Physics Informed Neural Networks and Higher-Order High-Resolution Schemes for Resolving Discontinuities and Shocks: A Comprehensive Study

Dealing with discontinuities in fluid flow problem problems is inherently challenging, especially when shocks are formed due to the nonlinear nature of the flow. Although addressing discontinuities is a well-established practice in computational fluid dynamics, it remains a significant challenge when using physics-informed neural networks (PINNs) for problems that involve discontinuities. In this study, we compare the shock-resolving abilities of traditional methods with those of PINNs, emphasizing the advantages of PINNs. Our results illustrate that PINNs exhibit less dissipative behavior compared to conventional techniques. We assessed the performance of both PINNs and traditional methods on linear and non-linear test cases, demonstrating that PINNs have superior shock-resolving properties. Notably, PINNs can effectively resolve inviscid shocks using only three grid points, whereas traditional methods require a minimum of eight grid points on a coarse grid. This indicates that PINNs are more effective at resolving shocks and discontinuities than the traditional methods on coarse grids.