ΦFlow Cookbook¶

This notebook lists useful code snippets.

Import for NumPy, TensorFlow, Jax, PyTorch¶

In [1]:
from phi.flow import *
from phi.tf.flow import *
from phi.jax.stax.flow import *
from phi.torch.flow import *

2026-01-01 14:03:38.968435: 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.
/opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/keras/src/export/tf2onnx_lib.py:8: FutureWarning: In the future `np.object` will be defined as the corresponding NumPy scalar.
  if not hasattr(np, "object"):
2026-01-01 14:03:41.131098: 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)

Select GPU or CPU¶

In [2]:
backend.default_backend().list_devices('GPU')
Out[2]:
[]
In [3]:
backend.default_backend().list_devices('CPU')
Out[3]:
[torch device 'CPU' (CPU 'cpu') | 15996 MB | 4 processors | ]
In [4]:
assert backend.default_backend().set_default_device('CPU')

Use 64 bit FP precision¶

In [5]:
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)

Sample Random Values¶

In [6]:
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
Out[6]:
(examplesᵇ=10, vectorᶜ=x,y) 0.493 ± 0.278 (7e-02...1e+00)

Slice a Tensor¶

In [7]:
data.examples[0]
Out[7]:
(x=0.504, y=0.449)

Print a Tensor¶

In [8]:
print(data)
print(f"{data:full:shape:dtype:color:.1f}")

(examplesᵇ=10, vectorᶜ=x,y) 0.493 ± 0.278 (7e-02...1e+00)
(examplesᵇ=10, vectorᶜ=x,y)
[[0.5, 0.4],
 [0.3, 0.8],
 [0.5, 0.4],
 [0.9, 0.9],
 [0.2, 1.0],
 [0.6, 0.6],
 [0.6, 0.1],
 [0.3, 0.2],
 [0.2, 0.1],
 [1.0, 0.3]]

Plot a Tensor¶

In [9]:
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
Out[9]:

Convert a Tensor to NumPy¶

In [10]:
data.numpy(order='x,y')
Out[10]:
array([[0.86076456, 0.6703891 , 0.9264835 , 0.92898655, 0.59064484,
        0.08103144],
       [0.3046475 , 0.11739182, 0.9487509 , 0.41053718, 0.47591877,
        0.6626666 ],
       [0.11831802, 0.13099313, 0.8727725 , 0.9008764 , 0.31461298,
        0.5797448 ],
       [0.1671769 , 0.4774536 , 0.55992484, 0.22749293, 0.07220328,
        0.59889746],
       [0.49140292, 0.93649167, 0.7418189 , 0.54731715, 0.52658325,
        0.27715594],
       [0.5404138 , 0.65778285, 0.67605394, 0.9889702 , 0.15994543,
        0.99108917],
       [0.86041063, 0.18264854, 0.35895813, 0.85692936, 0.05468988,
        0.29867953],
       [0.86620307, 0.24771738, 0.02713335, 0.48176414, 0.4676721 ,
        0.42801386]], dtype=float32)
In [11]:
math.reshaped_native(data, ['extra', data.shape], to_numpy=True)

/tmp/ipykernel_2556/2990683702.py:1: DeprecationWarning: phiml.math.reshaped_native() is deprecated. Use Tensor.native() instead.
  math.reshaped_native(data, ['extra', data.shape], to_numpy=True)
Out[11]:
array([[0.86076456, 0.6703891 , 0.9264835 , 0.92898655, 0.59064484,
        0.08103144, 0.3046475 , 0.11739182, 0.9487509 , 0.41053718,
        0.47591877, 0.6626666 , 0.11831802, 0.13099313, 0.8727725 ,
        0.9008764 , 0.31461298, 0.5797448 , 0.1671769 , 0.4774536 ,
        0.55992484, 0.22749293, 0.07220328, 0.59889746, 0.49140292,
        0.93649167, 0.7418189 , 0.54731715, 0.52658325, 0.27715594,
        0.5404138 , 0.65778285, 0.67605394, 0.9889702 , 0.15994543,
        0.99108917, 0.86041063, 0.18264854, 0.35895813, 0.85692936,
        0.05468988, 0.29867953, 0.86620307, 0.24771738, 0.02713335,
        0.48176414, 0.4676721 , 0.42801386]], dtype=float32)

Compute Pair-wise Distances¶

In [12]:
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_2556/2195475714.py:3: DeprecationWarning: phiml.math.length is deprecated in favor of phiml.math.norm
  math.print(math.vec_length(distances))

Construct a CenteredGrid¶

In [13]:
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))
Out[13]:

Construct a StaggeredGrid¶

In [14]:
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))

/opt/hostedtoolcache/Python/3.12.12/x64/lib/python3.12/site-packages/phiml/math/_shape.py:2849: RuntimeWarning: Stacking shapes with incompatible labels will result in labels being lost. For vector Got ('x', 'y') and ('x',)
  warnings.warn(f"Stacking shapes with incompatible labels will result in labels being lost. For {dim.name} Got {prev.slice_names} and {dim.slice_names}", RuntimeWarning)
Out[14]:

Construct 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!

In [15]:
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)
Out[15]:
Field[(xˢ=32, yˢ=32, vectorᶜ=x,y)]

BFGS Optimization¶

In [16]:
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))
Out[16]:
(1.574, -1.574) along batchᵇ

Linear Solve¶

In [17]:
def f(x):
    return 2 * x

math.solve_linear(f, 84, Solve('CG', 1e-5, x0=0))
Out[17]:
tensor([42.])

Sparse Matrix Construction¶

In [18]:
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

Sampling a Function¶

In [19]:
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)
Out[19]:

Plot Optimization Trajectories¶

In [20]:
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)

/tmp/ipykernel_2556/1489350838.py:9: UserWarning: bounds argument is deprecated since 2.5 and will be ignored.
  segments.append(PointCloud(start, end - start, bounds=Box(x=2*PI, y=2*PI)))
Out[20]:
Your browser does not support the video tag. 
<Figure size 640x480 with 0 Axes>

Neural Network Training¶

In [21]:
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) 0.220 ± 0.204 (1e-04...6e-01)
Final loss: (batchᵇ=100) 0.034 ± 0.033 (1e-06...1e-01)

Parameter Count¶

In [22]:
parameter_count(net)
Out[22]:
97