View on TensorFlow.org | Run in Google Colab | View source on GitHub | Download notebook |

This guide provides a quick overview of *TensorFlow basics*. Each section of this doc is an overview of a larger topic—you can find links to full guides at the end of each section.

TensorFlow is an end-to-end platform for machine learning. It supports the following:

- Multidimensional-array based numeric computation (similar to NumPy.)
- GPU and distributed processing
- Automatic differentiation
- Model construction, training, and export
- And more

## Tensors

TensorFlow operates on multidimensional arrays or *tensors* represented as `tf.Tensor`

objects. Here is a two-dimensional tensor:

```
import tensorflow as tf
x = tf.constant([[1., 2., 3.],
[4., 5., 6.]])
print(x)
print(x.shape)
print(x.dtype)
```

2024-10-03 01:24:28.897515: E external/local_xla/xla/stream_executor/cuda/cuda_fft.cc:485] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered 2024-10-03 01:24:28.918497: E external/local_xla/xla/stream_executor/cuda/cuda_dnn.cc:8454] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered 2024-10-03 01:24:28.924727: E external/local_xla/xla/stream_executor/cuda/cuda_blas.cc:1452] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered WARNING: All log messages before absl::InitializeLog() is called are written to STDERR I0000 00:00:1727918671.501067 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.504962 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.508951 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.512662 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.523975 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS ha tf.Tensor( [[1. 2. 3.] [4. 5. 6.]], shape=(2, 3), dtype=float32) (2, 3) <dtype: 'float32'> d negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.527575 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.531095 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.534452 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.537818 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.541314 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.544748 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918671.548158 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.811574 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.813683 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.815800 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.817784 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.819747 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.822369 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.824361 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.826310 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.828145 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.830199 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.832189 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.834056 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.872509 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.874567 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.876725 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.878642 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.880502 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.882472 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.884483 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.886365 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.888219 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.891799 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.894304 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355 I0000 00:00:1727918672.896575 10533 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355

The most important attributes of a `tf.Tensor`

are its `shape`

and `dtype`

:

`Tensor.shape`

: tells you the size of the tensor along each of its axes.`Tensor.dtype`

: tells you the type of all the elements in the tensor.

TensorFlow implements standard mathematical operations on tensors, as well as many operations specialized for machine learning.

For example:

```
x + x
```

<tf.Tensor: shape=(2, 3), dtype=float32, numpy= array([[ 2., 4., 6.], [ 8., 10., 12.]], dtype=float32)>

```
5 * x
```

<tf.Tensor: shape=(2, 3), dtype=float32, numpy= array([[ 5., 10., 15.], [20., 25., 30.]], dtype=float32)>

```
x @ tf.transpose(x)
```

<tf.Tensor: shape=(2, 2), dtype=float32, numpy= array([[14., 32.], [32., 77.]], dtype=float32)>

```
tf.concat([x, x, x], axis=0)
```

<tf.Tensor: shape=(6, 3), dtype=float32, numpy= array([[1., 2., 3.], [4., 5., 6.], [1., 2., 3.], [4., 5., 6.], [1., 2., 3.], [4., 5., 6.]], dtype=float32)>

```
tf.nn.softmax(x, axis=-1)
```

<tf.Tensor: shape=(2, 3), dtype=float32, numpy= array([[0.09003057, 0.24472848, 0.6652409 ], [0.09003057, 0.24472848, 0.6652409 ]], dtype=float32)>

```
tf.reduce_sum(x)
```

<tf.Tensor: shape=(), dtype=float32, numpy=21.0>

```
tf.convert_to_tensor([1,2,3])
```

<tf.Tensor: shape=(3,), dtype=int32, numpy=array([1, 2, 3], dtype=int32)>

```
tf.reduce_sum([1,2,3])
```

<tf.Tensor: shape=(), dtype=int32, numpy=6>

Running large calculations on CPU can be slow. When properly configured, TensorFlow can use accelerator hardware like GPUs to execute operations very quickly.

```
if tf.config.list_physical_devices('GPU'):
print("TensorFlow **IS** using the GPU")
else:
print("TensorFlow **IS NOT** using the GPU")
```

TensorFlow **IS** using the GPU

Refer to the Tensor guide for details.

