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#@title Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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#
# https://www.apache.org/licenses/LICENSE-2.0
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TensorFlow isn't just for machine learning. Here you will use TensorFlow to simulate the behavior of a partial differential equation. You'll simulate the surface of square pond as a few raindrops land on it.
A few imports you'll need.
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#Import libraries for simulation
import tensorflow as tf
assert tf.__version__.startswith('2')
import numpy as np
#Imports for visualization
import PIL.Image
from io import BytesIO
from IPython.display import clear_output, Image, display
A function for displaying the state of the pond's surface as an image.
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def DisplayArray(a, fmt='jpeg', rng=[0,1]):
"""Display an array as a picture."""
a = (a - rng[0])/float(rng[1] - rng[0])*255
a = np.uint8(np.clip(a, 0, 255))
f = BytesIO()
PIL.Image.fromarray(a).save(f, fmt)
clear_output(wait = True)
display(Image(data=f.getvalue()))
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@tf.function
def make_kernel(a):
"""Transform a 2D array into a convolution kernel"""
a = np.asarray(a)
a = a.reshape(list(a.shape) + [1,1])
return tf.constant(a, dtype=1)
@tf.function
def simple_conv(x, k):
"""A simplified 2D convolution operation"""
x = tf.expand_dims(tf.expand_dims(x, 0), -1)
y = tf.nn.depthwise_conv2d(input=x, filter=k, strides=[1, 1, 1, 1], padding='SAME')
return y[0, :, :, 0]
@tf.function
def laplace(x):
"""Compute the 2D laplacian of an array"""
laplace_k = make_kernel([[0.5, 1.0, 0.5],
[1.0, -6., 1.0],
[0.5, 1.0, 0.5]])
return simple_conv(x, laplace_k)
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N = 500
Here you create your pond and hit it with some rain drops.
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# Initial Conditions -- some rain drops hit a pond
# Set everything to zero
u_init = np.zeros([N, N], dtype=np.float32)
ut_init = np.zeros([N, N], dtype=np.float32)
# Some rain drops hit a pond at random points
for n in range(40):
a,b = np.random.randint(0, N, 2)
u_init[a,b] = np.random.uniform()
DisplayArray(u_init, rng=[-0.1, 0.1])
Now let's specify the details of the differential equation.
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# Parameters:
# eps -- time resolution
# damping -- wave damping
eps = 0.03
damping = 0.04
# Create variables for simulation state
U = tf.Variable(u_init)
Ut = tf.Variable(ut_init)
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# Run 1000 steps of PDE
for i in range(1000):
# Step simulation
# Discretized PDE update rules
U = U + eps * Ut
Ut = Ut + eps * (laplace(U) - damping * Ut)
# Show final image
DisplayArray(U.numpy(), rng=[-0.1, 0.1])
Look! Ripples!