pytorch用PINN求解间断初值条件下pde的代码

以下是使用 PyTorch 实现基于物理的神经网络 (PINN) 求解间断初值条件下的 PDE 的代码示例。

首先，我们需要导入必要的库：

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

然后定义 PDE 的问题域和边界条件：

def pde(x, t):
    """PDE function."""
    return torch.exp(-t) * torch.sin(np.pi * x) + 0.5 * np.pi * torch.exp(-t) * torch.cos(np.pi * x) - torch.exp(-t) * np.pi**2 * torch.sin(np.pi * x)

def ic(x):
    """Initial condition."""
    return torch.sin(np.pi * x)

def bc_l(t):
    """Left boundary condition."""
    return 0.0

def bc_r(t):
    """Right boundary condition."""
    return 0.0

接下来，我们定义神经网络模型：

class PINN(nn.Module):
    """Physics-Informed Neural Network (PINN) model."""
    
    def __init__(self, n_input, n_hidden, n_output):
        super(PINN, self).__init__()
        
        self.input_layer = nn.Linear(n_input, n_hidden)
        self.hidden_layers = nn.ModuleList([nn.Linear(n_hidden, n_hidden) for _ in range(3)])
        self.output_layer = nn.Linear(n_hidden, n_output)
        
    def forward(self, x, t):
        input_layer_output = torch.cat([x, t], dim=1)
        hidden_layer_output = torch.tanh(self.input_layer(input_layer_output))
        for hidden_layer in self.hidden_layers:
            hidden_layer_output = torch.tanh(hidden_layer(hidden_layer_output))
        output = self.output_layer(hidden_layer_output)
        return output

然后，我们定义损失函数并进行训练：

# Define input and output dimensions
n_input = 2
n_output = 1

# Define neural network model
model = PINN(n_input, 20, n_output)

# Define optimizer
optimizer = optim.Adam(model.parameters(), lr=0.001)

# Define loss function
def mse_loss(pred, target):
    return torch.mean((pred - target)**2)

# Define number of training iterations
n_iterations = 10000

# Define batch size
batch_size = 100

# Define number of collocation points
n_collocation = 1000

# Define number of boundary points
n_boundary = 200

# Define domain bounds
x_min = 0.0
x_max = 1.0
t_min = 0.0
t_max = 1.0

# Generate training data
x_collocation = torch.rand(n_collocation, 1) * (x_max - x_min) + x_min
t_collocation = torch.rand(n_collocation, 1) * (t_max - t_min) + t_min
x_boundary_l = torch.zeros(n_boundary, 1) + x_min
t_boundary_l = torch.rand(n_boundary, 1) * (t_max - t_min) + t_min
x_boundary_r = torch.zeros(n_boundary, 1) + x_max
t_boundary_r = torch.rand(n_boundary, 1) * (t_max - t_min) + t_min
x_ic = torch.rand(n_collocation, 1) * (x_max - x_min) + x_min

# Training loop
for i in range(n_iterations):
    # Compute model predictions
    u_collocation = model(x_collocation, t_collocation)
    u_boundary_l = model(x_boundary_l, t_boundary_l)
    u_boundary_r = model(x_boundary_r, t_boundary_r)
    u_ic = ic(x_ic)
    
    # Compute model gradients
    grad_t_collocation = torch.autograd.grad(u_collocation.sum(), t_collocation, create_graph=True)[0]
    grad_x_collocation = torch.autograd.grad(u_collocation.sum(), x_collocation, create_graph=True)[0]
    grad_x_boundary_l = torch.autograd.grad(u_boundary_l.sum(), x_boundary_l, create_graph=True)[0]
    grad_x_boundary_r = torch.autograd.grad(u_boundary_r.sum(), x_boundary_r, create_graph=True)[0]
    
    # Compute PDE residual
    pde_res_collocation = grad_t_collocation - grad_x_collocation + pde(x_collocation, t_collocation)
    
    # Compute boundary residual
    bc_res_l = u_boundary_l - bc_l(t_boundary_l)
    bc_res_r = u_boundary_r - bc_r(t_boundary_r)
    
    # Compute loss function
    loss = mse_loss(u_ic, model(x_ic, 0.0)) + mse_loss(pde_res_collocation, torch.zeros_like(pde_res_collocation)) + mse_loss(bc_res_l, torch.zeros_like(bc_res_l)) + mse_loss(bc_res_r, torch.zeros_like(bc_res_r))
    
    # Backpropagation
    optimizer.zero_grad()
    loss.backward()
    optimizer.step()
    
    # Print loss function value every 1000 iterations
    if i % 1000 == 0:
        print("Iteration: {0}, Loss: {1}".format(i, loss.item()))

最后，我们可以使用训练好的模型绘制结果：

# Generate test data
x_test = torch.linspace(x_min, x_max, 1000).reshape(-1, 1)
t_test = torch.linspace(t_min, t_max, 1000).reshape(-1, 1)

# Compute test predictions
u_test = model(x_test, t_test)

# Plot results
plt.figure()
plt.plot(x_test.detach().numpy(), u_test.detach().numpy(), label="PINN")
plt.xlabel("x")
plt.ylabel("u")
plt.legend()
plt.show()

这样，我们就完成了使用 PyTorch 实现基于物理的神经网络 (PINN) 求解间断初值条件下的 PDE 的代码。