## Variables

Normal `tf.Tensor`

objects are immutable. To store model weights (or other mutable state) in TensorFlow use a `tf.Variable`

.

```
var = tf.Variable([0.0, 0.0, 0.0])
```

```
var.assign([1, 2, 3])
```

<tf.Variable 'UnreadVariable' shape=(3,) dtype=float32, numpy=array([1., 2., 3.], dtype=float32)>

```
var.assign_add([1, 1, 1])
```

<tf.Variable 'UnreadVariable' shape=(3,) dtype=float32, numpy=array([2., 3., 4.], dtype=float32)>

Refer to the Variables guide for details.

## Automatic differentiation

*Gradient descent* and related algorithms are a cornerstone of modern machine learning.

To enable this, TensorFlow implements automatic differentiation (autodiff), which uses calculus to compute gradients. Typically you'll use this to calculate the gradient of a model's *error* or *loss* with respect to its weights.

```
x = tf.Variable(1.0)
def f(x):
y = x**2 + 2*x - 5
return y
```

```
f(x)
```

<tf.Tensor: shape=(), dtype=float32, numpy=-2.0>

At `x = 1.0`

, `y = f(x) = (1**2 + 2*1 - 5) = -2`

.

The derivative of `y`

is `y' = f'(x) = (2*x + 2) = 4`

. TensorFlow can calculate this automatically:

```
with tf.GradientTape() as tape:
y = f(x)
g_x = tape.gradient(y, x) # g(x) = dy/dx
g_x
```

<tf.Tensor: shape=(), dtype=float32, numpy=4.0>

This simplified example only takes the derivative with respect to a single scalar (`x`

), but TensorFlow can compute the gradient with respect to any number of non-scalar tensors simultaneously.

Refer to the Autodiff guide for details.

## Graphs and tf.function

While you can use TensorFlow interactively like any Python library, TensorFlow also provides tools for:

**Performance optimization**: to speed up training and inference.**Export**: so you can save your model when it's done training.

These require that you use `tf.function`

to separate your pure-TensorFlow code from Python.

```
@tf.function
def my_func(x):
print('Tracing.\n')
return tf.reduce_sum(x)
```

The first time you run the `tf.function`

, although it executes in Python, it captures a complete, optimized graph representing the TensorFlow computations done within the function.

```
x = tf.constant([1, 2, 3])
my_func(x)
```

Tracing. <tf.Tensor: shape=(), dtype=int32, numpy=6>

On subsequent calls TensorFlow only executes the optimized graph, skipping any non-TensorFlow steps. Below, note that `my_func`

doesn't print *tracing* since `print`

is a Python function, not a TensorFlow function.

```
x = tf.constant([10, 9, 8])
my_func(x)
```

<tf.Tensor: shape=(), dtype=int32, numpy=27>

A graph may not be reusable for inputs with a different *signature* (`shape`

and `dtype`

), so a new graph is generated instead:

```
x = tf.constant([10.0, 9.1, 8.2], dtype=tf.float32)
my_func(x)
```

Tracing. <tf.Tensor: shape=(), dtype=float32, numpy=27.3>

These captured graphs provide two benefits:

- In many cases they provide a significant speedup in execution (though not this trivial example).
- You can export these graphs, using
`tf.saved_model`

, to run on other systems like a server or a mobile device, no Python installation required.

Refer to Intro to graphs for more details.

## Modules, layers, and models

`tf.Module`

is a class for managing your `tf.Variable`

objects, and the `tf.function`

objects that operate on them. The `tf.Module`

class is necessary to support two significant features:

- You can save and restore the values of your variables using
`tf.train.Checkpoint`

. This is useful during training as it is quick to save and restore a model's state. - You can import and export the
`tf.Variable`

values*and*the`tf.function`

graphs using`tf.saved_model`

. This allows you to run your model independently of the Python program that created it.

