serket 0.0.10

JAX NN library.

The ✨Magical✨ JAX Scientific ML Library.

*Serket is the goddess of magic in Egyptian mythology

Installation |Description |Quick Example |Freezing/Fine tuning |Filtering

🛠️ Installation

pip install serket

Install development version

pip install git+https://github.com/ASEM000/serket

📖 Description

• serket aims to be the most intuitive and easy-to-use physics-based Neural network library in JAX.
• serket is built on top of pytreeclass
• serket is fully transparent to jax transformation (e.g. vmap,grad,jit,...)

➖➕Finite difference package: serket.fd➕➖

Group	Function/Layer
Finite difference layer	- Difference: apply finite difference to input array to any derivative order and accuracy
Finite difference functions	- difference: finite difference of array with any accuracy and derivative order - generate_finitediff_coeffs : generate coeffs using sample points and derivative order - fgrad: differentiate functions (similar to jax.grad) with custom accuracy and derivative order
Vector operator layers	- Curl, Divergence, Gradient, Laplacian, Jacobian, Hessian
Vector operator function	- curl, divergence, gradient, laplacian, jacobian, hessian

🧠 Neural network package: serket.nn 🧠

Group	Layers
Linear	- Linear, Bilinear, Multilinear, GeneralLinear, Identity
Densely connected	- FNN (Fully connected network), - PFNN (Parallel fully connected network)
Convolution	- Conv1D, Conv2D, Conv3D, - Conv1DTranspose , Conv2DTranspose, Conv3DTranspose, - DepthwiseConv1D, DepthwiseConv2D, DepthwiseConv3D, - SeparableConv1D, SeparableConv2D, SeparableConv3D, - Conv1DLocal, Conv2DLocal, Conv3DLocal
Containers	- Sequential, Lambda
Pooling (kernex backend)	- MaxPool1D, MaxPool2D, MaxPool3D, - AvgPool1D, AvgPool2D, AvgPool3D - GlobalMaxPool1D, GlobalMaxPool2D, GlobalMaxPool3D, - GlobalAvgPool1D, GlobalAvgPool2D, GlobalAvgPool3D - LPPool1D, LPPool2D,LPPool3D , - AdaptivePool1D, AdaptivePool2D, AdaptivePool3D, - AdaptiveConcatPool1D,AdaptiveConcatPool2D,AdaptiveConcatPool3D
Reshaping	- Flatten, Unflatten, - FlipLeftRight2D, FlipUpDown2D, - Repeat1D, Repeat2D, Repeat3D, - Resize1D, Resize2D, Resize3D, - Upsample1D, Upsample2D, Upsample3D, - Pad1D, Pad2D, Pad3D
Crop	- Crop1D, Crop2D,
Normalization	- LayerNorm, InstanceNorm, GroupNorm
Blurring	- AvgBlur2D, GaussianBlur2D
Dropout	- Dropout, , - Dropout1D, Dropout2D, Dropout3D,
Random transforms	- RandomCrop1D, RandomCrop2D, - RandomApply, - RandomCutout1D, RandomCutout2D, - RandomZoom2D, - RandomContrast2D
Misc	- HistogramEqualization2D, AdjustContrast2D, Filter2D, PixelShuffle2D
Activations	- AdaptiveLeakyReLU,AdaptiveReLU,AdaptiveSigmoid,AdaptiveTanh, - CeLU,ELU,GELU,GLU - HardSILU,HardShrink,HardSigmoid,HardSwish,HardTanh, - LeakyReLU,LogSigmoid,LogSoftmax,Mish,PReLU, - ReLU,ReLU6,SILU,SeLU,Sigmoid,SoftPlus,SoftShrink, - SoftSign,Swish,Tanh,TanhShrink, ThresholdedReLU, Snake
Recurrent	- SimpleRNNCell, LSTMCell, GRUCell, - ConvLSTM1D, ConvLSTM2D, ConvLSTM3D, - SeparableConvLSTM1DCell, SeparableConvLSTM2DCell, SeparableConvLSTM3DCell - ConvGRU1DCell,ConvGRU2DCell,ConvGRU3DCell, - SeparableConvGRU1DCell,SeparableConvGRU2DCell,SeparableConvGRU3DCell
Blocks	- VGG16Block, VGG19Block, UNetBlock

