We present GoRINNs: numerical analysis-informed (shallow) neural networks for the solution of inverse problems of non-linear systems of conservation laws. GoRINNs is a hybrid/blended machine learning scheme 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 or just parameters of conservative Finite Volume methods, relying on deep neural networks (that may lead to poor approximations due to the computational complexity involved in their training), GoRINNs learn the closures of the conservation laws per se based on "intelligently" numerical-assisted shallow neural networks. Due to their structure, in particular, GoRINNs provide explainable, conservative schemes, that solve the inverse problem for hyperbolic PDEs, 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：一种基于数值分析信息的（浅层）神经网络，用于求解非线性守恒律系统的反问题。GoRINNs是一种混合/融合机器学习方案，其基础是用于求解双曲偏微分方程（PDEs）中黎曼问题的高分辨率Godunov格式。与现有其他机器学习方法（这些方法学习数值通量或仅学习保守有限体积方法的参数，依赖于深度神经网络——其训练涉及的计算复杂性可能导致近似效果不佳）不同，GoRINNs基于“智能”数值辅助的浅层神经网络，直接学习守恒律本身的闭合关系。特别是由于其结构，GoRINNs提供了可解释的、保守的格式，这些格式基于满足Rankine-Hugoniot条件的近似黎曼求解器，解决了双曲PDEs的反问题。GoRINNs的性能通过四个基准问题进行了评估，即Burgers方程、浅水波方程、Lighthill-Whitham-Richards交通流模型和Payne-Whitham交通流模型。这些PDE的解在有限时间内表现出激波、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。