Neural Network-Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

作     者：Min Wang Jianfeng Lu 

作者机构：Duke UniversityDurhamNCUSA 

出 版 物：《Communications in Mathematics and Statistics》 (数学与统计通讯（英文）)

年 卷 期：2023年第11卷第1期

页      面：21-57页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：supported in part by National Science Foundation via grant DMS-2012286 by Department of Energy via grant DE-SC0019449. 

主　　题：Quadratic porous medium equation High-dimensional nonlinear PDE,Neural network Variational formulation Deep learning 

摘      要：In this paper,we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation(QPME).Three variational formulations of this nonlinear PDE are presented:a strong formulation and two weak formulations.For the strong formulation,the solution is directly parameterized with a neural network and optimized by minimizing the PDEresidual.It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the L^(1)sense.Theweak formulations are derived following(Brenier in Examples of hidden convexity in nonlinear PDEs,2020)which characterizes the very weak solutions of QPME.Specifically speaking,the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations.Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions.This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods,which we hope can provide some useful experience for future investigations.