A Two-stage Adaptive Lifting PINN Framework for Solving Viscous Approximations to Hyperbolic Conservation Laws

TL;DR

This paper introduces a two-stage adaptive lifting PINN framework that enhances the training stability and accuracy of neural networks solving hyperbolic conservation laws with small viscosity, especially near shock discontinuities.

Contribution

The paper proposes a novel two-stage adaptive lifting PINN that uses learned coordinate transformations to improve training stability and accuracy without prior interface knowledge.

Findings

Accelerated convergence and improved stability in training.

More accurate reconstructions near discontinuities.

Theoretical insights into error and variance reduction mechanisms.

Abstract

Training physics informed neural networks PINNs for hyperbolic conservation laws near the inviscid limit presents considerable difficulties because strong form residuals become ill posed at shock discontinuities, while small viscosity regularization introduces narrow boundary layers that exacerbate spectral bias. To address these issues this paper proposes a novel two stage adaptive lifting PINN, a lifting based framework designed to mitigate such challenges without requiring a priori knowledge of the interface geometry. The key idea is to augment the physical coordinates by introducing a learned auxiliary field generated through r adaptive coordinate transformations. Theoretically we first derive an a posteriori L2 error estimate to quantify how training difficulty depends on viscosity. Secondly we provide a statistical interpretation revealing that embedded sampling induces variance reduction analogous to importance sampling. Finally we perform an NTK and gradient flow analysis, demonstrating that input augmentation improves conditioning and accelerates residual decay. Supported by these insights our numerical experiments show accelerated and more stable convergence as well as accurate reconstructions near discontinuities.