  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 PINNs
that use regularized PDE-based losses as well as standard FVM-based losses, as
tested on the 2D Riemann problem in the context of time-stepping and
super-resolution reconstruction.