Here is a complete example exporting a simple `tf.Module`

object:

```
class MyModule(tf.Module):
def __init__(self, value):
self.weight = tf.Variable(value)
@tf.function
def multiply(self, x):
return x * self.weight
```

```
mod = MyModule(3)
mod.multiply(tf.constant([1, 2, 3]))
```

<tf.Tensor: shape=(3,), dtype=int32, numpy=array([3, 6, 9], dtype=int32)>

Save the `Module`

:

```
save_path = './saved'
tf.saved_model.save(mod, save_path)
```

INFO:tensorflow:Assets written to: ./saved/assets INFO:tensorflow:Assets written to: ./saved/assets

The resulting SavedModel is independent of the code that created it. You can load a SavedModel from Python, other language bindings, or TensorFlow Serving. You can also convert it to run with TensorFlow Lite or TensorFlow JS.

```
reloaded = tf.saved_model.load(save_path)
reloaded.multiply(tf.constant([1, 2, 3]))
```

<tf.Tensor: shape=(3,), dtype=int32, numpy=array([3, 6, 9], dtype=int32)>

The `tf.keras.layers.Layer`

and `tf.keras.Model`

classes build on `tf.Module`

providing additional functionality and convenience methods for building, training, and saving models. Some of these are demonstrated in the next section.

Refer to Intro to modules for details.

## Training loops

Now put this all together to build a basic model and train it from scratch.

First, create some example data. This generates a cloud of points that loosely follows a quadratic curve:

```
import matplotlib
from matplotlib import pyplot as plt
matplotlib.rcParams['figure.figsize'] = [9, 6]
```

```
x = tf.linspace(-2, 2, 201)
x = tf.cast(x, tf.float32)
def f(x):
y = x**2 + 2*x - 5
return y
y = f(x) + tf.random.normal(shape=[201])
plt.plot(x.numpy(), y.numpy(), '.', label='Data')
plt.plot(x, f(x), label='Ground truth')
plt.legend();
```

Create a quadratic model with randomly initialized weights and a bias:

```
class Model(tf.Module):
def __init__(self):
# Randomly generate weight and bias terms
rand_init = tf.random.uniform(shape=[3], minval=0., maxval=5., seed=22)
# Initialize model parameters
self.w_q = tf.Variable(rand_init[0])
self.w_l = tf.Variable(rand_init[1])
self.b = tf.Variable(rand_init[2])
@tf.function
def __call__(self, x):
# Quadratic Model : quadratic_weight * x^2 + linear_weight * x + bias
return self.w_q * (x**2) + self.w_l * x + self.b
```

First, observe your model's performance before training:

```
quad_model = Model()
```

```
def plot_preds(x, y, f, model, title):
plt.figure()
plt.plot(x, y, '.', label='Data')
plt.plot(x, f(x), label='Ground truth')
plt.plot(x, model(x), label='Predictions')
plt.title(title)
plt.legend()
```

```
plot_preds(x, y, f, quad_model, 'Before training')
```

Now, define a loss for your model:

Given that this model is intended to predict continuous values, the mean squared error (MSE) is a good choice for the loss function. Given a vector of predictions, \(\hat{y}\), and a vector of true targets, \(y\), the MSE is defined as the mean of the squared differences between the predicted values and the ground truth.

\(MSE = \frac{1}{m}\sum_{i=1}^{m}(\hat{y}_i -y_i)^2\)

```
def mse_loss(y_pred, y):
return tf.reduce_mean(tf.square(y_pred - y))
```

Write a basic training loop for the model. The loop will make use of the MSE loss function and its gradients with respect to the input in order to iteratively update the model's parameters. Using mini-batches for training provides both memory efficiency and faster convergence. The `tf.data.Dataset`