⏩ Examples:

Finite difference examples

import jax

jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
import numpy.testing as npt

import serket as sk


# lets first define a vector valued function F: R^3 -> R^3
# F = F1, F2
# F1 = x^2 + y^3
# F2 = x^4 + y^3
# F3 = 0
# F = [x**2 + y**3, x**4 + y**3, 0]

x, y, z = [jnp.linspace(0, 1, 100)] * 3
dx, dy, dz = x[1] - x[0], y[1] - y[0], z[1] - z[0]
X, Y, Z = jnp.meshgrid(x, y, z, indexing="ij")
F1 = X**2 + Y**3
F2 = X**4 + Y**3
F3 = jnp.zeros_like(F1)
F = jnp.stack([F1, F2, F3], axis=0)

# ∂F1/∂x : differentiate F1 with respect to x (i.e axis=0)
dF1dx = sk.fd.difference(F1, axis=0, step_size=dx, accuracy=6)
dF1dx_exact = 2 * X
npt.assert_allclose(dF1dx, dF1dx_exact, atol=1e-7)

# ∂F2/∂y : differentiate F2 with respect to y (i.e axis=1)
dF2dy = sk.fd.difference(F2, axis=1, step_size=dy, accuracy=6)
dF2dy_exact = 3 * Y**2
npt.assert_allclose(dF2dy, dF2dy_exact, atol=1e-7)

# ∇.F : the divergence of F
divF = sk.fd.divergence(F, step_size=(dx, dy, dz), keepdims=False, accuracy=6)
divF_exact = 2 * X + 3 * Y**2
npt.assert_allclose(divF, divF_exact, atol=1e-7)

# ∇F1 : the gradient of F1
gradF1 = sk.fd.gradient(F1, step_size=(dx, dy, dz), accuracy=6)
gradF1_exact = jnp.stack([2 * X, 3 * Y**2, 0 * X], axis=0)
npt.assert_allclose(gradF1, gradF1_exact, atol=1e-7)

# ΔF1 : laplacian of F1
lapF1 = sk.fd.laplacian(F1, step_size=(dx, dy, dz), accuracy=6)
lapF1_exact = 2 + 6 * Y
npt.assert_allclose(lapF1, lapF1_exact, atol=1e-7)

# ∇xF : the curl of F
curlF = sk.fd.curl(F, step_size=(dx, dy, dz), accuracy=6)
curlF_exact = jnp.stack([F1 * 0, F1 * 0, 4 * X**3 - 3 * Y**2], axis=0)
npt.assert_allclose(curlF, curlF_exact, atol=1e-7)

# Jacobian of F
JF = sk.fd.jacobian(F, accuracy=4, step_size=(dx, dy, dz))
JF_exact = jnp.array(
    [
        [2 * X, 3 * Y**2, jnp.zeros_like(X)],
        [4 * X**3, 3 * Y**2, jnp.zeros_like(X)],
        [jnp.zeros_like(X), jnp.zeros_like(X), jnp.zeros_like(X)],
    ]
)
npt.assert_allclose(JF, JF_exact, atol=1e-7)

# Hessian of F1
HF1 = sk.fd.hessian(F1, accuracy=4, step_size=(dx, dy, dz))
HF1_exact = jnp.array(
    [
        [
            2 * jnp.ones_like(X),  # ∂2F1/∂x2
            0 * jnp.ones_like(X),  # ∂2F1/∂xy
            0 * jnp.ones_like(X),  # ∂2F1/∂xz
        ],
        [
            0 * jnp.ones_like(X),  # ∂2F1/∂yx
            6 * Y**2,              # ∂2F1/∂y2
            0 * jnp.ones_like(X),  # ∂2F1/∂yz
        ],
        [
            0 * jnp.ones_like(X),  # ∂2F1/∂zx
            0 * jnp.ones_like(X),  # ∂2F1/∂zy
            0 * jnp.ones_like(X),  # ∂2F1/∂z2
        ],
    ]
)
npt.assert_allclose(JF, JF_exact, atol=1e-7)

