Discontinuity Computing with Physics-Informed Neural Network

How to simulate shock waves and other discontinuities is a long history topic. As a new and booming method, the physics-informed neural network (PINN) is still weak in calculating shock waves than traditional shock-capturing methods. In this paper, we propose a ‘retreat in order to advance’ way to improve the shock capturing ability of PINN by using a weighted equations (WE) method with PINN. The primary strategy of the method is to weaken the expression of the network in high compressible regions by adding a local positive and compression-dependent weight into governing equations at each interior point. With this strategy, the network will focus on training smooth parts of the solutions. Then automatically affected by the compressible property near shock waves, a sharp discontinuity appears with the wrong inside-shock-points ‘compressed’ into well-trained smooth regions just like passive particles. In this paper, we study one-dimensional and two-dimensional Euler equations. And illustrated by the comparisons with high-order classical WENO-Z method in numerical examples, the proposed method can significantly improve the discontinuity computing ability