python PINN求解一维Sod的预测解并给出精确解以及代码

Sod问题是一个经典的气体动力学问题，它描述了一个强激波穿过气体的过程。在这里，我们将使用物理信息神经网络（PINN）来预测一维Sod问题的解，并将其与精确解进行比较。

首先，定义问题的初始和边界条件：

$$\begin{aligned} \rho(x,0) &= \begin{cases} 1 & x<0.5 \\ 0.125 & x\geq 0.5 \end{cases} \\ u(x,0) &= 0 \\ p(x,0) &= \begin{cases} 1 & x<0.5 \\ 0.1 & x\geq 0.5 \end{cases} \\ \rho(0,t) &= 1 \\ u(0,t) &= 0 \\ p(0,t) &= 1 \\ \rho(1,t) &= 0.125 \\ u(1,t) &= 0 \\ p(1,t) &= 0.1 \end{aligned} $$

接下来，我们将使用PINN来求解这个问题。首先，我们需要导入必要的库：

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

然后，我们定义网络的结构和损失函数：

class SodPINN(tf.keras.Model):
    def __init__(self):
        super(SodPINN, self).__init__()

        self.dense1 = tf.keras.layers.Dense(50, activation=tf.nn.tanh, input_shape=(1,))
        self.dense2 = tf.keras.layers.Dense(50, activation=tf.nn.tanh)
        self.dense3 = tf.keras.layers.Dense(50, activation=tf.nn.tanh)
        self.dense4 = tf.keras.layers.Dense(50, activation=tf.nn.tanh)
        self.dense5 = tf.keras.layers.Dense(50, activation=tf.nn.tanh)
        self.dense6 = tf.keras.layers.Dense(50, activation=tf.nn.tanh)
        self.dense7 = tf.keras.layers.Dense(3)

    def call(self, inputs):
        x = inputs
        x = self.dense1(x)
        x = self.dense2(x)
        x = self.dense3(x)
        x = self.dense4(x)
        x = self.dense5(x)
        x = self.dense6(x)
        return self.dense7(x)

def pinn_loss(model, x, rho, u, p, gamma):
    with tf.GradientTape(persistent=True) as g:
        g.watch(x)

        # Forward pass
        inputs = tf.concat([x, rho, u, p], axis=1)
        pred = model(inputs)

        # Extracting predictions
        rho_pred = pred[:, 0:1]
        u_pred = pred[:, 1:2]
        p_pred = pred[:, 2:3]

        # Computing gradients
        drho_dx = g.gradient(rho_pred, x)
        du_dx = g.gradient(u_pred, x)
        dp_dx = g.gradient(p_pred, x)

    # Computing residual terms
    rho_res = drho_dx + rho_pred * (u_pred - u_pred[:, 0:1]) / (x - x[:, 0:1])
    u_res = du_dx + (p_pred - p_pred[:, 0:1]) / (rho_pred * (x - x[:, 0:1])) - (gamma - 1) / gamma * (u_pred - u_pred[:, 0:1]) * (dp_dx / p_pred)
    p_res = dp_dx + gamma * p_pred * (u_pred - u_pred[:, 0:1]) / (x - x[:, 0:1])

    # Computing loss
    loss = tf.reduce_mean(tf.square(rho_res) + tf.square(u_res) + tf.square(p_res))

    return loss

接下来，我们将定义一些必要的变量，并初始化模型：

# Defining constants
gamma = 1.4
N = 10000
x = np.linspace(0, 1, N)[:, None]
rho = np.zeros((N, 1))
u = np.zeros((N, 1))
p = np.zeros((N, 1))
for i in range(N):
    if x[i] < 0.5:
        rho[i] = 1
        u[i] = 0
        p[i] = 1
    else:
        rho[i] = 0.125
        u[i] = 0
        p[i] = 0.1

# Initializing model
model = SodPINN()
optimizer = tf.keras.optimizers.Adam(learning_rate=1e-3)

然后，我们将训练模型：

# Training model
for epoch in range(10000):
    with tf.GradientTape() as tape:
        loss = pinn_loss(model, x, rho, u, p, gamma)

    grads = tape.gradient(loss, model.trainable_variables)
    optimizer.apply_gradients(zip(grads, model.trainable_variables))

    if epoch % 1000 == 0:
        print("Epoch: {}, Loss: {}".format(epoch, loss.numpy()))

最后，我们将绘制预测解和精确解的比较图：

# Computing predictions
inputs = tf.concat([x, rho, u, p], axis=1)
pred = model(inputs)

rho_pred = pred[:, 0:1].numpy()
u_pred = pred[:, 1:2].numpy()
p_pred = pred[:, 2:3].numpy()

# Computing exact solution
rho_exact = np.zeros((N, 1))
u_exact = np.zeros((N, 1))
p_exact = np.zeros((N, 1))
for i in range(N):
    if x[i] < 0.5:
        rho_exact[i] = 1
        u_exact[i] = 0
        p_exact[i] = 1
    else:
        rho_exact[i] = 0.125
        u_exact[i] = 0
        p_exact[i] = 0.1

# Plotting results
plt.figure(1)
plt.plot(x, rho_pred, 'r-', label='PINN')
plt.plot(x, rho_exact, 'k--', label='Exact')
plt.xlabel('x')
plt.ylabel('Density')
plt.legend()

plt.figure(2)
plt.plot(x, u_pred, 'r-', label='PINN')
plt.plot(x, u_exact, 'k--', label='Exact')
plt.xlabel('x')
plt.ylabel('Velocity')
plt.legend()

plt.figure(3)
plt.plot(x, p_pred, 'r-', label='PINN')
plt.plot(x, p_exact, 'k--', label='Exact')
plt.xlabel('x')
plt.ylabel('Pressure')
plt.legend()

plt.show()

完整代码如下：