用神经网络修正近似黎曼求解器，提升超时方程计算精度。
Neural network-based Godunov corrections for approximate Riemann solvers using bi-fidelity learning
- 用监督学习训练神经网络，从状态变量预测精确通量。
- 双精度学习方法显著提升复杂场景下的计算准确性。
- 适合需要高精度模拟的流体动力学与科学计算研究者。
黎曼问题在双曲型偏微分方程的数值模拟中至关重要，支撑着稳定高效的迎风格式发展。尽管精确求解器能提供可靠的迎风通量，但其计算成本过高，因此常采用近似求解器。然而，近似求解器在某些情况下会产生不准确结果。为此，我们提出构建基于神经网络的代理模型，通过监督学习将内部与外部守恒状态变量映射为对应精确通量。具体提出两种方法：一种使用基础神经网络，另一种采用双精度神经网络。所提方法在一维和二维偏微分方程中的应用展示了其鲁棒性与高精度。
原文摘要 · Abstract (English)
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.
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