提出新方法强加边界条件，提升物理神经算子的稳定性与精度。
Enforcing boundary conditions for physics-informed neural operators
- 基于正交投影重构解结构，实现强边界条件精确满足。
- 在非光滑边界上仍保持稳定，避免传统方法的发散问题。
- 适用于达西流与纳维-斯托克斯方程，适合高精度物理模拟场景。
基于机器学习的物理信息神经网络和物理信息神经算子在求解复杂偏微分方程系统方面日益成熟。边界条件可弱化通过损失函数惩罚偏差，或强化通过构造满足预设值和导数的解结构来实现。前者实现简单，后者在精度和训练时间上更具优势。然而，此前对诺伊曼或罗宾边界条件的强加方法要求区域具有完全 $C^1$ 边界；我们证明，当边界为分段 $C^1$ 但整体仅 $C^0$ 时，该方法会导致不稳定。本文推广了 Sukumar & Srivastava（doi: 10.1016/j.cma.2021.114333）的方法，并提出一种基于正交投影的新方法以克服此限制。新方法在标量达西流方程和定常纳维-斯托克斯方程上与弱化及半弱化边界条件进行对比，验证其有效性。
原文摘要 · Abstract (English)
Machine-learning based methods like physics-informed neural networks and physics-informed neural operators are becoming increasingly adept at solving even complex systems of partial differential equations. Boundary conditions can be enforced either weakly by penalizing deviations in the loss function or strongly by training a solution structure that inherently matches the prescribed values and derivatives. The former approach is easy to implement but the latter can provide benefits with respect to accuracy and training times. However, previous approaches to strongly enforcing Neumann or Robin boundary conditions require a domain with a fully $C^1$ boundary and, as we demonstrate, can lead to instability if those boundary conditions are posed on a segment of the boundary that is piecewise $C^1$ but only $C^0$ globally. We introduce a generalization of the approach by Sukumar \& Srivastava (doi: 10.1016/j.cma.2021.114333), and a new approach based on orthogonal projections that overcome this limitation. The performance of these new techniques is compared against weakly and semi-weakly enforced boundary conditions for the scalar Darcy flow equation and the stationary Navier-Stokes equations.
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