Train Bidirectional-LSTM

import jax
import jax.numpy as jnp
import jax.random as jr
import matplotlib.pyplot as plt
import optax  # for gradient optimization

import serket as sk

x = jnp.linspace(0, 1, 101).reshape(-1, 1)  # 101 samples of 1D data
y = jnp.sin(2 * jnp.pi * x)
y += 0.1 * jr.normal(jr.PRNGKey(0), y.shape)

# we will use 2 time steps to predict the next time step
x_batched = jnp.stack([x[:-1], x[1:]], axis=1)
x_batched = jnp.reshape(x_batched, (100, 1, 2, 1))  # 100 minibatches x 1 sample x 2 time steps x 1D data
y_batched = jnp.reshape(y[1:], (100, 1, 1))  # 100 minibatches x 1 samples x 1D data

model = sk.nn.Sequential(
    [
        # first cell is the forward cell, second cell is the backward cell for bidirectional RNN
        # we return the full sequence of outputs for each cell by setting return_sequences=True
        # we use None in place of `in_features` to infer the input shape from the input
        sk.nn.ScanRNN(
            sk.nn.LSTMCell(None, 64),
            backward_cell=sk.nn.LSTMCell(None, 64),
            return_sequences=True,
        ),
        # here the in_features is inferred from the previous layer by setting it to None
        # or simply we can set it to 64*2 (64 for each cell from previous layer)
        # we set return_sequences=False to return only the last output of the sequence
        sk.nn.ScanRNN(sk.nn.LSTMCell(None, 1), return_sequences=False),
    ]
)


@jax.value_and_grad
def loss_func(NN, batched_x, batched_y):
    # use jax.vmap to apply the model to each minibatch
    # in our case single x minibatch has shape (1, 2, 1)
    # and single y minibatch has shape (1, 1)
    # then vmap will be applied to the leading axis
    batched_preds = jax.vmap(NN)(batched_x)
    return jnp.mean((batched_preds - batched_y) ** 2)


@jax.jit
def batch_step(NN, batched_x, batched_y, opt_state):
    loss, grads = loss_func(NN, batched_x, batched_y)
    updates, optim_state = optim.update(grads, opt_state)
    NN = optax.apply_updates(NN, updates)
    return NN, optim_state, loss


# dry run to infer the in_features (i.e. replace None with in_features)
# if you want restrict the model to a specific input shape or to avoid
# confusion you can manually specify the in_features as a consequence
# dry run is not necessary in this case
model(x_batched[0, 0])

optim = optax.adam(1e-3)
opt_state = optim.init(model)

epochs = 100

for i in range(1, epochs + 1):
    epoch_loss = []
    for x_b, y_b in zip(x_batched, y_batched):
        model, opt_state, loss = batch_step(model, x_b, y_b, opt_state)
        epoch_loss.append(loss)

    epoch_loss = jnp.mean(jnp.array(epoch_loss))

    if i % 10 == 0:
        print(f"Epoch {i:3d} Loss {epoch_loss:.4f}")

# Epoch  10 Loss 0.0880
# Epoch  20 Loss 0.0796
# Epoch  30 Loss 0.0620
# Epoch  40 Loss 0.0285
# Epoch  50 Loss 0.0205
# Epoch  60 Loss 0.0187
# Epoch  70 Loss 0.0182
# Epoch  80 Loss 0.0176
# Epoch  90 Loss 0.0171
# Epoch 100 Loss 0.0166

y_pred = jax.vmap(model)(x_batched.reshape(-1, 2, 1))
plt.plot(x[1:], y[1:], "--k", label="data")
plt.plot(x[1:], y_pred, label="prediction")
plt.legend()

Lazy initialization

In cases where in_features needs to be inferred from input, use None instead of in_features to infer the value at runtime. However, since the lazy module initialize it's state after the first call (i.e. mutate it's state) jax transformation ex: vmap, grad ... is not allowed before initialization. Using any jax transformation before initialization will throw a ValueError.

