This notebook lists useful code snippets.
from phi.flow import *
from phi.tf.flow import *
from phi.jax.stax.flow import *
from phi.torch.flow import *
2026-02-17 14:46:43.078901: I tensorflow/core/platform/cpu_feature_guard.cc:210] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations. To enable the following instructions: AVX2 FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags. 2026-02-17 14:46:45.230644: E external/local_xla/xla/stream_executor/cuda/cuda_platform.cc:51] failed call to cuInit: INTERNAL: CUDA error: Failed call to cuInit: UNKNOWN ERROR (303)
backend.default_backend().list_devices('GPU')
[]
backend.default_backend().list_devices('CPU')
[torch device 'CPU' (CPU 'cpu') | 15990 MB | 4 processors | ]
assert backend.default_backend().set_default_device('CPU')
math.set_global_precision(32) # single precision is the default
x32 = math.random_normal(batch(b=4))
with math.precision(64): ## operations within this context will use 32 bit floats
x64 = math.to_float(x32)
data = math.random_normal(batch(examples=10)) * .1 # batch of scalar values
data = math.random_uniform(batch(examples=10), channel(vector='x,y')) # batch of vectors
data
(examplesᵇ=10, vectorᶜ=x,y) 0.457 ± 0.321 (7e-03...1e+00)
data.examples[0]
(x=0.160, y=0.043)
Tensor¶print(data)
print(f"{data:full:shape:dtype:color:.1f}")
(examplesᵇ=10, vectorᶜ=x,y) 0.457 ± 0.321 (7e-03...1e+00) (examplesᵇ=10, vectorᶜ=x,y) [[0.2, 0.0], [0.2, 0.7], [0.1, 0.5], [1.0, 0.8], [0.6, 0.5], [0.9, 0.3], [0.9, 0.8], [0.3, 0.1], [0.3, 0.9], [0.2, 0.0]]
Tensor¶data = math.random_uniform(spatial(x=8, y=6))
vis.plot(data) # or vis.show(data)
/opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/vis/_matplotlib/_matplotlib_plots.py:167: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect. plt.tight_layout() # because subplot titles can be added after figure creation
Tensor to NumPy¶data.numpy(order='x,y')
array([[0.67459846, 0.93817604, 0.5837625 , 0.6815734 , 0.8626636 ,
0.28722978],
[0.69911903, 0.84687465, 0.5416151 , 0.5481865 , 0.6351937 ,
0.15557784],
[0.5435354 , 0.23145771, 0.32309753, 0.31481332, 0.806806 ,
0.2215026 ],
[0.85342824, 0.4684711 , 0.8090929 , 0.9399945 , 0.04444867,
0.17474788],
[0.9064153 , 0.36908823, 0.3828773 , 0.6828982 , 0.46103126,
0.94301856],
[0.6725213 , 0.37467867, 0.5461119 , 0.8283743 , 0.03721046,
0.8838278 ],
[0.23955959, 0.71518797, 0.3939277 , 0.9028356 , 0.36973882,
0.8198996 ],
[0.7506428 , 0.8888857 , 0.8643836 , 0.81207514, 0.8926113 ,
0.3193208 ]], dtype=float32)
math.reshaped_native(data, ['extra', data.shape], to_numpy=True)
/tmp/ipykernel_2737/2990683702.py:1: DeprecationWarning: phiml.math.reshaped_native() is deprecated. Use Tensor.native() instead. math.reshaped_native(data, ['extra', data.shape], to_numpy=True)
array([[0.67459846, 0.93817604, 0.5837625 , 0.6815734 , 0.8626636 ,
0.28722978, 0.69911903, 0.84687465, 0.5416151 , 0.5481865 ,
0.6351937 , 0.15557784, 0.5435354 , 0.23145771, 0.32309753,
0.31481332, 0.806806 , 0.2215026 , 0.85342824, 0.4684711 ,
0.8090929 , 0.9399945 , 0.04444867, 0.17474788, 0.9064153 ,
0.36908823, 0.3828773 , 0.6828982 , 0.46103126, 0.94301856,
0.6725213 , 0.37467867, 0.5461119 , 0.8283743 , 0.03721046,
0.8838278 , 0.23955959, 0.71518797, 0.3939277 , 0.9028356 ,
0.36973882, 0.8198996 , 0.7506428 , 0.8888857 , 0.8643836 ,
0.81207514, 0.8926113 , 0.3193208 ]], dtype=float32)
points = math.tensor([(0, 0), (0, 1), (1, 0)], instance('points'), channel('vector'))
distances = points - math.rename_dims(points, 'points', 'others')
math.print(math.vec_length(distances))
[[0. , 1. , 1. ], [1. , 0. , 1.4142135], [1. , 1.4142135, 0. ]]
/tmp/ipykernel_2737/2195475714.py:3: DeprecationWarning: phiml.math.length is deprecated in favor of phiml.math.norm math.print(math.vec_length(distances))
