Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations
This addresses a limitation in computational fluid dynamics for researchers and engineers dealing with complex physical phenomena like multi-phase flows, but it appears incremental as it focuses on evaluating an existing method rather than introducing a new one.
The paper investigates how Physics-Informed Neural Networks (PINNs) perform with conservative versus non-conservative PDE formulations in solving problems with shocks and discontinuities, such as the Burgers and Euler equations, to assess their limitations and capabilities.
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.