DeepMartNet -- A Martingale Based Deep Neural Network Learning Method for Dirichlet BVPs and Eigenvalue Problems of Elliptic PDEs in R^d

2023/11/15 by Wei Cai, Cai, Wei, Andrew Qing He +3 · 2 citations
Computer Science · Engineering · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2311.09456

openalex publication_date 2023/11/15 · openalex created_date 2023/11/18 · openalex updated_date 2026/07/28

Abstract

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 or domains with complex geometries. The method is based on Varadhan's Martingale problem formulation for the BVPs/eigenvalue problems where a loss function enforcing the Martingale property for the PDE solution is used for an efficient optimization by sampling the stochastic processes associated with corresponding elliptic operators. High dimensional numerical results for BVPs of the linear and nonlinear Poisson-Boltzmann equation and eigenvalue problems of the Laplace equation and a Fokker-Planck equation demonstrate the capability of the proposed DeepMartNet learning method in solving high dimensional PDE problems.

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