A Multi-Block Neural Networks for General and Approximate Riemann problems.

The Riemann solver plays very important role on shock capturing method in computational fluid dynamics. Although significant breakthroughs have been conducted in the study of Riemann problems, only one-dimensional standard Riemann problem could be solved analytically. In this study, we proposed Multi-Block Neural Networks (MBNNs) to approximate the exact solution of general Riemann problem, based on the Physically Informed Neural Networks (PINNs) and the Shock Position Neural Networks (SPNNs). 

With the advantage of SPNNs, the spatio-temporal domain could be divided into multiple smooth spaces, which avoids the difficulties in representing non-analytic and discontinuous functions for neural network. Therefore, the MBNNs could be trained with lower depth and breadth given the number and types of discontinuities or shocks. The one-dimensional generalized Riemann problem has been approximated and the predictions of MBNNs agree very well with the numerical simulations for the one-dimensional shock problems.