import serket as sk 
import jax
import jax.numpy as jnp 

model = sk.nn.Sequential(
    [
        sk.nn.Conv2D(None, 128, 3),
        sk.nn.ReLU(),
        sk.nn.MaxPool2D(2, 2),
        sk.nn.Conv2D(128, 64, 3),
        sk.nn.ReLU(),
        sk.nn.MaxPool2D(2, 2),
        sk.nn.Flatten(),
        sk.nn.Linear(None, 128),
        sk.nn.ReLU(),
        sk.nn.Linear(128, 1),
    ]
)

# print the first `Conv2D` layer before initialization
print(model[0].__repr__())
# Conv2D(
#   weight=None,
#   bias=None,
#   *in_features=None,
#   *out_features=None,
#   *kernel_size=None,
#   *strides=None,
#   *padding=None,
#   *input_dilation=None,
#   *kernel_dilation=None,
#   weight_init_func=None,
#   bias_init_func=None,
#   *groups=None
# )

try :
    jax.vmap(model)(jnp.ones((10, 1,28, 28)))
except ValueError:
    print("***** Not initialized *****")
# ***** Not initialized *****

# dry run to initialize the model
model(jnp.empty([3,128,128]))

print(model[0].__repr__())
# Conv2D(
#   weight=f32[128,3,3,3],
#   bias=f32[128,1,1],
#   *in_features=3,
#   *out_features=128,
#   *kernel_size=(3,3),
#   *strides=(1,1),
#   *padding=((1,1),(1,1)),
#   *input_dilation=(1,1),
#   *kernel_dilation=(1,1),
#   weight_init_func=Partial(glorot_uniform(key,shape,dtype)),
#   bias_init_func=Partial(zeros(key,shape,dtype)),
#   *groups=1
# )

Train MNIST

We will use tensorflow datasets for dataloading. for more on interface of jax/tensorflow dataset see here

# imports
import tensorflow as tf
# Ensure TF does not see GPU and grab all GPU memory.
tf.config.set_visible_devices([], device_type="GPU")
import tensorflow_datasets as tfds
import tensorflow.experimental.numpy as tnp
import jax
import jax.numpy as jnp
import jax.random as jr 
import optax  # for gradient optimization
import serket as sk
import matplotlib.pyplot as plt
import functools as ft

# Construct a tf.data.Dataset
batch_size = 128

# convert the samples from integers to floating-point numbers
# and channel first format
def preprocess_data(x):
    # convert to channel first format
    image = tnp.moveaxis(x["image"], -1, 0)
    # normalize to [0, 1]
    image = tf.cast(image, tf.float32) / 255.0

    # one-hot encode the labels
    label = tf.one_hot(x["label"], 10) / 1.0
    return {"image": image, "label": label}


ds_train, ds_test = tfds.load("mnist", split=["train", "test"], shuffle_files=True)
# (batches, batch_size, 1, 28, 28)
ds_train = ds_train.shuffle(1024).map(preprocess_data).batch(batch_size).prefetch(tf.data.AUTOTUNE)

# (batches, 1, 28, 28)
ds_test = ds_test.map(preprocess_data).prefetch(tf.data.AUTOTUNE)

🏗️ Model definition

We will use jax.vmap(model) to apply model on batches.

@sk.treeclass
class CNN:
    def __init__(self):
        self.conv1 = sk.nn.Conv2D(1, 32, (3, 3), padding="valid")
        self.relu1 = sk.nn.ReLU()
        self.pool1 = sk.nn.MaxPool2D((2, 2), strides=(2, 2))
        self.conv2 = sk.nn.Conv2D(32, 64, (3, 3), padding="valid")
        self.relu2 = sk.nn.ReLU()
        self.pool2 = sk.nn.MaxPool2D((2, 2), strides=(2, 2))
        self.flatten = sk.nn.Flatten(start_dim=0)
        self.dropout = sk.nn.Dropout(0.5)
        self.linear = sk.nn.Linear(5*5*64, 10)

    def __call__(self, x):
        x = self.conv1(x)
        x = self.relu1(x)
        x = self.pool1(x)
        x = self.conv2(x)
        x = self.relu2(x)
        x = self.pool2(x)
        x = self.flatten(x)
        x = self.dropout(x)
        x = self.linear(x)
        return x

model = CNN()

