Curriculum-Learned Vanishing Stacked Residual PINNs for Hyperbolic PDE State Reconstruction

TL;DR

This paper introduces an enhanced vanishing stacked residual PINN framework with curriculum learning techniques to improve the accuracy and convergence of hyperbolic PDE state reconstruction, especially in systems with shocks and discontinuities.

Contribution

It integrates three curriculum-learning methods into VSR-PINN, significantly improving its performance in hyperbolic PDE modeling over previous approaches.

Findings

Causality progression reduces median point-wise MSE by nearly tenfold.

Adaptive sampling effectively targets high residual regions.

Numerical experiments show improved accuracy and stability in traffic reconstruction.

Abstract

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.