PINN求解一维Sod代码

好的，以下是使用 PINN 求解一维 Sod 问题的代码示例，注释中有详细的说明：

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

# 定义问题参数
gamma = 1.4
x_left = 0.0
x_right = 1.0
t_final = 0.2
num_points = 1000

# 定义模型参数
num_hidden_layers = 4
num_neurons_per_layer = 20
learning_rate = 0.001

# 定义模型
class SodPINN(tf.keras.Model):
    def __init__(self, num_hidden_layers, num_neurons_per_layer):
        super(SodPINN, self).__init__()
        self.num_hidden_layers = num_hidden_layers
        self.num_neurons_per_layer = num_neurons_per_layer
        
        # 定义输入层
        self.input_layer = tf.keras.layers.InputLayer(input_shape=(2,))
        
        # 定义隐藏层
        self.hidden_layers = []
        for i in range(self.num_hidden_layers):
            self.hidden_layers.append(tf.keras.layers.Dense(self.num_neurons_per_layer, activation=tf.nn.tanh))
            
        # 定义输出层
        self.output_layer = tf.keras.layers.Dense(3)
        
    def call(self, inputs):
        x = self.input_layer(inputs)
        for hidden_layer in self.hidden_layers:
            x = hidden_layer(x)
        return self.output_layer(x)

# 定义边界条件
def initial_value(x):
    rho = tf.where(x < 0.5, 1.0, 0.125)
    u = tf.where(x < 0.5, 0.0, 0.0)
    p = tf.where(x < 0.5, 1.0, 0.1)
    return tf.concat([rho[:, None], u[:, None], p[:, None]], axis=1)

def left_boundary(x):
    rho = tf.ones_like(x[:, 0:1])
    u = tf.zeros_like(x[:, 0:1])
    p = tf.ones_like(x[:, 0:1])
    return tf.concat([rho, u, p], axis=1)

def right_boundary(x):
    rho = tf.where(x[:, 0:1] < 0.5, 1.0, 0.125)
    u = tf.zeros_like(x[:, 0:1])
    p = tf.where(x[:, 0:1] < 0.5, 1.0, 0.1)
    return tf.concat([rho, u, p], axis=1)

# 定义损失函数
def get_loss(model, inputs, outputs):
    x = inputs[:, 0:1]
    t = inputs[:, 1:2]
    rho = outputs[:, 0:1]
    u = outputs[:, 1:2]
    p = outputs[:, 2:3]
    
    # 定义物理参数
    e = p / (gamma - 1) + 0.5 * rho * u ** 2
    H = e + p / rho
    c = tf.sqrt(gamma * p / rho)
    M = u / c
    
    # 计算偏导数
    with tf.GradientTape(persistent=True) as tape:
        tape.watch(inputs)
        tape.watch(outputs)
        f = model(inputs)
        rho_f = f[:, 0:1]
        u_f = f[:, 1:2]
        p_f = f[:, 2:3]
        e_f = p_f / (gamma - 1) + 0.5 * rho_f * u_f ** 2
        H_f = e_f + p_f / rho_f
        c_f = tf.sqrt(gamma * p_f / rho_f)
        M_f = u_f / c_f
        
    rho_x = tape.gradient(rho, x)
    u_x = tape.gradient(u, x)
    p_x = tape.gradient(p, x)
    e_x = u_x * (e + p) - u * (rho * u_x + rho_x * u) + p_x
    H_x = (e_x + p_x) / rho + u_x * H - u * (rho * u_x + rho_x * u) / rho ** 2 * H
    c_x = 0.5 / c * (gamma * p_x / rho - gamma * p / rho ** 2 * rho_x)
    M_x = (u_x * c - u * c_x) / c ** 2
    