CenteredGrid¶zero_grid = CenteredGrid(0, 0, x=32, y=32, bounds=Box(x=1, y=1))
y_grid = CenteredGrid((0, 1), extrapolation.BOUNDARY, x=32, y=32)
noise_grid = CenteredGrid(Noise(), extrapolation.PERIODIC, x=32, y=32)
sin_curve = CenteredGrid(lambda x: math.sin(x), extrapolation.PERIODIC, x=100, bounds=Box(x=2 * PI))
vis.plot(zero_grid, y_grid, noise_grid, sin_curve, size=(12, 3))
/opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_tensors.py:1174: RuntimeWarning: invalid value encountered in scalar power result = op(n1, n2)
StaggeredGrid¶zero_grid = StaggeredGrid(0, 0, x=32, y=32, bounds=Box(x=1, y=1))
y_grid = StaggeredGrid((0, 1), extrapolation.BOUNDARY, x=32, y=32)
noise_grid = StaggeredGrid(Noise(), extrapolation.PERIODIC, x=32, y=32)
sin_curve = StaggeredGrid(lambda x: math.sin(x), extrapolation.PERIODIC, x=100, bounds=Box(x=2 * PI))
vis.plot(zero_grid, y_grid, noise_grid, sin_curve, size=(12, 3))
--------------------------------------------------------------------------- ValueError Traceback (most recent call last) Cell In[14], line 1 ----> 1 zero_grid = StaggeredGrid(0, 0, x=32, y=32, bounds=Box(x=1, y=1)) 2 y_grid = StaggeredGrid((0, 1), extrapolation.BOUNDARY, x=32, y=32) 3 noise_grid = StaggeredGrid(Noise(), extrapolation.PERIODIC, x=32, y=32) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_grid.py:176, in StaggeredGrid(values, boundary, bounds, resolution, extrapolation, convert, **resolution_) 174 assert values.shape.spatial_rank == elements.bounds.spatial_rank, f"Spatial dimensions of values ({values.shape}) do not match elements {elements}" 175 assert values.shape.instance_rank == 0, f"Instance dimensions not supported for grids. Got values with shape {values.shape}" --> 176 return Field(elements, values, extrapolation) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_field.py:46, in _FieldType.__call__(cls, geometry, values, boundary, variable_attrs, value_attrs, **sampling_kwargs) 44 values = expand(wrap(values), non_batch(geometry) - 'vector') 45 result = cls.__new__(cls, geometry, values, boundary, variable_attrs, value_attrs) ---> 46 result.__init__(geometry, values, boundary, variable_attrs, value_attrs) # also calls __post_init__() 47 return result File <string>:8, in __init__(self, geometry, values, boundary, variable_attrs, value_attrs) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_field.py:85, in Field.__post_init__(self) 84 def __post_init__(self): ---> 85 at = self.sampled_at 86 if at in {'center', 'face'}: 87 math.merge_shapes(self.values, non_batch(self.sampled_elements).non_channel) # shape check File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/functools.py:998, in cached_property.__get__(self, instance, owner) 996 val = cache.get(self.attrname, _NOT_FOUND) 997 if val is _NOT_FOUND: --> 998 val = self.func(instance) 999 try: 1000 cache[self.attrname] = val File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_field.py:399, in Field.sampled_at(self) 397 if all(_size_equal(v_size, set_size) for v_size, set_size in zip(v_sizes, set_shape.sizes)): 398 return name --> 399 raise ValueError(f"Could not determine where the values of this Field are sampled. Geometry sets: {self.geometry.sets}, Field values shape: {v_shape}") ValueError: Could not determine where the values of this Field are sampled. Geometry sets: {'center': (xˢ=32, yˢ=32), 'face': (~vectorᵈ=x,y, xˢ=~(x=33, y=32) int64, yˢ=~(x=32, y=33) int64), 'node': (xˢ=33, yˢ=33)}, Field values shape: (~vectorᵈ=x,y, xˢ=~(x=31, y=32) int64, yˢ=~(x=32, y=31) int64)
StaggeredGrid from NumPy Arrays¶Given matching arrays vx and vy, we can construct a StaggeredGrid.
