TensorFlow isn't just for machine learning. Here we give a (somewhat pedestrian) example of using TensorFlow for simulating the behavior of a partial differential equation. We'll simulate the surface of square pond as a few raindrops land on it.

A few imports we'll need.

#Import libraries for simulation import tensorflow as tf 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.

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()))

Here we start an interactive TensorFlow session for convenience in playing around. A regular session would work as well if we were doing this in an executable .py file.

sess = tf.InteractiveSession()

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) 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(x, k, [1, 1, 1, 1], padding='SAME') return y[0, :, :, 0] 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)

Our pond is a perfect 500 x 500 square, as is the case for most ponds found in nature.

N = 500

Here we create our pond and hit it with some rain drops.

# 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.

# Parameters: # eps -- time resolution # damping -- wave damping eps = tf.placeholder(tf.float32, shape=()) damping = tf.placeholder(tf.float32, shape=()) # Create variables for simulation state U = tf.Variable(u_init) Ut = tf.Variable(ut_init) # Discretized PDE update rules U_ = U + eps * Ut Ut_ = Ut + eps * (laplace(U) - damping * Ut) # Operation to update the state step = tf.group( U.assign(U_), Ut.assign(Ut_))

This is where it gets fun -- running time forward with a simple for loop.

# Initialize state to initial conditions tf.global_variables_initializer().run() # Run 1000 steps of PDE for i in range(1000): # Step simulation step.run({eps: 0.03, damping: 0.04}) DisplayArray(U.eval(), rng=[-0.1, 0.1])

Look! Ripples!

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Licensed under the Creative Commons Attribution License 3.0.

Code samples licensed under the Apache 2.0 License.

https://www.tensorflow.org/tutorials/pdes