🎨 Visualize model

Model summary

print(model.summary(show_config=False, array=jnp.empty((1, 28, 28))))  
┌───────┬─────────┬─────────┬──────────────┬─────────────┬─────────────┐
│Name   │Type     │Param #  │Size          │Input        │Output       │
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│conv1  │Conv2D   │320(0)   │1.25KB(0.00B) │f32[1,28,28] │f32[32,26,26]│
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│relu1  │ReLU     │0(0)     │0.00B(0.00B)  │f32[32,26,26]│f32[32,26,26]│
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│pool1  │MaxPool2D│0(0)     │0.00B(0.00B)  │f32[32,26,26]│f32[32,13,13]│
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│conv2  │Conv2D   │18,496(0)│72.25KB(0.00B)│f32[32,13,13]│f32[64,11,11]│
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│relu2  │ReLU     │0(0)     │0.00B(0.00B)  │f32[64,11,11]│f32[64,11,11]│
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│pool2  │MaxPool2D│0(0)     │0.00B(0.00B)  │f32[64,11,11]│f32[64,5,5]  │
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│flatten│Flatten  │0(0)     │0.00B(0.00B)  │f32[64,5,5]  │f32[1600]    │
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│dropout│Dropout  │0(0)     │0.00B(0.00B)  │f32[1600]    │f32[1600]    │
├───────┼─────────┼─────────┼──────────────┼─────────────┼─────────────┤
│linear │Linear   │16,010(0)│62.54KB(0.00B)│f32[1600]    │f32[10]      │
└───────┴─────────┴─────────┴──────────────┴─────────────┴─────────────┘
Total count :	34,826(0)
Dynamic count :	34,826(0)
Frozen count :	0(0)
------------------------------------------------------------------------
Total size :	136.04KB(0.00B)
Dynamic size :	136.04KB(0.00B)
Frozen size :	0.00B(0.00B)
========================================================================

tree diagram

print(model.tree_diagram())
CNN
    ├── conv1=Conv2D
    │   ├── weight=f32[32,1,3,3]
    │   ├── bias=f32[32,1,1]
    │   ├*─ in_features=1
    │   ├*─ out_features=32
    │   ├*─ kernel_size=(3, 3)
    │   ├*─ strides=(1, 1)
    │   ├*─ padding=((0, 0), (0, 0))
    │   ├*─ input_dilation=(1, 1)
    │   ├*─ kernel_dilation=(1, 1)
    │   ├── weight_init_func=Partial(init(key,shape,dtype))
    │   ├── bias_init_func=Partial(zeros(key,shape,dtype))
    │   └*─ groups=1    
    ├── relu1=ReLU  
    ├*─ pool1=MaxPool2D
    │   ├*─ kernel_size=(2, 2)
    │   ├*─ strides=(2, 2)
    │   └*─ padding='valid' 
    ├── conv2=Conv2D
    │   ├── weight=f32[64,32,3,3]
    │   ├── bias=f32[64,1,1]
    │   ├*─ in_features=32
    │   ├*─ out_features=64
    │   ├*─ kernel_size=(3, 3)
    │   ├*─ strides=(1, 1)
    │   ├*─ padding=((0, 0), (0, 0))
    │   ├*─ input_dilation=(1, 1)
    │   ├*─ kernel_dilation=(1, 1)
    │   ├── weight_init_func=Partial(init(key,shape,dtype))
    │   ├── bias_init_func=Partial(zeros(key,shape,dtype))
    │   └*─ groups=1    
    ├── relu2=ReLU  
    ├*─ pool2=MaxPool2D
    │   ├*─ kernel_size=(2, 2)
    │   ├*─ strides=(2, 2)
    │   └*─ padding='valid' 
    ├*─ flatten=Flatten
    │   ├*─ start_dim=0
    │   └*─ end_dim=-1  
    ├── dropout=Dropout
    │   ├*─ p=0.5
    │   └── eval=None   
    └── linear=Linear
        ├── weight=f32[1600,10]
        ├── bias=f32[10]
        ├*─ in_features=1600
        └*─ out_features=10  
    

