We present GoRINNs: numerical analysis-informed neural networks for the solution of inverse problems of non-linear systems of conservation laws. GoRINNs are based on high-resolution Godunov schemes for the solution of the Riemann problem in hyperbolic Partial Differential Equations (PDEs). In contrast to other existing machine learning methods that learn the numerical fluxes of conservative Finite Volume methods, GoRINNs learn the physical flux function per se. Due to their structure, GoRINNs provide interpretable, conservative schemes, that learn the solution operator on the basis of approximate Riemann solvers that satisfy the Rankine-Hugoniot condition. The performance of GoRINNs is assessed via four benchmark problems, namely the Burgers', the Shallow Water, the Lighthill-Whitham-Richards and the Payne-Whitham traffic flow models. The solution profiles of these PDEs exhibit shock waves, rarefactions and/or contact discontinuities at finite times. We demonstrate that GoRINNs provide a very high accuracy both in the smooth and discontinuous regions.

翻译：我们提出GoRINNs：一种用于求解非线性守恒律系统反问题的数值分析信息神经网络。GoRINNs基于求解双曲偏微分方程中黎曼问题的高分辨率Godunov格式。与现有其他学习守恒型有限体积法数值通量的机器学习方法不同，GoRINNs直接学习物理通量函数本身。得益于其结构，GoRINNs提供了可解释的守恒格式，该格式基于满足Rankine-Hugoniot条件的近似黎曼求解器来学习解算子。GoRINNs的性能通过四个基准问题进行评估，即Burgers方程、浅水方程、Lighthill-Whitham-Richards交通流模型和Payne-Whitham交通流模型。这些偏微分方程的解剖面在有限时间内呈现激波、稀疏波和/或接触间断。我们证明GoRINNs在光滑区域和间断区域均能提供极高的精度。