发表机构
University of Notre Dame（圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出特征跟踪PINN（FT-PINN），通过联合优化非线性变形流形与解网络，使配点沿任意几何激波集中，在有限配点预算下准确解析激波位置，优于普通PINN。
AI 中文摘要
物理信息神经网络（PINNs）在求解含激波的守恒律时，往往收敛到不精确的解，因为均匀分布的配点对局部特征欠采样，使得残差被已经很好解析的区域所主导。我们提出了一种特征跟踪PINN（FT-PINN），其中解网络定义在固定的参考域上，并与来自参数化非线性流形的微分同胚变形映射复合。通过最小化拉回（pulled-back）的守恒律残差，变形和网络参数被联合训练。这使得配点能够沿着任意几何形状的特征集中，包括弯曲、倾斜和合并的激波，而无需事先知道它们的位置。该框架对参数化的选择是无关的（agnostic）。边界保持通过位移的切向投影精确强制实现，折叠则通过雅可比行列式上的单侧惩罚来抑制。在四个测试问题（时空粘性Burgers方程合并激波、减速Burgers激波、时空Euler激波管和稳态二维Euler规则激波反射）上，FT-PINN在有限的配点预算下，在正确位置解析了激波。具有相同架构、预算和训练的普通PINN要么错置激波，要么无法形成激波。
英文摘要
Physics-informed neural networks (PINNs) often converge to inaccurate solutions for conservation laws with shocks, because uniformly distributed collocation points undersample localized features and let the residual be dominated by regions that are already well resolved. We propose a feature-tracking PINN (FT-PINN) in which the solution network is defined on a fixed reference domain and composed with a diffeomorphic deformation map from a parameterized nonlinear manifold. The deformation and solution-network parameters are trained jointly by minimizing the pulled-back conservation-law residual. This lets collocation points concentrate along features of essentially arbitrary geometry, including curved, oblique, and merging shocks, without prior knowledge of their locations. The framework is agnostic to the choice of parameterization. Boundary preservation is enforced exactly through a tangential projection of the displacement, and folding is discouraged by a one-sided penalty on the Jacobian determinant. On four test problems (space-time viscous Burgers with merging shocks, a decelerating Burgers shock, the space-time Euler shock tube, and steady 2D Euler regular shock reflection), FT-PINN resolves shocks at their correct locations with a limited collocation budget. A vanilla PINN with the same architecture, budget, and training either misplaces the shocks or fails to form them.
