A DEEP LEARNING BASED DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS WITH DISCONTINUOUS SOLUTIONS AND RANDOM UNCERTAINTIES  

作　　者：Jingrun Chen Shi Jin Liyao Lyu 

机构地区：[1]School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China [2]Suzhou Institute for Advanced Research,University of Science and Technology of China,Suzhou 215123,China [3]School of Mathematical Sciences,Institute of Natural Sciences,and MOE-LSC,Shanghai Jiao Tong University,Shanghai,200240,China [4]Department of Computational Mathematics,Science,and Engineering,Michigan State University,East Lansing,MI,48824,USA

出　　处：《Journal of Computational Mathematics》2023年第6期1281-1304,共24页计算数学（英文）

基　　金：supported by National Key R&D Program of China under grants 2018YFA0701700,2018YFA0701701,and NSFC grant 11971021;supported by the Natural Science Foundation of China under grant 12031013.

摘　　要：We propose a deep learning based discontinuous Galerkin method(D2GM)to solve hyperbolic equations with discontinuous solutions and random uncertainties.The main computational challenges for such problems include discontinuities of the solutions and the curse of dimensionality due to uncertainties.Deep learning techniques have been favored for high-dimensional problems but face difficulties when the solution is not smooth,thus have so far been mainly used for viscous hyperbolic system that admits only smooth solutions.We alleviate this difficulty by setting up the loss function using discrete shock capturing schemes–the discontinous Galerkin method as an example–since the solutions are smooth in the discrete space.The convergence of D2GM is established via the Lax equivalence theorem kind of argument.The high-dimensional random space is handled by the Monte-Carlo method.Such a setup makes the D2GM approximate high-dimensional functions over the random space with satisfactory accuracy at reasonable cost.The D2GM is found numerically to be first-order and second-order accurate for(stochastic)linear conservation law with smooth solutions using piecewise constant and piecewise linear basis functions,respectively.Numerous examples are given to verify the efficiency and the robustness of D2GM with the dimensionality of random variables up to 200 for(stochastic)linear conservation law and(stochastic)Burgers’equation.

关 键 词：Discontinuous Galerkin method Loss function Convergence analysis Deep learning Hyperbolic equation 

分 类 号：O175.2[理学—数学] TP183[理学—基础数学]