kernex 0.2.0

Stencil computations in JAX.

Differentiable Stencil computations in JAX

Installation |Description |Quick example |Function mesh |More Examples |Benchmarking

🛠️ Installation

pip install kernex

📖 Description

Kernex extends jax.vmap/jax.lax.map/jax.pmap with kmap and jax.lax.scan with kscan for general stencil computations.

The prime motivation for this package is to blend the solution process of PDEs into a NN setting.

⏩ Quick Example

kmap	kscan
import kernex as kex import jax.numpy as jnp @kex.kmap(kernel_size=(3,)) def sum_all(x): return jnp.sum(x) >>> x = jnp.array([1,2,3,4,5]) >>> print(sum_all(x)) [ 6 9 12]	import kernex as kex import jax.numpy as jnp @kex.kscan(kernel_size=(3,)) def sum_all(x): return jnp.sum(x) > > > x = jnp.array([1,2,3,4,5]) > > > print(sum_all(x)) > > > [ 6 13 22]

`jax.vmap` is used to sum each window content.

`lax.scan` is used to update the array and the window sum is calculated sequentially. the first three rows represents the three sequential steps used to get the solution in the last row.

🕸️ Function mesh concept

The objective is to apply f(x) = x^2 at index=0 and f(x) = x^3 at index=(1,10)

To achieve the following operation with jax.lax.switch , we need a list of 10 functions correspoing to each cell of the example array. For this reason , kernex adopts a modified version of jax.lax.switch to reduce the number of branches required.

# function applies x^2 at boundaries, and applies x^3 to to the interior

        ┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
  f =   │ x^2 │ x^3 │ x^3 │ x^3 │ x^3 │ x^3 │ x^3 │ x^3 │ x^3 │ x^3 │
        └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘

        ┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
 f(     │  1  │  2  │  3  │  4  │  5  │  6  │  7  │  8  │  9  │ 10  │ ) =
        └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘
        ┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
        │  1  │  8  │  27 │  64 │ 125 │ 216 │ 343 │ 512 │ 729 │1000 │
        └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘

# Gradient of this function
        ┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
df/dx = │ 2x  │3x^2 │3x^2 │3x^2 │3x^2 │3x^2 │3x^2 │3x^2 │3x^2 │3x^2 │
        └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘


        ┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
 df/dx( │  1  │  2  │  3  │  4  │  5  │  6  │  7  │  8  │  9  │ 10  │ ) =
        └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘
        ┌─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┬─────┐
        │  2  │  12 │ 27  │  48 │ 75  │ 108 │ 147 │ 192 │ 243 │ 300 │
        └─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘

Function mesh	Array equivalent
F = kex.kmap(kernel_size=(1,)) F[0] = lambda x:x[0]**2 F[1:] = lambda x:x[0]**3 array = jnp.arange(1,11).astype('float32') print(F(array)) >>> [1., 8., 27., 64., 125., ... 216., 343., 512., 729., 1000.] print(jax.grad(lambda x:jnp.sum(F(x)))(array)) >>> [2.,12.,27.,48.,75., ... 108.,147.,192.,243.,300.]	def F(x): f1 = lambda x:x**2 f2 = lambda x:x**3 x = x.at[0].set(f1(x[0])) x = x.at[1:].set(f2(x[1:])) return x array = jnp.arange(1,11).astype('float32') print(F(array)) >>> [1., 8., 27., 64., 125., ... 216., 343., 512., 729., 1000.] print(jax.grad(lambda x: jnp.sum(F(x)))(array)) >>> [2.,12.,27.,48.,75., ... 108.,147.,192.,243.,300.]

