通过因果性训练提升双曲型PDE状态重建精度，误差降低近一个数量级。
Curriculum-Learned Vanishing Stacked Residual PINNs for Hyperbolic PDE State Reconstruction
- 引入因果推进策略，按时间与梯度演化顺序训练网络
- 在交通重建任务中，点对点均方误差降低近一个数量级
- 适合需要高精度物理模拟的科学计算与工程建模场景
由双曲型偏微分方程（PDE）支配的分布式动力系统建模仍具挑战性，因间断与激波会阻碍传统物理信息神经网络（PINNs）的收敛。近期提出的消失粘性堆叠残差PINN（VSR-PINN）在堆叠残差优化中嵌入消失粘性机制，实现从抛物型到双曲型的平滑过渡。本文将三种课程学习方法——对偶优化、因果推进与自适应采样——整合进VSR-PINN：对偶策略平衡物理与数据损失，因果方案通过尊重时间与梯度演化实现更深网络堆叠，自适应采样聚焦高残差区域。在交通重建的数值实验中，强制因果性系统性地降低了中位点对点均方误差及其跨运行的变异性，相较于非因果训练，在基线与对偶优化变体中均实现近一个数量级的提升。
原文摘要 · Abstract (English)
Modeling distributed dynamical systems governed by hyperbolic partial differential equations (PDEs) remains challenging due to discontinuities and shocks that hinder the convergence of traditional physics-informed neural networks (PINNs). The recently proposed vanishing stacked residual PINN (VSR-PINN) embeds a vanishing-viscosity mechanism within stacked residual refinements to enable a smooth transition from the parabolic to hyperbolic regime. This paper integrates three curriculum-learning methods as primal-dual (PD) optimization, causality progression, and adaptive sampling into the VSR-PINN. The PD strategy balances physics and data losses, the causality scheme unlocks deeper stacks by respecting temporal and gradient evolution, and adaptive sampling targets high residuals. Numerical experiments on traffic reconstruction confirm that enforcing causality systematically reduces the median point-wise MSE and its variability across runs, yielding improvements of nearly one order of magnitude over non-causal training in both the baseline and PD variants.
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