Recent research works for solving partial differential equations (PDEs) with deep neural networks (DNNs) have demonstrated that spatiotemporal function approximators defined by auto-differentiation are effective for approximating nonlinear problems, e.g. the Burger's equation, heat conduction equations, Allen-Cahn and other reaction-diffusion equations, and Navier-Stokes equation. Meanwhile, researchers apply automatic differentiation in physics-informed neural network (PINN) to solve nonlinear hyperbolic systems based on conservation laws with highly discontinuous transition, such as Riemann problem, by inverse problem formulation in data-driven approach. However, it remains a challenge for forward methods using DNNs without knowing part of the solution to resolve discontinuities in nonlinear conservation laws. In this study, we incorporate 1st order numerical schemes into DNNs to set up the loss functional approximator instead of auto-differentiation from traditional deep learning framework, e.g. TensorFlow package, which improves the effectiveness of capturing discontinuities in Riemann problems. In particular, the 2-Coarse-Grid neural network (2CGNN) and 2-Diffusion-Coefficient neural network (2DCNN) are introduced in this work. We use 2 solutions of a conservation law from a converging sequence, computed from a low-cost numerical scheme, and in a domain of dependence of a space-time grid point as the input for a neural network to predict its high-fidelity solution at the grid point. Despite smeared input solutions, they output sharp approximations to solutions containing shocks and contacts and are efficient to use once trained.
On general and complete multidimensional Riemann solvers for nonlinear systems of hyperbolic conservation laws
University of Verona（维罗纳大学）；Inria, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（法国国家信息与自动化研究所，波尔多大学，法国国家科学研究中心，波尔多国立理工学院，IMB）；Laboratory of Applied Mathematics, DICAM, University of Trento（特伦托大学应用数学实验室）
GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws
Modelling Engineering Risk and Complexity, Scuola Superiore Meridionale（建模工程风险与复杂性，南方高级学校）；Dept. of Electrical Engineering and Information Technology, University of Naples Federico II（电气工程与信息技术系，那不勒斯费德里科二世大学）；Institute of Science and Technology for Energy and Sustainable Mobility, Consiglio Nazionale delle Ricerche（能源与可持续交通科学与技术研究所，意大利国家研究委员会）
A Two-Stage Fourth Order Time-Accurate Discretization for Lax-Wendroff Type Flow Solvers. I. Hyperbolic Conservation Laws
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（计算物理实验室，应用物理与计算数学研究所）；School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）
Central schemes for networked scalar conservation laws
我们提出了一个新的方案，数值求解网络上的标量守恒律，而无需解决连接处的黎曼问题。该格式是利用在[Jin和Xin，Comm. Pure Appl.Math.48（3），235-276（1995）]中介绍的松弛系统并在网络的节点处取松弛极限而导出的。该方案是质量守恒的，并产生良好的定义和易于计算的耦合条件，即使是一般的网络。讨论了该格式的高阶推广及其在交通流和两相流中的应用。在前者中，我们与文献中得到的结果进行比较。
A Hermite WENO reconstruction for fourth order temporal accurate schemes based on the GRP solver for hyperbolic conservation laws
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所计算物理实验室）
A central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere
University of Ottawa（渥太华大学）；Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室与法国国家科学研究中心，皮埃尔和玛丽居里大学（巴黎第六大学））
