Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations

TL;DR

This paper investigates how Physics-Informed Neural Networks (PINNs) perform when solving PDEs in conservative versus non-conservative forms, especially in the presence of shocks and discontinuities in fluid dynamics.

Contribution

It provides a comprehensive analysis of PINNs' sensitivity to PDE formulation choices across benchmark fluid dynamics problems involving shocks.

Findings

PINNs show varying accuracy depending on PDE form used.

Non-conservative forms can lead to significant errors near discontinuities.

The study highlights the importance of PDE formulation in PINN-based simulations.

Abstract

In the realm of computational fluid dynamics, traditional numerical methods, which heavily rely on discretization, typically necessitate the formulation of partial differential equations (PDEs) in conservative form to accurately capture shocks and other discontinuities in compressible flows. Conversely, utilizing non-conservative forms often introduces significant errors near these discontinuities or results in smeared shocks. This dependency poses a considerable limitation, particularly as many PDEs encountered in complex physical phenomena, such as multi-phase flows, are inherently non-conservative. This inherent non-conservativity restricts the direct applicability of standard numerical solvers designed for conservative forms. This work aims to thoroughly investigate the sensitivity of Physics-Informed Neural Networks (PINNs) to the choice of PDE formulation (conservative vs. non-conservative) when solving problems involving shocks and discontinuities. We have conducted this investigation across a range of benchmark problems, specifically the Burgers equation and both steady and unsteady Euler equations, to provide a comprehensive understanding of PINNs capabilities in this critical area.