Additionally , we can combine the function mesh concept with stencil computation for scientific computing. See Linear convection in More examples section

🔢 More examples

1️⃣ Convolution operation

import jax
import jax.numpy as jnp
import kernex as kex

@jax.jit
@kex.kmap(
    kernel_size= (3,3,3),
    padding = ('valid','same','same'))
def kernex_conv2d(x,w):
    # JAX channel first conv2d with 3x3x3 kernel_size
    return jnp.sum(x*w)

2️⃣ Laplacian operation

# see also
# https://numba.pydata.org/numba-doc/latest/user/stencil.html#basic-usage
import jax
import jax.numpy as jnp
import kernex as kex

@kex.kmap(
    kernel_size=(3,3),
    padding= 'valid',
    relative=True) # `relative`= True enables relative indexing
def laplacian(x):
    return ( 0*x[1,-1]  + 1*x[1,0]   + 0*x[1,1] +
             1*x[0,-1]  +-4*x[0,0]   + 1*x[0,1] +
             0*x[-1,-1] + 1*x[-1,0]  + 0*x[-1,1] )

# apply laplacian
>>> print(laplacian(jnp.ones([10,10])))
DeviceArray(
    [[0., 0., 0., 0., 0., 0., 0., 0.],
    [0., 0., 0., 0., 0., 0., 0., 0.],
    [0., 0., 0., 0., 0., 0., 0., 0.],
    [0., 0., 0., 0., 0., 0., 0., 0.],
    [0., 0., 0., 0., 0., 0., 0., 0.],
    [0., 0., 0., 0., 0., 0., 0., 0.],
    [0., 0., 0., 0., 0., 0., 0., 0.],
    [0., 0., 0., 0., 0., 0., 0., 0.]], dtype=float32)

3️⃣ Get Patches of an array

import jax
import jax.numpy as jnp
import kernex as kex

@kex.kmap(kernel_size=(3,3),relative=True)
def identity(x):
    # similar to numba.stencil
    # this function returns the top left cell in the padded/unpadded kernel view
    # or center cell if `relative`=True
    return x[0,0]

# unlike numba.stencil , vector output is allowed in kernex
# this function is similar to
# `jax.lax.conv_general_dilated_patches(x,(3,),(1,),padding='same')`
@jax.jit
@kex.kmap(kernel_size=(3,3),padding='same')
def get_3x3_patches(x):
    # returns 5x5x3x3 array
    return x

mat = jnp.arange(1,26).reshape(5,5)
>>> print(mat)
[[ 1  2  3  4  5]
 [ 6  7  8  9 10]
 [11 12 13 14 15]
 [16 17 18 19 20]
 [21 22 23 24 25]]


# get the view at array index = (0,0)
>>> print(get_3x3_patches(mat)[0,0])
[[0 0 0]
 [0 1 2]
 [0 6 7]]

4️⃣ Linear convection

$\Large {\partial u \over \partial t} + c {\partial u \over \partial x} = 0$

$\Large u_i^{n} = u_i^{n-1} - c \frac{\Delta t}{\Delta x}(u_i^{n-1}-u_{i-1}^{n-1})$

Problem setup	Stencil view

import jax
import jax.numpy as jnp
import kernex as kex
import matplotlib.pyplot as plt

# see https://nbviewer.org/github/barbagroup/CFDPython/blob/master/lessons/01_Step_1.ipynb

tmax,xmax = 0.5,2.0
nt,nx = 151,51
dt,dx = tmax/(nt-1) , xmax/(nx-1)
u = jnp.ones([nt,nx])
c = 0.5

# kscan moves sequentially in row-major order and updates in-place using lax.scan.

F = kernex.kscan(
        kernel_size = (3,3),
        padding = ((1,1),(1,1)),
        named_axis={0:'n',1:'i'},  # n for time axis , i for spatial axis (optional naming)
        relative=True
        )


# boundary condtion as a function
def bc(u):
    return 1

# initial condtion as a function
def ic1(u):
    return 1

def ic2(u):
    return 2

def linear_convection(u):
    return ( u['i','n-1'] - (c*dt/dx) * (u['i','n-1'] - u['i-1','n-1']) )


F[:,0]  = F[:,-1] = bc # assign 1 for left and right boundary for all t

# square wave initial condition
F[:,:int((nx-1)/4)+1] = F[:,int((nx-1)/2):] = ic1
F[0:1, int((nx-1)/4)+1 : int((nx-1)/2)] = ic2

