用PINN求解arctanx的二阶偏导数的代码

PINN（Physics-Informed Neural Networks）是一种结合了机器学习和数值计算的方法，用于解决包含物理约束的问题。在这里，我们想用PINN来求解函数 arctan(x) 的二阶偏导数。首先，我们需要定义一个神经网络模型，然后定义损失函数，包括数据点误差、物理方程和边界条件。

以下是一个简单的Python示例，使用PyTorch库和torchdiffeq模块：

import torch
from torch import nn
import torch.nn.functional as F
from torchdiffeq import odeint

# 定义神经网络模型
class ArcTanNet(nn.Module):
    def __init__(self, n_hidden=64):
        super(ArcTanNet, self).__init__()
        self.l1 = nn.Linear(1, n_hidden)
        self.relu = nn.ReLU()
        self.l2 = nn.Linear(n_hidden, n_hidden)
        self.out = nn.Linear(n_hidden, 1)

    def forward(self, x):
        x = self.l1(x)
        x = self.relu(x)
        x = self.l2(x)
        out = self.out(x)
        return out

# 定义物理方程和边界条件
def physics_law(t, y):
    x = y[:, 0]
    dydt = -1 / (1 + x**2)
    return dydt

# 数据点
x_data = torch.linspace(-10, 10, 1000).reshape(-1, 1)
y_data = torch.atan(x_data)

# 初始化神经网络并设置优化器
net = ArcTanNet()
optimizer = torch.optim.Adam(net.parameters(), lr=0.001)

# 训练过程
for epoch in range(1000):  # 可能需要调整迭代次数
    t = torch.tensor([0], dtype=torch.float32)
    y0 = torch.zeros_like(x_data)
    sol = odeint(physics_law, y0, t, method='rk4', rtol=1e-3, atol=1e-3)

    # 损失函数：数据拟合 + 物理方程满足度
    data_loss = F.mse_loss(sol[-1], y_data)
    phys_loss = F.mse_loss(sol, y_data)
    loss = data_loss + phys_loss

    optimizer.zero_grad()
    loss.backward()
    optimizer.step()

# 输出神经网络的近似结果
with torch.no_grad():
    x_test = torch.linspace(-10, 10, 100).reshape(-1, 1)
    arc_tan_approx = net(x_test)
    second_derivative = -2 * net(lin_interpolate(torch.tensor([-1, 1]), x_test))

# 极限处理二阶导数
second_derivative[abs(x_test) == 1] = 0

print(f"Approximated second derivative at x: {second_derivative.item()}")