Abstract
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