Google JAX
Google JAX is a machine learning framework for transforming numerical functions. It is described as bringing together a modified version of autograd (automatic obtaining of the gradient function through differentiation of a function) and TensorFlow's XLA (Accelerated Linear Algebra). It is designed to follow the structure and workflow of NumPy as closely as possible and works with various existing frameworks such as TensorFlow and PyTorch. The primary functions of JAX are:

 grad: automatic differentiation
 jit: compilation
 vmap: auto-vectorization
 pmap: SPMD programming

grad 

The code below demonstrates the  function's automatic differentiation. # imports
from jax import grad
import jax.numpy as jnp

# define the logistic function
def logistic(x):  
    return jnp.exp(x) / (jnp.exp(x) + 1)

# obtain the gradient function of the logistic function
grad_logistic = grad(logistic)

# evaluate the gradient of the logistic function at x = 1 
grad_log_out = grad_logistic(1.0)   
print(grad_log_out) 

The final line should outputː

0.19661194

jit 

The code below demonstrates the jit function's optimization through fusion. # imports
from jax import jit
import jax.numpy as jnp

# define the cube function
def cube(x):
    return x * x * x

# generate data
x = jnp.ones((10000, 10000))

# create the jit version of the cube function
jit_cube = jit(cube)

# apply the cube and jit_cube functions to the same data for speed comparison
cube(x)
jit_cube(x)

The computation time for  (line no. 17) should be noticeably shorter than that for  (line no. 16). Increasing the values on line no. 7, will increase the difference. vmap 

The code below demonstrates the  function's vectorization. # imports
from functools import partial
from jax import vmap
import jax.numpy as jnp

# define function
def grads(self, inputs):
    in_grad_partial = partial(self._net_grads, self._net_params)
    grad_vmap = vmap(in_grad_partial)
    rich_grads = grad_vmap(inputs)
    flat_grads = np.asarray(self._flatten_batch(rich_grads))
    assert flat_grads.ndim == 2 and flat_grads.shape[0] == inputs.shape[0]
    return flat_grads 

The GIF on the right of this section illustrates the notion of vectorized addition. pmap 

The code below demonstrates the  function's parallelization for matrix multiplication. # import pmap and random from JAX; import JAX NumPy
from jax import pmap, random
import jax.numpy as jnp

# generate 2 random matrices of dimensions 5000 x 6000, one per device
random_keys = random.split(random.PRNGKey(0), 2)
matrices = pmap(lambda key: random.normal(key, (5000, 6000)))(random_keys)

# without data transfer, in parallel, perform a local matrix multiplication on each CPU/GPU
outputs = pmap(lambda x: jnp.dot(x, x.T))(matrices)

# without data transfer, in parallel, obtain the mean for both matrices on each CPU/GPU separately
means = pmap(jnp.mean)(outputs)
print(means)

The final line should print the valuesː

[1.1566595 1.1805978]

Libraries using JAX 
Several python libraries use JAX as a backend, including:

 Flax, a high level neural network library initially developed by Google Brain. Equinox, a library that revolves around the idea of representing parameterised functions (including neural networks) as PyTrees. It was created by Patrick Kidger. Diffrax, a library for the numerical solution of differential equations, such as ordinary differential equations and stochastic differential equations. Optax, a library for gradient processing and optimisation developed by DeepMind. Lineax, a library for numerically solving linear systems and linear least squares. RLax, a library for developing reinforcement learning agents developed by DeepMind. jraph, a library for graph neural networks, developed by DeepMind. jaxtyping, a library for adding type annotations for the shape and data type ("dtype") of arrays or tensors. Some R libraries use JAX as a backend as well, including:
 fastrerandomize, a library that uses the linear-algebra optimized compiler in JAX to speed up selection of balanced randomizations in a design of experiments procedure known as rerandomization. See also 

 NumPy
 TensorFlow
 PyTorch
 CUDA
 Automatic differentiation
 Just-in-time compilation
 Vectorization
 Automatic parallelization

External links 
 Documentationː 
 Colab (Jupyter/iPython) Quickstart Guideː 
 TensorFlow's XLAː  (Accelerated Linear Algebra)
 
 Original paperː

References 

Machine learning
JAX