DeepMartNet -- A Martingale Based Deep Neural Network Learning Method for Dirichlet BVP and Eigenvalue Problems of Elliptic PDEs
Wei CaiAndrew HeDaniel Margolis
Wei CaiAndrew HeDaniel Margolis
Nov 2023
0被引用
0笔记
摘要原文
In this paper, we propose DeepMartNet - a Martingale based deep neural network learning method for solving Dirichlet boundary value problems (BVPs) and eigenvalue problems for elliptic partial differential equations (PDEs) in high dimensions. The method is based on Varadhan's Martingale problem formulation for the BVP/eigenvalue problems where a loss function enforcing the Martingale property for the PDE solution is used for efficient optimization by sampling the stochastic processes associated with elliptic operators. High dimensional numerical results for BVPs of the Poisson-Boltzmann equation and eigenvalue problems of a Fokker-Planck equation demonstrate the capability of the proposed DeepMartNet learning method for solving high dimensional PDE problems.
