Rimnet: a deep finite volume method with hybrid neural–numerical Riemann flux approximation in the 1D shallow water equations

ABSTRACT This study introduces a deep-learned finite volume method (DFVM) – a hybrid neural-numerical scheme that embeds neural flux modules within a Godunov-type finite volume solution framework (RimNet) for solving the 1D shallow water equations. Acting as a modular surrogate for a classical Riemann solver, RimNet estimates intercell numerical fluxes using a neural network supervised by HLL-generated fluxes from locally reconstructed Riemann states. Unlike global neural surrogates, RimNet operates at the flux level, preserving the conservative properties and time-stepping structure of the finite volume scheme. Systematic numerical experiments demonstrate the accuracy and robustness of the hybrid scheme across diverse flow regimes, including shock waves, wet-dry transitions, and oscillatory flows. The model maintains predictive fidelity over varying grid discretization resolution and unseen inflow conditions. By bridging the computational potential of neural networks with the conservative properties of the finite volume methods, this study provides a consistent, efficient, and transferable framework for modelling shallow environmental flows and offers methodological insight into combining deep learning with established schemes for solving hyperbolic conservation laws.