HCP-PINN：用硬约束投影层求解一维 Burgers 方程
Hard Constraint Projection in a Physics Informed Neural Network (Burgers)
给 PINN 加一层不可学习的投影，误差直接低一个数量级，做物理机器学习的可以看看这个思路。
该论文将 Chen 等人 2021 年在 Journal of Computational Physics 上提出的硬约束投影（HCP）方法从线性 PDE 扩展到非线性的一维 Burgers 方程。方法先用有限差分把 Burgers 方程离散为约束矩阵和预测矩阵，再在 PINN 末端加一个不可学习的投影层，把速度预测投射到离散方程精确解的超平面上。实验显示 HCP-PINN 在 PINN 损失趋于平台后仍继续下降，在未见过的均匀网格上数据误差比可比 PINN 低约一个数量级。作者还观察到 HCP 层让网络更严格遵守控制方程，并对配点选择和数量不那么敏感。
Hard Constraint Projection in a Physics Informed Neural Network (Burgers)
Physics informed machine learning provides a framework to study and simulate fluid dynamics, potentially at a fraction of the computational cost of traditional numerical methods, with the goal of improving the explainability, accuracy and robustness of purely data driven machine learning methods. In this work, we extend the hard constraint projection (HCP) method developed by Chen et al. [Journal of Computational Physics, 445, 110624, 2021] from a linear partial differential equation (PDE) to a nonlinear PDE: the 1D Burgers' equation. To strictly impose a discrete approximation of the governing equation, the Burgers' equation is first locally discretised (using a finite difference stencil) then decomposed into a constraint matrix and a prediction matrix. Using tools from linear algebra, the constraint matrix is used to construct a projection matrix. A physics informed neural network (PINN) [Raissi et al. Journal of Computational Physics, 378, 686, 2019] is used to predict the velocity, and a final unlearnable layer projects the velocity prediction to the hyperplane of exact solutions to a discretised form of the Burgers' equation. Our initial results indicate that the HCP-PINN initially trains similarly to an equivalent PINN, and as the PINN loss plateaus, the HCP-PINN loss continues to decline. When tested on a previously unseen uniform grid of points covering the domain, the HCP-PINN exhibits higher accuracy, converging to a data error approximately an order of magnitude lower than the comparable PINN. We also observe that the inclusion of the HCP layer results in better observance of the governing PDE. Our findings suggest that the HCP-PINN can be more robust to the selection of physics collocation points, and less dependent on collocation point quantity.