TensorFlow 不仅仅是用来机器学习，它更可以用来模拟仿真。在这里，我们将通过模拟仿真几滴落入一块方形水池的雨点的例子，来引导您如何使用 TensorFlow 中的偏微分方程来模拟仿真的基本使用方法。
首先,我们需要导入一些必要的引用。
| #导入模拟仿真需要的库 |
| import tensorflow as tf |
| import numpy as np |
| #导入可视化需要的库 |
| 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() |
| 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) |
首先,我们需要创建一个完美的 500 × 500 的正方形池塘,就像是我们在现实中找到的一样。
| 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]) |
现在，让我们来指定该微分方程的一些详细参数。
| # 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_)) |
为了能看清仿真效果，我们可以用一个简单的 for 循环来远行我们的仿真程序。
| # Initialize state to initial conditions |
| tf.initialize_all_variables().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]) |
看！！ 雨点落在池塘中,和现实中一样的泛起了涟漪。
