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

Yameng Zhu, Weibing Deng, Ran Bi

TL;DR

The paper tackles training physics-informed neural networks for hyperbolic conservation laws in the inviscid limit, where shocks and narrow viscous layers impede learning. It introduces TAL-PINN, a two-stage adaptive lifting framework that learns geometry-aware, $r$-adaptive coordinates to serve as the lifting variable, enabling PDE residuals to be enforced on a lifted manifold $\mathcal{M}$ and reducing spectral bias without prior interface knowledge. A posteriori $L^2$ error estimates, an importance-sampling interpretation of residuals, and a gradient-flow NTK analysis establish the theoretical benefits of lifting for training stability and faster convergence. Numerical experiments on 1D/2D Burgers and 1D Euler Lax shock-tube demonstrate accelerated convergence, improved NTK conditioning, and accurate reconstruction of shocks and discontinuities as viscosity decreases, highlighting TAL-PINN’s practical impact for inviscid regimes.

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.

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

TL;DR

The paper tackles training physics-informed neural networks for hyperbolic conservation laws in the inviscid limit, where shocks and narrow viscous layers impede learning. It introduces TAL-PINN, a two-stage adaptive lifting framework that learns geometry-aware, -adaptive coordinates to serve as the lifting variable, enabling PDE residuals to be enforced on a lifted manifold and reducing spectral bias without prior interface knowledge. A posteriori error estimates, an importance-sampling interpretation of residuals, and a gradient-flow NTK analysis establish the theoretical benefits of lifting for training stability and faster convergence. Numerical experiments on 1D/2D Burgers and 1D Euler Lax shock-tube demonstrate accelerated convergence, improved NTK conditioning, and accurate reconstruction of shocks and discontinuities as viscosity decreases, highlighting TAL-PINN’s practical impact for inviscid regimes.

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.

Paper Structure

This paper contains 19 sections, 2 theorems, 74 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic conservation laws and viscous regularization
  4. Physics-Informed Neural Networks (PINNs)
  5. Lifting-based strategies
  6. Proposed method
  7. Combining the lifting representation and viscous regularization
  8. $R$-adaptivity and the equidistribution principle
  9. Using $r$-adaptive coordinates as the lifting variable
  10. Theoretical analysis
  11. A posteriori error estimate
  12. Importance sampling to reduce statistical error
  13. Gradient flow perspective: lifting accelerates training
  14. Numerical experiments
  15. 1D Burgers equation with a stationary shock
  16. ...and 4 more sections

Key Result

Lemma 4.1

Suppose $u \in L^2(0,T; H^1(\Omega))$, $u_t \in L^2(0,T; H^{-1}(\Omega))$, and $\mathcal{R}(u) \in L^2(0,T; L^2(\Omega)).$ Then, for all $t\in[0,T]$, the error $e:=u^{\nu}-u$ satisfies where $C_P > 0$ is the Poincaré constant depending only on $\Omega$ and the boundary condition, and $L_f := \sup_{s \in I} \left\| f'(s) \right\|_{\mathbb{R}^d}$ with $I := \mathrm{range}(u) \cup \mathrm{range}(u^{

Figures (12)

  • Figure 1: Illustration of the Heaviside function $H(x - 0.5)$ and its lifting representation $U(x, z)$ with $z = H(x - 0.5)$.
  • Figure 2: Left: manually constructed auxiliary variable $z(x)$. Right: adaptive coordinate $\xi(x)$ obtained from a Burgers solution.
  • Figure 3: Visualization of the reference solution and the adaptive coordinate transformation: (a) reference solution $u^*(t,x)$; (b) adaptive sampling points in the physical domain, colored by the corresponding value of the adaptive coordinate $\xi(t,x)$; (c) profiles of $u^*$ and $\xi$ at $t=1.0$.
  • Figure 4: Comparison among the three training strategies. (a) Eigenvalue spectra of the total NTK $K^*$; (b) eigenvalue spectra of the residual NTK $K_{r,r}^*$; (c) evolution of the $L^2$ test error during training.
  • Figure 5: Spatio–temporal distribution of $|\mathcal{R}(t,x)|$: strong localization along the (vertical) shock trajectory at $x=0.5$, supporting adaptive (importance) sampling near the discontinuity as predicted by Section \ref{['sec:sec42']}.
  • ...and 7 more figures

Theorems & Definitions (4)

  • Lemma 4.1
  • proof
  • Theorem 4.2
  • proof