Godunov Loss Functions for Modelling of Hyperbolic Conservation Laws

Abstract: Machine learning techniques are being used as an alternative to traditional numerical discretization methods for solving hyperbolic partial differential equations (PDEs) relevant to fluid flow. Whilst numerical methods are higher fidelity, they are computationally expensive. Machine learning methods on the other hand are lower fidelity but can provide significant speed-ups. The emergence of physics-informed neural networks (PINNs) in fluid dynamics has allowed scientists to directly use PDEs for evaluating loss functions. The downfall of this approach is that the differential form of systems is invalid at regions of shock inherent in hyperbolic PDEs such as the compressible Euler equations. To circumvent this problem we propose the Godunov loss function: a loss based on the finite volume method (FVM) that crucially incorporates the flux of Godunov-type methods. These Godunov-type methods are also known as approximate Riemann solvers and evaluate intercell fluxes in an entropy-satisfying and non-oscillatory manner, yielding more physically accurate shocks. Our approach leads to superior performance compared to standard PINNs that use regularized PDE-based losses as well as FVM-based losses, as tested on the 2D Riemann problem in the context of time-stepping and super-resolution reconstruction.