用残差的海森矩阵指导采样点选择，提升PINNs求解PDE的精度
Méthode de quadrature pour les PINNs fondée théoriquement sur la hessienne des résiduels
- 基于残差海森矩阵设计新积分采样方法
- 采样点自适应聚焦于高误差区域，提升收敛性
- 适用于对精度要求高的物理方程求解场景
物理信息神经网络（PINNs）通过将物理模型嵌入损失函数，并利用自动微分在称为采样点的位置最小化残差，高效学习偏微分方程（PDE）的代理神经求解器。最初采样点为均匀分布，近年来研究致力于自适应优化采样策略。本文提出一种基于函数海森矩阵的新积分近似方法，并将其用于训练过程中采样点的选择，以更精准地捕捉解的复杂变化特征。
原文摘要 · Abstract (English)
Physics-informed Neural Networks (PINNs) have emerged as an efficient way to learn surrogate neural solvers of PDEs by embedding the physical model in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements. In this paper, we propose a new quadrature method for approximating definite integrals based on the hessian of the considered function, and that we leverage to guide the selection of the collocation points during the training process of PINNs.
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