API has useful functions for batching and shuffling.

```
batch_size = 32
dataset = tf.data.Dataset.from_tensor_slices((x, y))
dataset = dataset.shuffle(buffer_size=x.shape[0]).batch(batch_size)
```

```
# Set training parameters
epochs = 100
learning_rate = 0.01
losses = []
# Format training loop
for epoch in range(epochs):
for x_batch, y_batch in dataset:
with tf.GradientTape() as tape:
batch_loss = mse_loss(quad_model(x_batch), y_batch)
# Update parameters with respect to the gradient calculations
grads = tape.gradient(batch_loss, quad_model.variables)
for g,v in zip(grads, quad_model.variables):
v.assign_sub(learning_rate*g)
# Keep track of model loss per epoch
loss = mse_loss(quad_model(x), y)
losses.append(loss)
if epoch % 10 == 0:
print(f'Mean squared error for step {epoch}: {loss.numpy():0.3f}')
# Plot model results
print("\n")
plt.plot(range(epochs), losses)
plt.xlabel("Epoch")
plt.ylabel("Mean Squared Error (MSE)")
plt.title('MSE loss vs training iterations');
```

Mean squared error for step 0: 57.050 Mean squared error for step 10: 10.417 Mean squared error for step 20: 4.270 Mean squared error for step 30: 2.171 Mean squared error for step 40: 1.429 Mean squared error for step 50: 1.162 Mean squared error for step 60: 1.062 Mean squared error for step 70: 1.027 Mean squared error for step 80: 1.017 Mean squared error for step 90: 1.014

Now, observe your model's performance after training:

```
plot_preds(x, y, f, quad_model, 'After training')
```

That's working, but remember that implementations of common training utilities are available in the `tf.keras`

module. So, consider using those before writing your own. To start with, the `Model.compile`

and `Model.fit`

methods implement a training loop for you:

Begin by creating a Sequential Model in Keras using `tf.keras.Sequential`

. One of the simplest Keras layers is the dense layer, which can be instantiated with `tf.keras.layers.Dense`

. The dense layer is able to learn multidimensional linear relationships of the form \(\mathrm{Y} = \mathrm{W}\mathrm{X} + \vec{b}\). In order to learn a nonlinear equation of the form, \(w_1x^2 + w_2x + b\), the dense layer's input should be a data matrix with \(x^2\) and \(x\) as features. The lambda layer, `tf.keras.layers.Lambda`

, can be used to perform this stacking transformation.

```
new_model = tf.keras.Sequential([
tf.keras.layers.Lambda(lambda x: tf.stack([x, x**2], axis=1)),
tf.keras.layers.Dense(units=1, kernel_initializer=tf.random.normal)])
```

```
new_model.compile(
loss=tf.keras.losses.MSE,
optimizer=tf.keras.optimizers.SGD(learning_rate=0.01))
history = new_model.fit(x, y,
epochs=100,
batch_size=32,
verbose=0)
new_model.save('./my_new_model.keras')
```

WARNING: All log messages before absl::InitializeLog() is called are written to STDERR I0000 00:00:1727918680.462363 10695 service.cc:146] XLA service 0x7f51a0006eb0 initialized for platform CUDA (this does not guarantee that XLA will be used). Devices: I0000 00:00:1727918680.462397 10695 service.cc:154] StreamExecutor device (0): Tesla T4, Compute Capability 7.5 I0000 00:00:1727918680.462401 10695 service.cc:154] StreamExecutor device (1): Tesla T4, Compute Capability 7.5 I0000 00:00:1727918680.462404 10695 service.cc:154] StreamExecutor device (2): Tesla T4, Compute Capability 7.5 I0000 00:00:1727918680.462407 10695 service.cc:154] StreamExecutor device (3): Tesla T4, Compute Capability 7.5 I0000 00:00:1727918680.790826 10695 device_compiler.h:188] Compiled cluster using XLA! This line is logged at most once for the lifetime of the process.

Observe your Keras model's performance after training:

```
plt.plot(history.history['loss'])
plt.xlabel('Epoch')
plt.ylim([0, max(plt.ylim())])
plt.ylabel('Loss [Mean Squared Error]')
plt.title('Keras training progress');
```

```
plot_preds(x, y, f, new_model, 'After Training: Keras')
```

Refer to Basic training loops and the Keras guide for more details.