PINNs 入门：物理信息神经网络的核心思想

什么是 PINNs#

物理信息神经网络（Physics-Informed Neural Networks, PINNs）由 Raissi et al. (2019) 提出，其核心思想是将物理定律（以 PDE 形式）嵌入神经网络的损失函数中。

数学框架#

考虑一般的 PDE 问题：

其中 $\mathcal{N}$ 是微分算子，$\mathcal{B}$ 是边界条件算子。

1
import torch
2

3
def pde_residual(model, x):
4
    """
5
    计算 PDE 残差
6
    model: 神经网络 u_theta(x)
7
    x: 配置点 [N, d]
8
    """
9
    x.requires_grad_(True)
10
    u = model(x)
11

12
    # 一阶导数
13
    du_dx = torch.autograd.grad(
14
        u, x,
15
        grad_outputs=torch.ones_like(u),
16
        create_graph=True
17
    )[0]
18

19
    # 二阶导数
20
    d2u_dx2 = torch.autograd.grad(
21
        du_dx, x,
22
        grad_outputs=torch.ones_like(du_dx),
23
        create_graph=True
24
    )[0]
25

26
    # 对于 Poisson 方程: -Δu = f
27
    residual = -d2u_dx2.sum(dim=1) - f(x)
28
    return residual

简单示例：1D Poisson 方程#

求解 $-u''(x) = \pi^2 \sin(\pi x), x \in [0, 1]$，边界条件 $u(0) = u(1) = 0$。

解析解：$u(x) = \sin(\pi x)$。

1
import torch
2
import torch.nn as nn
3

4
class PINN(nn.Module):
5
    def __init__(self, layers):
6
        super().__init__()
7
        self.net = nn.Sequential()
8
        for i in range(len(layers) - 1):
9
            self.net.append(nn.Linear(layers[i], layers[i+1]))
10
            if i < len(layers) - 2:
11
                self.net.append(nn.Tanh())
12

13
    def forward(self, x):
14
        return self.net(x)
15

16
# 网络结构: [1, 32, 32, 32, 1]
17
model = PINN([1, 32, 32, 32, 1])
18

19
# 训练配置
20
optimizer = torch.optim.Adam(model.parameters(), lr=1e-3)
21
lambda_phys = 1.0

PINNs 的优势与挑战#

参考文献#

1. Raissi, M., Perdikaris, P., & Karniadakis, G. E. “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.” Journal of Computational Physics, 2019.
2. Lu, L., Meng, X., Mao, Z., & Karniadakis, G. E. “DeepXDE: A deep learning library for solving differential equations.” SIAM Review, 2021.
3. Karniadakis, G. E., et al. “Physics-informed machine learning.” Nature Reviews Physics, 2021.