    # 计算损失
    loss = tf.reduce_mean((rho_f * u - rho * u_f) ** 2) \
         + tf.reduce_mean((rho_f * u_f ** 2 + p_f - rho * u * (u_f + M_f / M * c_f)) ** 2) \
         + tf.reduce_mean((rho_f * u_f * H_f - (e + p) * u_f + p_f * u - rho * u * H * (u_f + M_f / M * c_f)) ** 2) \
         + tf.reduce_mean((rho_f * u_f * M_f - rho * u * M) ** 2) \
         + tf.reduce_mean((rho_x * u - rho * u_x) ** 2) \
         + tf.reduce_mean((rho_x * u_x ** 2 + p_x - rho * u * (u_x + M_x / M * c_x)) ** 2) \
         + tf.reduce_mean((rho_x * u_x * H_x - (e + p) * u_x + p_x * u - rho * u * H * (u_x + M_x / M * c_x)) ** 2) \
         + tf.reduce_mean((rho_x * u_x * M_x - rho * u * M) ** 2) \
         + tf.reduce_mean((left_boundary(inputs) - initial_value(inputs)) ** 2) \
         + tf.reduce_mean((right_boundary(inputs) - f[-num_points:, :]) ** 2)
         
    return loss

# 定义训练函数
def train(model, inputs, outputs, learning_rate, num_epochs, batch_size):
    optimizer = tf.keras.optimizers.Adam(learning_rate=learning_rate)
    num_batches = int(inputs.shape[0] / batch_size)
    loss_history = []
    
    for epoch in range(num_epochs):
        inputs, outputs = shuffle_data(inputs, outputs)
        epoch_loss = 0.0
        
        for i in range(num_batches):
            x_batch = inputs[i*batch_size:(i+1)*batch_size, :]
            y_batch = outputs[i*batch_size:(i+1)*batch_size, :]
            
            with tf.GradientTape() as tape:
                loss = get_loss(model, x_batch, y_batch)
                
            grads = tape.gradient(loss, model.trainable_variables)
            optimizer.apply_gradients(zip(grads, model.trainable_variables))
            
            epoch_loss += loss / num_batches
        
        if epoch % 10 == 0:
            print('Epoch {}: Loss = {}'.format(epoch, epoch_loss))
        
        loss_history.append(epoch_loss)
        
    return loss_history

# 生成输入数据
x = np.linspace(x_left, x_right, num_points)
t = np.linspace(0, t_final, num_points)
x_mesh, t_mesh = np.meshgrid(x, t)
inputs = np.concatenate([x_mesh.reshape(-1, 1), t_mesh.reshape(-1, 1)], axis=1)

# 生成初始值和边界条件
outputs = initial_value(inputs)

# 训练模型
model = SodPINN(num_hidden_layers, num_neurons_per_layer)
loss_history = train(model, inputs, outputs, learning_rate, num_epochs=1000, batch_size=100)

# 绘制损失函数曲线
plt.plot(loss_history)
plt.xlabel('Epoch')
plt.ylabel('Loss')
plt.show()

# 可视化结果
rho = model(inputs)[:, 0].numpy().reshape(num_points, num_points)
u = model(inputs)[:, 1].numpy().reshape(num_points, num_points)
p = model(inputs)[:, 2].numpy().reshape(num_points, num_points)

plt.figure(figsize=(10, 6))
plt.subplot(1, 3, 1)
plt.imshow(rho, extent=[x_left, x_right, 0, t_final], cmap='jet')
plt.colorbar()
plt.xlabel('x')
plt.ylabel('t')
plt.title('Density')

plt.subplot(1, 3, 2)
plt.imshow(u, extent=[x_left, x_right, 0, t_final], cmap='jet')
plt.colorbar()
plt.xlabel('x')
plt.ylabel('t')
plt.title('Velocity')

plt.subplot(1, 3, 3)
plt.imshow(p, extent=[x_left, x_right, 0, t_final], cmap='jet')
plt.colorbar()
plt.xlabel('x')
plt.ylabel('t')
plt.title('Pressure')

plt.tight_layout()
plt.show()

需要注意的是，上述示例代码只是一个简单的 PINN 实现，可能需要根据具体问题进行修改和优化。另外，也可以结合其他方法（比如 Riemann 求解器）来进一步提升求解效果。