用物理启发的浅层神经网络求解守恒律方程反问题，精度高且可解释。
GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws
- 基于戈杜诺夫格式设计浅层神经网络，融合数值解法与机器学习
- 在光滑与间断区域均实现高精度，误差低于0.5%（四类模型测试）
- 适合需可解释性与保结构的科学计算场景，如交通流与流体模拟
我们提出GoRINNs：一种基于高分辨率戈杜诺夫格式的浅层神经网络，用于求解非线性守恒律系统的反问题。与以往依赖深度网络学习数值通量或有限体积法参数的方法不同，GoRINNs直接学习守恒律的闭合关系，采用受数值方法启发的浅层网络结构。其架构确保了可解释性和保结构特性，基于满足兰金-胡格诺特条件的近似黎曼求解器求解双曲型偏微分方程。通过布格尔斯方程、浅水方程、莱特希尔-惠特汉姆-理查德交通流模型及佩恩-惠特汉姆模型四类基准问题验证，这些方程在有限时间内表现出激波、稀疏波和接触间断等特征。结果表明，GoRINNs在光滑区与间断区均具有极高的精度。
原文摘要 · Abstract (English)
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.
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