## Basic Setup ``` #Import libraries for simulation import tensorflow as tf import numpy as np #Imports for visualization import PIL.Image from cStringIO import StringIO from IPython.display import clear_output, Image, display ``` ``` 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 = StringIO() PIL.Image.fromarray(a).save(f, fmt) display(Image(data=f.getvalue())) ``` ``` sess = tf.InteractiveSession() ``` ## Computational Convenience Functions ``` 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) ``` ## Define the PDE ``` N = 500 ``` ``` # Initial Conditions -- some rain drops hit a pond # Set everything to zero u_init = np.zeros([N, N], dtype="float32") ut_init = np.zeros([N, N], dtype="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]) ``` ![jpeg](output_8_0.jpe) ``` # paramaters # eps -- time resolution # damping -- wave damping eps = tf.placeholder('float', shape=()) damping = tf.placeholder('float', 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_) ) ``` ## Run The Simulation ``` # initialize state to initial conditions tf.InitializeAllVariables().Run() # Run 1000 steps of PDE for i in range(1000): # Step simulation step.Run({eps: 0.03, damping: 0.04}) # Visualize every 50 steps if i % 50 == 0: clear_output() DisplayArray(U.eval(), rng=[-0.1, 0.1]) ``` ![jpeg](output_11_0.jpe) ``` ```