pytorch PINN求解欧拉方程的间断问题的预测解和真实解以及误差图的代码

以下是基于PyTorch的PINN求解欧拉方程的间断问题的预测解和真实解以及误差图的代码示例：

import torch
import torch.nn as nn
import torch.optim as optim
import numpy as np
import matplotlib.pyplot as plt

# 定义欧拉方程的PINN模型
class EulerPINN(nn.Module):
    def __init__(self):
        super(EulerPINN, self).__init__()
        self.fc1_u = nn.Linear(2, 20)
        self.fc2_u = nn.Linear(20, 20)
        self.fc3_u = nn.Linear(20, 1)
        self.fc1_f = nn.Linear(2, 20)
        self.fc2_f = nn.Linear(20, 20)
        self.fc3_f = nn.Linear(20, 1)
    
    def forward(self, x):
        u = torch.sin(self.fc1_u(x))
        u = torch.sin(self.fc2_u(u))
        u = self.fc3_u(u)
        f = torch.sin(self.fc1_f(x))
        f = torch.sin(self.fc2_f(f))
        f = self.fc3_f(f)
        return u, f
    
# 定义PINN求解欧拉方程的函数
def solve_euler_pinn(x, t, x_l, x_r, t_f, model, optimizer, max_epochs):
    loss_list = []
    for epoch in range(max_epochs):
        optimizer.zero_grad()
        x.requires_grad = True
        t.requires_grad = True
        x_l.requires_grad = True
        x_r.requires_grad = True
        t_f.requires_grad = True
        X = torch.cat([x, t], dim=1)
        U, F = model(X)
        U_x = torch.autograd.grad(U, x, grad_outputs=torch.ones_like(U), create_graph=True)[0]
        U_t = torch.autograd.grad(U, t, grad_outputs=torch.ones_like(U), create_graph=True)[0]
        F_x = torch.autograd.grad(F, x, grad_outputs=torch.ones_like(F), create_graph=True)[0]
        F_t = torch.autograd.grad(F, t, grad_outputs=torch.ones_like(F), create_graph=True)[0]
        loss = nn.MSELoss()(U_x + F, torch.zeros_like(U_x)) + \
               nn.MSELoss()(U_t + U * F_x, torch.zeros_like(U_t)) + \
               nn.MSELoss()(model(torch.cat([x_l, t_f], dim=1))[0], torch.zeros_like(x_l)) + \
               nn.MSELoss()(model(torch.cat([x_r, t_f], dim=1))[0], torch.ones_like(x_r))
        loss_list.append(loss.item())
        loss.backward()
        optimizer.step()
        if epoch % 100 == 0:
            print('Epoch [{}/{}], Loss: {:.4f}'.format(epoch+1, max_epochs, loss.item()))
    return loss_list

# 定义真实解函数
def true_solution(x, t):
    u = np.zeros((x.shape[0], t.shape[0]))
    for i in range(x.shape[0]):
        for j in range(t.shape[0]):
            if x[i, 0] >= 0 and x[i, 0] <= 0.5*t[j]:
                u[i, j] = x[i, 0] / t[j]
            else:
                u[i, j] = 1
    return u

# 定义画图函数
def plot_results(x, t, u_pred, u_true):
    plt.figure(figsize=(10, 10))
    plt.subplot(2, 1, 1)
    plt.pcolor(x, t, u_pred, cmap='jet')
    plt.colorbar()
    plt.xlabel('x')
    plt.ylabel('t')
    plt.title('Predicted solution')
    plt.subplot(2, 1, 2)
    plt.pcolor(x, t, u_true, cmap='jet')
    plt.colorbar()
    plt.xlabel('x')
    plt.ylabel('t')
    plt.title('True solution')
    plt.show()
    
# 设置模型参数和求解参数
model = EulerPINN()
optimizer = optim.Adam(model.parameters(), lr=0.001)
max_epochs = 10000

# 设置求解区域和边界条件
x_l = torch.linspace(0, 1, 100).reshape(-1, 1)
x_r = torch.linspace(0, 1, 100).reshape(-1, 1)
t_f = torch.linspace(0, 1, 100).reshape(-1, 1)
x = torch.linspace(0, 1, 100).reshape(-1, 1)
t = torch.linspace(0, 1, 100).reshape(-1, 1)

# 求解欧拉方程
loss_list = solve_euler_pinn(x, t, x_l, x_r, t_f, model, optimizer, max_epochs)

# 计算预测解和真实解并画图
X, T = np.meshgrid(x.detach().numpy().flatten(), t.detach().numpy().flatten())
U_pred = model(torch.cat([x, t], dim=1))[0].detach().numpy().reshape(-1, 1)
U_true = true_solution(np.concatenate([X.reshape(-1, 1), T.reshape(-1, 1)], axis=1), T.reshape(-1, 1))
U_pred = np.transpose(U_pred.reshape((x.shape[0], t.shape[0])))
U_true = np.transpose(U_true.reshape((x.shape[0], t.shape[0])))
plot_results(X, T, U_pred, U_true)

# 画出误差图
plt.figure(figsize=(10, 5))
plt.plot(loss_list)
plt.xlabel('Epoch')
plt.ylabel('Loss')
plt.title('PINN loss')
plt.show()

在上面的代码中，我们定义了一个名为`EulerPINN`的类，该类继承自`nn.Module`，并包含了欧拉方程的PINN模型。我们还定义了一个名为`solve_euler_pinn`的函数，该函数使用PINN求解欧拉方程，并返回损失列表。真实解函数`true_solution`用于计算欧拉方程的真实解。`plot_results`函数用于将预测解和真实解可视化。在主函数中，我们首先设置模型参数和求解参数。然后，我们定义求解区域和边界条件。接下来，我们使用`solve_euler_pinn`函数求解欧拉方程，并计算预测解和真实解。最后，我们使用`plot_results`函数将预测解和真实解可视化，并画出损失图。