Capturing Shock Waves by Relaxation Neural Networks

TL;DR

The paper presents RelaxNN, a relaxation-system–based extension of PINN designed to capture shocks in nonlinear hyperbolic conservation laws by learning both the macroscopic variable $\bm{u}$ and the relaxation variable $\bm{v}$ with separate neural networks. By replacing the stiff dissipative term with a flux-consistency loss, RelaxNN achieves stable, accurate shock capturing across Burgers', shallow water, and Euler equations, and demonstrates substantial improvements over standard PINN in a wide range of test problems. It also extends to uncertainty quantification, showing that RelaxNN can handle high-dimensional stochastic inputs while preserving accuracy. The approach preserves the simplicity and generality of PINN, enabling the reuse of existing sampling and training strategies and offering a scalable path to physics-informed learning for hyperbolic systems.

Abstract

In this paper, we put forward a neural network framework to solve the nonlinear hyperbolic systems. This framework, named relaxation neural networks(RelaxNN), is a simple and scalable extension of physics-informed neural networks(PINN). It is shown later that a typical PINN framework struggles to handle shock waves that arise in hyperbolic systems' solutions. This ultimately results in the failure of optimization that is based on gradient descent in the training process. Relaxation systems provide a smooth asymptotic to the discontinuity solution, under the expectation that macroscopic problems can be solved from a microscopic perspective. Based on relaxation systems, the RelaxNN framework alleviates the conflict of losses in the training process of the PINN framework. In addition to the remarkable results demonstrated in numerical simulations, most of the acceleration techniques and improvement strategies aimed at the standard PINN framework can also be applied to the RelaxNN framework.

Figures (22)

• Figure 1: Burgers' equation with Sine initial condition: training a PINN model for approximating the solution to the Burgers' equation with sine initial condition : (a)loss curves along the training process. (b) absolute error between the prediction of PINN and the reference solution
• Figure 2: An overview diagram for different improvement ideas
• Figure 3: Brief illustration about the relationship of different systems mentioned above. Here the dashed line represents the low-order approximation.
• Figure 4: The schematic diagram of the RelaxNN to solve the hyperbolic systems with initial condition (IC) $\bm{u}_{0}$ and flux function $\bm{F}(\cdot)$.
• Figure 5: Burgers' equation: Comparison of the loss curves of RelaxNN against the one of PINN during the training process: (a)Riemann initial condition. (b) Sine initial condition.