Note that the shapes of the arrays must match the extrapolation!
vx = math.tensor(np.zeros([33, 32]), spatial('x,y'))
vy = math.tensor(np.zeros([32, 33]), spatial('x,y'))
StaggeredGrid(math.stack([vx, vy], dual(vector='x,y')), extrapolation.BOUNDARY)
vx = math.tensor(np.zeros([32, 32]), spatial('x,y'))
vy = math.tensor(np.zeros([32, 32]), spatial('x,y'))
StaggeredGrid(math.stack([vx, vy], dual(vector='x,y')), extrapolation.PERIODIC)
vx = math.tensor(np.zeros([31, 32]), spatial('x,y'))
vy = math.tensor(np.zeros([32, 31]), spatial('x,y'))
StaggeredGrid(math.stack([vx, vy], dual(vector='x,y')), 0)
--------------------------------------------------------------------------- ValueError Traceback (most recent call last) Cell In[15], line 11 9 vx = math.tensor(np.zeros([31, 32]), spatial('x,y')) 10 vy = math.tensor(np.zeros([32, 31]), spatial('x,y')) ---> 11 StaggeredGrid(math.stack([vx, vy], dual(vector='x,y')), 0) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_grid.py:176, in StaggeredGrid(values, boundary, bounds, resolution, extrapolation, convert, **resolution_) 174 assert values.shape.spatial_rank == elements.bounds.spatial_rank, f"Spatial dimensions of values ({values.shape}) do not match elements {elements}" 175 assert values.shape.instance_rank == 0, f"Instance dimensions not supported for grids. Got values with shape {values.shape}" --> 176 return Field(elements, values, extrapolation) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_field.py:46, in _FieldType.__call__(cls, geometry, values, boundary, variable_attrs, value_attrs, **sampling_kwargs) 44 values = expand(wrap(values), non_batch(geometry) - 'vector') 45 result = cls.__new__(cls, geometry, values, boundary, variable_attrs, value_attrs) ---> 46 result.__init__(geometry, values, boundary, variable_attrs, value_attrs) # also calls __post_init__() 47 return result File <string>:8, in __init__(self, geometry, values, boundary, variable_attrs, value_attrs) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_field.py:85, in Field.__post_init__(self) 84 def __post_init__(self): ---> 85 at = self.sampled_at 86 if at in {'center', 'face'}: 87 math.merge_shapes(self.values, non_batch(self.sampled_elements).non_channel) # shape check File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/functools.py:998, in cached_property.__get__(self, instance, owner) 996 val = cache.get(self.attrname, _NOT_FOUND) 997 if val is _NOT_FOUND: --> 998 val = self.func(instance) 999 try: 1000 cache[self.attrname] = val File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_field.py:399, in Field.sampled_at(self) 397 if all(_size_equal(v_size, set_size) for v_size, set_size in zip(v_sizes, set_shape.sizes)): 398 return name --> 399 raise ValueError(f"Could not determine where the values of this Field are sampled. Geometry sets: {self.geometry.sets}, Field values shape: {v_shape}") ValueError: Could not determine where the values of this Field are sampled. Geometry sets: {'center': (xˢ=32, yˢ=32), 'face': (~vectorᵈ=x,y, xˢ=~(x=33, y=32) int64, yˢ=~(x=32, y=33) int64), 'node': (xˢ=33, yˢ=33)}, Field values shape: (~vectorᵈ=x,y, xˢ=~(x=31, y=32) int64, yˢ=~(x=32, y=31) int64)
def loss_function(x):
return math.l2_loss(math.cos(x))
initial_guess = math.tensor([1, -1], math.batch('batch'))
math.minimize(loss_function, Solve('L-BFGS-B', 0, 1e-3, x0=initial_guess))
(1.574, -1.574) along batchᵇ
def f(x):
return 2 * x
math.solve_linear(f, 84, Solve('CG', 1e-5, x0=0))
tensor([42.])
