords in zip(shape.spatial.unstack(), coordinates):
            dim_extrapolations = self.ext[dim.name]
            if dim_extrapolations[0] == dim_extrapolations[1]:
                result.append(dim_extrapolations[0].transform_coordinates(dim_coords, dim))
            else:  # separate boundary for lower and upper face
                lower = dim_extrapolations[0].transform_coordinates(dim_coords, dim)
                upper = dim_extrapolations[1].transform_coordinates(dim_coords, dim)
                result.append(math.where(dim_coords <= 0, lower, upper))
        if 'vector' in result[0].shape:
            return math.concat(result, channel('vector'))
        else:
            return math.stack(result, channel('vector'))

    def __getitem__(self, item):
        if isinstance(item, dict):
            return combine_sides(**{dim: (e1[item], e2[item]) for dim, (e1, e2) in self.ext.items()})
        else:
            dim, face = item
            return self.ext[dim][face]

    def __add__(self, other):
        return self._op2(other, lambda e1, e2: e1 + e2)

    def __radd__(self, other):
        return self._op2(other, lambda e1, e2: e2 + e1)

    def __sub__(self, other):
        return self._op2(other, lambda e1, e2: e1 - e2)

    def __rsub__(self, other):
        return self._op2(other, lambda e1, e2: e2 - e1)

    def __mul__(self, other):
        return self._op2(other, lambda e1, e2: e1 * e2)

    def __rmul__(self, other):
        return self._op2(other, lambda e1, e2: e2 * e1)

    def _op2(self, other, operator):
        if isinstance(other, _MixedExtrapolation):
            assert self.ext.keys() == other.ext.keys()
            return combine_sides(**{ax: (operator(lo, other.ext[ax][False]), operator(hi, other.ext[ax][True])) for ax, (lo, hi) in self.ext.items()})
        else:
            return combine_sides(**{ax: (operator(lo, other), operator(hi, other)) for ax, (lo, hi) in self.ext.items()})


def from_dict(dictionary: dict) -> Extrapolation:
    """
    Loads an `Extrapolation` object from a dictionary that was created using `Extrapolation.to_dict()`.

    Args:
        dictionary: serializable dictionary holding all extrapolation properties

    Returns:
        Loaded extrapolation
    """
    etype = dictionary['type']
    if etype == 'constant':
        return ConstantExtrapolation(dictionary['value'])
    elif etype == 'periodic':
        return PERIODIC
    elif etype == 'boundary':
        return BOUNDARY
    elif etype == 'symmetric':
        return SYMMETRIC
    elif etype == 'reflect':
        return REFLECT
    elif etype == 'mixed':
        dims: Dict[str, tuple] = dictionary['dims']
        extrapolations = {dim: (from_dict(lo_up[0]), from_dict(lo_up[1])) for dim, lo_up in dims.items()}
        return _MixedExtrapolation(extrapolations)
    else:
        raise ValueError(dictionary)


def order_by_shape(shape: Shape, sequence, default=None) -> tuple or list:
    """
    If sequence is a dict with dimension names as keys, orders its values according to this shape.

    Otherwise, the sequence is returned unchanged.

    Args:
      sequence: Sequence or dict to be ordered
      default: default value used for dimensions not contained in sequence

    Returns:
      ordered sequence of values
    """
    if isinstance(sequence, dict):
        result = [sequence.get(name, default) for name in shape.names]
        return result
    elif isinstance(sequence, (tuple, list)):
        assert len(sequence) == shape.rank
        return sequence
    else:  # just a constant
        return sequence