Plot sample predictions before training

 
# set all dropout off
test_model = model.at[model == "eval"].set(True, is_leaf=lambda x: x is None)

def show_images_with_predictions(model, images, one_hot_labels):
    logits = jax.vmap(model)(images)
    predictions = jnp.argmax(logits, axis=-1)
    fig, axes = plt.subplots(5, 5, figsize=(10, 10))
    for i, ax in enumerate(axes.flat):
        ax.imshow(images[i].reshape(28, 28), cmap="binary")
        ax.set(title=f"Prediction: {predictions[i]}\nLabel: {jnp.argmax(one_hot_labels[i], axis=-1)}")
        ax.set_xticks([])
        ax.set_yticks([])
    plt.show()

example = ds_test.take(25).as_numpy_iterator()
example = list(example)
sample_test_images = jnp.stack([x["image"] for x in example])
sample_test_labels = jnp.stack([x["label"] for x in example])

show_images_with_predictions(test_model, sample_test_images, sample_test_labels)

🏃 Train the model

@ft.partial(jax.value_and_grad, has_aux=True)
def loss_func(model, batched_images, batched_one_hot_labels):
    logits = jax.vmap(model)(batched_images)
    loss = jnp.mean(optax.softmax_cross_entropy(logits=logits, labels=batched_one_hot_labels))
    return loss, logits


# using optax for gradient updates
optim = optax.adam(1e-3)
optim_state = optim.init(model)


@jax.jit
def batch_step(model, batched_images, batched_one_hot_labels, optim_state):
    (loss, logits), grads = loss_func(model, batched_images, batched_one_hot_labels)
    updates, optim_state = optim.update(grads, optim_state)
    model = optax.apply_updates(model, updates)
    accuracy = jnp.mean(jnp.argmax(logits, axis=-1) == jnp.argmax(batched_one_hot_labels, axis=-1))
    return model, optim_state, loss, accuracy


epochs = 5

for i in range(epochs):
    epoch_accuracy = []
    epoch_loss = []

    for example in ds_train.as_numpy_iterator():
        image, label = example["image"], example["label"]
        model, optim_state, loss, accuracy = batch_step(model, image, label, optim_state)
        epoch_accuracy.append(accuracy)
        epoch_loss.append(loss)

    epoch_loss = jnp.mean(jnp.array(epoch_loss))
    epoch_accuracy = jnp.mean(jnp.array(epoch_accuracy))

    print(f"epoch:{i+1:00d}\tloss:{epoch_loss:.4f}\taccuracy:{epoch_accuracy:.4f}")
    
# epoch:1	loss:0.2706	accuracy:0.9268
# epoch:2	loss:0.0725	accuracy:0.9784
# epoch:3	loss:0.0533	accuracy:0.9836
# epoch:4	loss:0.0442	accuracy:0.9868
# epoch:5	loss:0.0368	accuracy:0.9889

🎨 Visualize After training

test_model = model.at[model == "eval"].set(True, is_leaf=lambda x: x is None)
show_images_with_predictions(test_model, sample_test_images, sample_test_labels)

PINN with Finite difference

We will try to estimate NN(x)~f(x), where df(x)/dx = cos(x) and df(x)/dx will be represented with finite difference scheme

import copy

import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt
import optax

import serket as sk

x = jnp.linspace(-jnp.pi, jnp.pi, 1000)[:, None]
y = jnp.sin(x)
dx = x[1] - x[0]
dydx = jnp.cos(x)

NN_fd = sk.nn.Sequential(
    [
        sk.nn.Linear(1, 128),
        sk.nn.ReLU(),
        sk.nn.Linear(128, 128),
        sk.nn.ReLU(),
        sk.nn.Linear(128, 1),
    ]
)

NN_ad = copy.copy(NN_fd)
optim = optax.adam(1e-3)


@jax.value_and_grad
def loss_func_fd(NN, x):
    y = NN(x)
    dydx = sk.fd.difference(y, axis=0, accuracy=5, step_size=dx)
    loss = jnp.mean((dydx - jnp.cos(x)) ** 2)
    loss += jnp.mean((NN(jnp.zeros_like(x))) ** 2)  # initial condition
    return loss


@jax.value_and_grad
def loss_func_ad(NN, x):
    loss = jnp.mean((sk.diff(NN)(x) - jnp.cos(x)) ** 2)
    loss += jnp.mean(NN(jnp.zeros_like(x)) ** 2)  # initial condition
    return loss


