将物理硬约束引入神经网络，精准求解非线性流体方程。
Hard Constraint Projection in a Physics Informed Neural Network
- 通过不可压缩流体的流函数与压力联合建模
- 设计不可学习的投影层，强制满足离散化方程
- 适用于高精度流体模拟，适合工程仿真场景
本文将硬约束方法扩展至二维不可压缩纳维-斯托克斯方程这一强非线性偏微分方程。采用物理信息神经网络（PINN）估计流体的流函数与压力，通过对流函数求导恢复无散速度场。引入一个不可学习的硬约束投影（HCP）层，将预测的速度和压力投影到仅包含精确离散化方程解的超平面上，确保数值解严格满足控制方程。
原文摘要 · Abstract (English)
In this work, we embed hard constraints in a physics informed neural network (PINN) which predicts solutions to the 2D incompressible Navier Stokes equations. We extend the hard constraint method introduced by Chen et al. (arXiv:2012.06148) from a linear PDE to a strongly non-linear PDE. The PINN is used to estimate the stream function and pressure of the fluid, and by differentiating the stream function we can recover an incompressible velocity field. An unlearnable hard constraint projection (HCP) layer projects the predicted velocity and pressure to a hyperplane that admits only exact solutions to a discretised form of the governing equations.
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