# assign linear convection function for
# interior spatial location [1:-1]
# and start from t>0  [1:]
F[1:,1:-1] = linear_convection

kx_solution = F(jnp.array(u))

plt.figure(figsize=(20,7))
for line in kx_solution[::20]:
    plt.plot(jnp.linspace(0,xmax,nx),line)

5️⃣ Gaussian blur

import jax
import jax.numpy as jnp
import kernex as kex

def gaussian_blur(image, sigma, kernel_size):
    x = jnp.linspace(-(kernel_size - 1) / 2.0, (kernel_size- 1) / 2.0, kernel_size)
    w = jnp.exp(-0.5 * jnp.square(x) * jax.lax.rsqrt(sigma))
    w = jnp.outer(w, w)
    w = w / w.sum()

    @kex.kmap(kernel_size=(kernel_size, kernel_size), padding="same")
    def conv(x):
        return jnp.sum(x * w)

    return conv(image)

6️⃣ Depthwise convolution

import jax
import jax.numpy as jnp
import kernex as kex

@jax.jit
@jax.vmap
@kex.kmap(
kernel_size= (3,3),
padding = ('same','same'))
def kernex_depthwise_conv2d(x,w): # Channel-first depthwise convolution # jax.debug.print("x=\n{a}\nw=\n{b} \n\n",a=x, b=w)
return jnp.sum(x\*w)

h,w,c = 5,5,2
k=3

x = jnp.arange(1,h*w*c+1).reshape(c,h,w)
w = jnp.arange(1,k*k*c+1).reshape(c,k,k)
print(kernex_depthwise_conv2d(x,w))</summary>

7️⃣ Maxpooling2D and Averagepooling2D

@jax.vmap # vectorize over the channel dimension
@kex.kmap(kernel_size=(3,3), strides=(2,2))
def maxpool_2d(x):
    # define the kernel for the Maxpool operation over the spatial dimensions
    return jnp.max(x)

@jax.vmap # vectorize over the channel dimension
@kex.kmap(kernel_size=(3,3), strides=(2,2))
def avgpool_2d(x):
    # define the kernel for the Average pool operation over the spatial dimensions
    return jnp.mean(x)

8️⃣ Runge-Kutta integration

# lets solve dydt = y, where y0 = 1 and y(t)=e^t
# using Runge-Kutta 4th order method
# f(t,y) = y
import jax.numpy as jnp
import matplotlib.pyplot as plt
import kernex as kex


t = jnp.linspace(0, 1, 5)
y = jnp.zeros(5)
x = jnp.stack([y, t], axis=0)
dt = t[1] - t[0]  # 0.1
f = lambda tn, yn: yn


def ic(x):
    """ initial condition y0 = 1 """
    return 1.


def rk4(x):
    """ runge kutta 4th order integration step """
    # ┌────┬────┬────┐      ┌──────┬──────┬──────┐
    # │ y0 │*y1*│ y2 │      │[0,-1]│[0, 0]│[0, 1]│
    # ├────┼────┼────┤ ==>  ├──────┼──────┼──────┤
    # │ t0 │ t1 │ t2 │      │[1,-1]│[1, 0]│[1, 1]│
    # └────┴────┴────┘      └──────┴──────┴──────┘
    t0 = x[1, -1]
    y0 = x[0, -1]
    k1 = dt * f(t0, y0)
    k2 = dt * f(t0 + dt / 2, y0 + 1 / 2 * k1)
    k3 = dt * f(t0 + dt / 2, y0 + 1 / 2 * k2)
    k4 = dt * f(t0 + dt, y0 + k3)
    yn_1 = y0 + 1 / 6 * (k1 + 2 * k2 + 2 * k3 + k4)
    return yn_1


F = kex.kscan(kernel_size=(2, 3), relative=True, padding=((0, 1)))  # kernel size = 3

F[0:1, 1:] = rk4
F[0, 0] = ic
# compile the solver
solver = jax.jit(F.__call__)
y = solver(x)[0, :]

plt.plot(t, y, '-o', label='rk4')
plt.plot(t, jnp.exp(t), '-o', label='analytical')
plt.legend()