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.

翻译：黎曼问题是双曲型偏微分方程计算建模中的基础问题，它使得构建稳定且精确的迎风格式成为可能。虽然精确求解器能提供鲁棒的迎风通量，但其高昂的计算成本使得近似求解器成为必要。尽管近似求解器在许多情况下能达到精度要求，但在某些特定情形下会产生不准确的解。为克服这一局限，我们提出构建基于神经网络的代理模型，该模型通过监督学习训练，旨在将内部和外部守恒状态变量映射至对应的精确通量。具体而言，我们提出两种不同的方法：一种采用普通神经网络，另一种则采用双保真度神经网络。通过在一维和二维偏微分方程中的应用，验证了所提方法的性能，展示了其鲁棒性与精确性。