Towards Exact Multidimensional Fluxes: A Structure-Preserving Data-Informed Framework for Riemann Solvers

Finite-volume and discontinuous Galerkin methods update cells using the space-time flux integral over their faces, which typically depends on the solutions to local Riemann problems. In the multidimensional Riemann problem setting, this integral is difficult to evaluate exactly because of complex wave interactions. In this work, we show that the exact space-time flux integral is a polynomial in the time step and face dimensions. This motivates a learning-based approach, as only the mapping from the initial states to the polynomial coefficients needs to be learned. We propose a structure-preserving, data-informed Riemann solver framework that uses a neural network trained on high-resolution simulation data to learn this mapping. Key structural properties, including consistency and symmetry, are enforced by construction. Provided that the high-resolution simulations converge to the exact flux integrals and the network accurately predicts the coefficients, this framework yields near-exact flux integrals for use in numerical simulations. We demonstrate the framework using classical implosion and explosion problems for the two-dimensional Euler equations. In an unsplit MUSCL-Hancock scheme, the learned Riemann solver remains stable at a CFL number of 0.9, whereas the same scheme using the exact one-dimensional Riemann solver fails. Compared with the latter scheme at CFL=0.45, the learned scheme is slightly more accurate and requires approximately 30% less computational time.