@jax.jit
def step_fd(NN, x, optim_state):
    loss, grads = loss_func_fd(NN, x)
    updates, optim_state = optim.update(grads, optim_state)
    NN = optax.apply_updates(NN, updates)
    return NN, optim_state, loss


def train_fd(NN_fd, optim_state_fd, epochs):
    for i in range(1, epochs + 1):
        NN_fd, optim_state_fd, loss_fd = step_fd(NN_fd, x, optim_state_fd)
    return NN_fd, optim_state_fd, loss_fd


@jax.jit
def step_ad(NN, x, optim_state):
    loss, grads = loss_func_ad(NN, x)
    updates, optim_state = optim.update(grads, optim_state)
    NN = optax.apply_updates(NN, updates)
    return NN, optim_state, loss


def train_ad(NN_ad, optim_state_ad, epochs):
    for i in range(1, epochs + 1):
        NN_ad, optim_state_ad, loss_ad = step_ad(NN_ad, x, optim_state_ad)
    return NN_ad, optim_state_ad, loss_ad


epochs = 1000


optim_state_fd = optim.init(NN_fd)
optim_state_ad = optim.init(NN_ad)


NN_fd, optim_state_fd, loss_fd = train_fd(NN_fd, optim_state_fd, epochs)
NN_ad, optim_state_ad, loss_ad = train_ad(NN_ad, optim_state_ad, epochs)
print(f"Loss_fd {loss_fd:.4f} \nLoss_ad {loss_ad:.4f}")
y_fd = NN_fd(x)
y_ad = NN_ad(x)
plt.plot(x, y, "--k", label="true")
plt.plot(x, y_fd, label="fd pred")
plt.plot(x, y_ad, label="ad pred")
plt.legend()

# Loss_fd 0.0012 
# Loss_ad 0.0235

Reconstructing a vector field F using ∇.F = 0 and ∇xF=2k condition

import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt
import optax

import serket as sk

x, y = [jnp.linspace(-1, 1,50)] * 2
dx, dy = [x[1] - x[0]] * 2
X, Y = jnp.meshgrid(x, y, indexing="ij")

F1 = -Y
F2 = +X
F = jnp.stack([F1, F2], axis=0)

NN = sk.nn.Sequential(
    [
        sk.nn.Conv2D(2, 32, kernel_size=3, padding="same"),
        sk.nn.ReLU(),
        sk.nn.Conv2D(32, 32, kernel_size=3, padding="same"),
        sk.nn.ReLU(),
        sk.nn.Conv2D(32, 2, kernel_size=3, padding="same"),
    ]
)

optim = optax.adam(1e-3)


@jax.value_and_grad
def loss_func(NN, F):
    F_pred = NN(F)
    div = sk.fd.divergence(F_pred, accuracy=5, step_size=(dx, dy))  
    loss = jnp.mean(div**2)  # divergence free condition
    curl = sk.fd.curl(F_pred, accuracy=2, step_size=(dx, dy))
    loss += jnp.mean((curl-jnp.ones_like(curl)*2)**2)  # curl condition 
    return loss


@jax.jit
def step(NN, F, optim_state):
    loss, grads = loss_func(NN, F)
    updates, optim_state = optim.update(grads, optim_state)
    NN = optax.apply_updates(NN, updates)
    return NN, optim_state, loss


def train(NN, Z, optim_state, epochs):
    for i in range(1, epochs + 1):
        NN, optim_state, loss = step(NN, Z, optim_state)
    return NN, optim_state, loss


Z = jnp.stack([X, Y], axis=0)  # collocation points
optim_state = optim.init(NN)  # initialise optimiser
epochs = 1_000
NN, _, loss = train(NN, Z, optim_state, epochs)

Fpred = NN(Z)  # predicted field

plt.figure(figsize=(10, 10))
plt.quiver(X, Y, Fpred[0], Fpred[1], color="r", label="pred")
plt.quiver(X, Y, F1, F2, color="k", alpha=0.5, label="true")
plt.legend()

🥶 Freezing parameters /Fine tuning

✨See here for more about freezing✨

🔘 Filtering by masking

✨See here for more about filterning ✨