S-PINN: physics-informed neural networks for solving weakly hyperbolic systems

Abstract In this article, we introduced a novel shock-capturing physics-informed neural network (S-PINN) framework tailored for solving weakly hyperbolic systems. In a weakly hyperbolic system, classical solutions may no longer exist, and the formation of shocks, contact discontinuities, delta shocks, and rarefaction waves within a finite time poses challenges for developing accurate numerical schemes. The S-PINN integrates the Rankine-Hugoniot (RH) condition with neural network architecture and employs a mask function to enable it to effectively capture discontinuities with high resolution. For comparison, we have developed a multi-level Weighted Essentially Non-Oscillatory (WENO) scheme, denoted as WENO-M(5,3,2), which demonstrates improved performance compared to the classical WENO scheme. The construction of the WENO-M(5,3,2) scheme involves blending of one fourth-degree polynomial, one quadratic polynomial, and two linear polynomials to capture both smooth and discontinuous features effectively. This multi-level formulation ensures better balance between accuracy and non-oscillatory properties. Through comprehensive numerical experiments, we highlight the superiority of the proposed S-PINN over traditional computational methods like WENO in terms of accuracy, resolving discontinuities with high resolution, and robustness.