Neural network-based Godunov corrections for approximate Riemann solvers using bi-fidelity learning
Akshay Thakur, Matthew J. Zahr
TL;DR
This work addresses the accuracy-cost gap in Riemann-solver flux evaluations by learning neural network surrogates that map interior and exterior conserved states $(U^+,U^-)$ to the Godunov flux. It presents two models: a vanilla FCNN surrogate and a bi-fidelity residual-corrected NN that leverages a cheaper approximate flux (e.g., Roe or HLL) to improve generalization. The methods are validated on 1D Burgers' and shallow water equations and extended to 2D problems via rotational symmetry, with the bi-fidelity NN consistently delivering flux predictions that closely match the Godunov flux and outperforming the vanilla NN and standard approximations in both a priori and a posteriori tests. The results suggest that data-driven flux surrogates can be integrated into finite-volume solvers to achieve robust, high-fidelity simulations of shocks and rarefactions while maintaining computational efficiency.
Abstract
The Riemann problem is fundamental in the computational modeling of hyperbolic partial differential equations, enabling the development of stable and accurate upwind schemes. While exact solvers provide robust upwinding fluxes, their high computational cost necessitates approximate solvers. Although approximate solvers achieve accuracy in many scenarios, they produce inaccurate solutions in certain cases. To overcome this limitation, we propose constructing neural network-based surrogate models, trained using supervised learning, designed to map interior and exterior conservative state variables to the corresponding exact flux. Specifically, we propose two distinct approaches: one utilizing a vanilla neural network and the other employing a bi-fidelity neural network. The performance of the proposed approaches is demonstrated through applications to one-dimensional and two-dimensional partial differential equations, showcasing their robustness and accuracy.