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

TL;DR

The paper tackles reconstructing the entropic solution of a 1D quasilinear hyperbolic PDE from limited noisy data, a setting where shocks challenge standard PINNs. It introduces Curriculum-Learned Vanishing Stacked Residual PINNs (C-VSR-PINN) that couple a vanishing-viscosity pathway with three curriculum components: stack-wise Primal-Dual optimization that adapts physics-data weighting via $\boldsymbol{\lambda}$; stack-wise causality that modulates residual training over time and stacks using $W(t)$, $g_i^k$, and $\omega_i^k$; and adaptive sampling that focuses collocation points where the PDE residuals are high. On traffic-reconstruction tests with the LWR model and Greenshields flux, the causal curricula reduce the median point-wise MSE and its variability by about an order of magnitude relative to non-causal baselines, with the final stacked solution $\hat{u}^{(n)}$ closely matching the entropic solution. The work demonstrates improved convergence robustness and fidelity in the hyperbolic regime and points to extensions to higher dimensions and real-world sensor data.

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.

Figures (3)

• Figure 1: Point-wise MSE distribution for all VSR-PINN variants over three independent runs, comparing causal and non-causal training.
• Figure 2: Reconstruction of the traffic state using the VSR-PINN equipped with causality and PD.
• Figure 3: Distribution of the residual block $\alpha_1 \mathcal{N}(\cdot; \theta_1)$..