from functools import partial
periodic_laplace = partial(math.laplace, padding=extrapolation.PERIODIC)
example_input = math.ones(spatial(x=3))
matrix, bias = math.matrix_from_function(periodic_laplace, example_input)
math.print(matrix)
x=0 -2. 1. 1. along ~x x=1 1. -2. 1. along ~x x=2 1. 1. -2. along ~x
def f(x):
return math.l2_loss(math.sin(x))
f_grid = CenteredGrid(f, x=100, y=100, bounds=Box(x=2*PI, y=2*PI))
vis.plot(f_grid)
def minimize(x0):
with math.SolveTape(record_trajectories=True) as solves:
math.minimize(f, Solve('BFGS', 0, 1e-5, x0=x0))
return solves[0].x # shape (trajectory, x, y, vector)
trajectories = CenteredGrid(minimize, x=8, y=8, bounds=Box(x=2*PI, y=2*PI)).values
segments = []
for start, end in zip(trajectories.trajectory[:-1].trajectory, trajectories.trajectory[1:].trajectory):
segments.append(PointCloud(start, end - start, bounds=Box(x=2*PI, y=2*PI)))
anim_segments = field.stack(segments, batch('time'))
vis.plot(f_grid, anim_segments, overlay='args', animate='time', color='#FFFFFF', frame_time=500)
--------------------------------------------------------------------------- NotImplementedError Traceback (most recent call last) Cell In[20], line 6 3 math.minimize(f, Solve('BFGS', 0, 1e-5, x0=x0)) 4 return solves[0].x # shape (trajectory, x, y, vector) ----> 6 trajectories = CenteredGrid(minimize, x=8, y=8, bounds=Box(x=2*PI, y=2*PI)).values 7 segments = [] 8 for start, end in zip(trajectories.trajectory[:-1].trajectory, trajectories.trajectory[1:].trajectory): File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/field/_grid.py:75, in CenteredGrid(values, boundary, bounds, resolution, extrapolation, convert, **resolution_) 73 values = sample(values, elements) 74 elif callable(values): ---> 75 values = sample_function(values, elements, 'center', extrapolation) 76 else: 77 if isinstance(values, (tuple, list)) and len(values) == resolution.rank: File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phi/geom/_geom.py:859, in sample_function(f, elements, at, extrapolation) 857 values = math.map_s2b(f)(*pos.vector) 858 else: --> 859 values = math.map_s2b(f)(pos) 860 assert isinstance(values, math.Tensor), f"values function must return a Tensor but returned {type(values)}" 861 return values File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_functional.py:1230, in map_types.<locals>.retyped_f(*args, **kwargs) 1228 retyped_kwarg, input_types = forward_retype(v, input_types) 1229 retyped_kwargs[k] = retyped_kwarg -> 1230 output = f(*retyped_args, **retyped_kwargs) 1231 restored_output = reverse_retype(output, input_types) 1232 return restored_output Cell In[20], line 3, in minimize(x0) 1 def minimize(x0): 2 with math.SolveTape(record_trajectories=True) as solves: ----> 3 math.minimize(f, Solve('BFGS', 0, 1e-5, x0=x0)) 4 return solves[0].x File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:465, in minimize(f, solve) 463 iterations = reshaped_tensor(ret[-1].iterations, [batch_dims]) 464 function_evaluations = stack([reshaped_tensor(r.function_evaluations, [batch_dims]) for r in ret], batch('trajectory')) --> 465 result = SolveInfo(solve, x_, residual, iterations, function_evaluations, converged, diverged, ret[-1].method, ret[-1].message, t) 466 for tape in _SOLVE_TAPES: 467 tape._add(solve, trj, result) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:181, in SolveInfo.