偏微分方程(PDE)是一个微分方程,它涉及具有几个自变量未知函数的偏导数.参考偏微分方程,我们将专注于创建新图.
让我们假设有一个尺寸为500 * 500平方 : 去的池塘;
N = 500
现在,我们将计算偏微分方程并使用它来形成相应的图形.考虑下面给出的计算图表的步骤.
步骤1 : 导入库进行模拟.
import tensorflow as tf import numpy as np import matplotlib.pyplot as plt
第2步 : 包括将2D数组转换为卷积内核的函数和简化的2D卷积运算.
def make_kernel(a): 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) sess = tf.InteractiveSession()
第3步 : 包括迭代次数并计算图形以相应地显示记录.
N = 500 # 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(100): a,b = np.random.randint(0, N, 2) u_init[a,b] = np.random.uniform() plt.imshow(u_init) plt.show() # 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_)) # 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 % 500 == 0: plt.imshow(U.eval()) plt.show()
图表如下图所示 :