Physical Informed Neural Network for Solving Conservation Laws Based on Relaxation Systems

Abstract

Solving partial differential equations (PDEs) with discontinuous solutions, such as shock waves in multiphase viscous flows through porous media, is critical for a wide range of scientific and engineering applications. These discontinuities represent sudden changes in physical quantities. In recent years, physics-informed neural networks (PINNs) have emerged as a promising method for solving PDEs but face significant challenges when modeling such problems. Specifically, neural networks struggle to compute gradients accurately near shock waves, leading to solutions that deviate from true physical phenomena. To address this issue, we propose a novel relaxation neural network method based on the conservation law relaxation model and its improved version, the relaxation limit neural network method. These two approaches employ auxiliary neural networks to approximate flux functions, enhancing the vanilla PINN framework's capability to simulate shock waves. The proposed methods retain the simplicity and extensibility of vanilla PINNs while avoiding the need for spatiotemporal discretization. Numerical experiments for one-dimensional and two-dimensional problems demonstrate the effectiveness of our approaches. The results show that the improved methods significantly outperform vanilla PINN in capturing shock wave dynamics.