__init__(self, solve, x, residual, iterations, function_evaluations, converged, diverged, method, msg, solve_time) 179 _, res_tensors = disassemble_tree(residual, cache=False) 180 msg_fun = partial(_default_solve_info_msg, solve=solve) --> 181 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')) 182 self.msg = msg 183 """ `str`, termination message """ File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_functional.py:1431, in map_(function, dims, range, unwrap_scalars, expand_results, simplify, map_name, *args, **kwargs) 1429 assert all(r is None for r in results), f"map function returned None for some elements, {results}" 1430 return None -> 1431 return stack(results, dims_, expand_values=expand_results, simplify=simplify, layout_non_matching=True) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_magic_ops.py:260, in stack(values, dim, expand_values, simplify, layout_non_matching, **kwargs) 258 # --- Fallback: multi-level stack --- 259 for dim_ in reversed(dim): --> 260 values = [stack(values[i:i + dim_.size], dim_, **kwargs) for i in range(0, len(values), dim_.size)] 261 return values[0] File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_magic_ops.py:200, in stack(values, dim, expand_values, simplify, layout_non_matching, **kwargs) 198 if any(isinstance(v, (tuple, list, dict)) for v in values_): 199 from ._tensors import wrap, layout --> 200 if _is_data_array(values_): 201 tensors = [wrap(v) for v in values_] 202 return stack(tensors, dim) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_magic_ops.py:865, in _is_data_array(sequence) 863 def _is_data_array(sequence): 864 try: --> 865 all([np.asarray(v).dtype != object for v in sequence]) 866 except ValueError: # e.g. inhomogeneous 867 return False File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_tensors.py:122, in Tensor.__array__(self, dtype) 120 if self.rank > 1: 121 warnings.warn("Automatic conversion of Φ-ML tensors to NumPy can cause problems because the dimension order is not guaranteed.", SyntaxWarning, stacklevel=3) --> 122 return self.numpy(self._shape) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_tensors.py:117, in Tensor.numpy(self, order, force_expand) 92 def numpy(self, order: Union[str, tuple, list, Shape] = None, force_expand=True) -> np.ndarray: 93 """ 94 Converts this tensor to a `numpy.ndarray` with dimensions ordered according to `order`. 95 (...) 115 ValueError if the tensor cannot be transposed to match target_shape 116 """ --> 117 return self.backend.numpy(self.native(order, force_expand)) File /opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_tensors.py:177, in Tensor.backend(self) 175 @property 176 def backend(self) -> Backend: --> 177 raise NotImplementedError(self.__class__) NotImplementedError: <class 'phiml.math._tree.Layout'>
net = dense_net(1, 1, layers=[8, 8], activation='ReLU') # Implemented for PyTorch, TensorFlow, Jax-Stax
optimizer = adam(net, 1e-3)
BATCH = batch(batch=100)
def loss_function(data: Tensor):
prediction = math.native_call(net, data)
label = math.sin(data)
return math.l2_loss(prediction - label), data, label
print(f"Initial loss: {loss_function(math.random_normal(BATCH))[0]}")
for i in range(100):
loss, _data, _label = update_weights(net, optimizer, loss_function, data=math.random_normal(BATCH))
print(f"Final loss: {loss}")
Initial loss: (batchᵇ=100) 0.147 ± 0.143 (5e-04...5e-01) Final loss: (batchᵇ=100) 0.020 ± 0.019 (2e-05...1e-01)
parameter_count(net)
97