Neural network-based Godunov corrections for approximate Riemann solvers using bi-fidelity learning

2025-03-17

AI Summary

To address the inherent trade-off between accuracy and efficiency in Riemann solvers for hyperbolic PDEs, this paper proposes a neural-network-based Godunov flux correction method: it maps low-cost approximate solver outputs directly to high-accuracy numerical fluxes. Innovatively integrating bi-fidelity learning with physical constraints from conservation laws, we construct a fully connected neural surrogate model trained jointly on both exact and approximate flux data. Evaluated on canonical 1D and 2D test problems—including challenging cases with strong shocks and contact discontinuities—the method reduces flux error by an order of magnitude, achieving accuracy comparable to exact Riemann solvers while retaining the computational efficiency of approximate solvers. This significantly enhances the robustness and applicability of upwind schemes without compromising stability or conservation properties.

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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.

Problem

• Improves accuracy of approximate Riemann solvers
• Reduces computational cost of exact solvers
• Uses neural networks for flux mapping

Innovation

• Neural network-based surrogate models for exact flux mapping
• Bi-fidelity learning to enhance solver accuracy
• Supervised training for robust and accurate upwind schemes