Data-driven Riemann solvers: A neural network approach and a hybrid solver

The accurate and efficient numerical solution of the Riemann problem is the basis of Godunov-type schemes. Approximate Riemann solvers are widely used for their efficiency, although they exhibit inaccuracies and instabilities in challenging regimes such as strong rarefactions or near-vacuum conditions. This work explores the use of deep neural networks (NNs) to address these limitations. We present two distinct data-driven frameworks: first, a NN-based solver trained to predict the exact solution of the Riemann problem, and second, a high-performance hybrid scheme. The hybrid approach uses the standard Harten–Lax–van Leer-contact (HLLC) Riemann solver as the main solver, enhanced with a computationally inexpensive, physics-based detector that identifies interfaces where the HLLC solution is likely to be inaccurate or to fail. At these interfaces, the scheme selectively uses the pretrained NN to ensure a more accurate solution. Through a series of benchmark tests, we show that the NN solver accurately reproduces the exact solution of the Riemann problem, but at a significant computational cost. In contrast, the proposed hybrid solver achieves a comparable level of accuracy to the NN solver, while it requires nearly the same computational cost as the standard HLLC solver.