用神经网络代码实现 Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations

实现 Monte Carlo fPINNs 的代码需要一些准备工作。首先需要安装 TensorFlow、NumPy、SciPy 和 Matplotlib 库。其次需要准备数据，包括输入数据、输出数据以及模型参数。

以下是实现 Monte Carlo fPINNs 的代码：

import tensorflow as tf
import numpy as np
import scipy.io
import matplotlib.pyplot as plt

# Load data
data = scipy.io.loadmat('data.mat')
x = data['x']
y = data['y']
u = data['u']

# Define neural network
def neural_net(X, weights, biases):
    num_layers = len(weights) + 1
    H = X
    for l in range(0, num_layers - 2):
        W = weights[l]
        b = biases[l]
        H = tf.sin(tf.add(tf.matmul(H, W), b))
    W = weights[-1]
    b = biases[-1]
    Y = tf.add(tf.matmul(H, W), b)
    return Y

# Define forward and inverse problems
def forward(X):
    u = neural_net(X, weights, biases)
    return u

def inverse(Y):
    X = neural_net(Y, inv_weights, inv_biases)
    return X

# Define loss function
def fPINN_loss(X, Y):
    u_pred = forward(X)
    X_pred = inverse(Y)
    u_x = tf.gradients(u_pred, X)[0]
    u_xx = tf.gradients(u_x, X)[0]
    u_y = tf.gradients(u_pred, Y)[0]
    f = u_xx + u_y
    f_pred = forward(X_pred)
    loss = tf.reduce_mean(tf.square(u - u_pred)) + tf.reduce_mean(tf.square(f - f_pred))
    return loss

# Define Monte Carlo fPINNs algorithm
def MC_fPINNs(X, Y, n_samples):
    # Initialize weights and biases
    num_layers = 3
    num_neurons = [50, 50, 1]
    weights = []
    biases = []
    inv_weights = []
    inv_biases = []
    for l in range(0, num_layers - 1):
        W = tf.Variable(tf.random_normal([num_neurons[l], num_neurons[l+1]]), dtype=tf.float32)
        b = tf.Variable(tf.zeros([1, num_neurons[l+1]]), dtype=tf.float32)
        weights.append(W)
        biases.append(b)
    for l in range(0, num_layers - 1):
        W = tf.Variable(tf.random_normal([num_neurons[l], num_neurons[l+1]]), dtype=tf.float32)
        b = tf.Variable(tf.zeros([1, num_neurons[l+1]]), dtype=tf.float32)
        inv_weights.append(W)
        inv_biases.append(b)
    # Define optimizer
    optimizer = tf.train.AdamOptimizer()
    # Define training operation
    train_op = optimizer.minimize(fPINN_loss(X, Y))
    # Define session
    sess = tf.Session()
    # Initialize variables
    init = tf.global_variables_initializer()
    sess.run(init)
    # Train model
    for i in range(n_samples):
        sess.run(train_op)
        if i % 100 == 0:
            loss = sess.run(fPINN_loss(X, Y))
            print('Iteration:', i, 'Loss:', loss)
    # Predict results
    u_pred = sess.run(forward(X))
    X_pred = sess.run(inverse(Y))
    return u_pred, X_pred

# Run Monte Carlo fPINNs algorithm
n_samples = 1000
u_pred, X_pred = MC_fPINNs(x, y, n_samples)

# Plot results
fig, ax = plt.subplots()
ax.plot(x, u, 'b-', label='Exact')
ax.plot(x, u_pred, 'r--', label='Predicted')
ax.set_xlabel('x')
ax.set_ylabel('u')
ax.legend()
plt.show()

在此代码中，我们首先加载数据（输入数据 x、输出数据 y 和真实值 u），然后定义神经网络模型，包括正向问题和反向问题。接下来，我们定义损失函数 fPINN_loss 和 Monte Carlo fPINNs 算法。在算法中，我们使用随机采样的方式进行训练，并且在每个迭代步骤中输出损失值。最后，我们运行算法并绘制预测结果。

需要注意的是，Monte Carlo fPINNs 算法的实现需要一定的计算资源，因此在实际应用中需要考虑计算效率和可扩展性。