DeepMartNet: A Martingale-Based Deep Neural Network Learning Method for Dirichlet BVPs and Eigenvalue Problems of Elliptic PDEs in \(\mathbb{R}\)\(^{d}\)
Abstract
Abstract. In this paper, we propose DeepMartNet, a martingale deep neural network learning method for solving Dirichlet boundary value problems (BVPs) and eigenvalue problems for elliptic 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 an efficient optimization by sampling the stochastic processes associated with the corresponding elliptic operators. Numerical results in high-dimensional BVPs of the linear and nonlinear Poisson–Boltzmann equation, as well as eigenvalue problems of the Laplace equation and a Fokker–Planck equation, demonstrate the capability of the proposed DeepMartNet method in solving high-dimensional PDE problems.
Fields of Study & Subjects
Model Reduction and Neural Networks
Polynomial and algebraic computation
Numerical methods in inverse problems
Publication details
- Journal / source
- SIAM Journal on Scientific Computing
- Publisher
- Society for Industrial & Applied Mathematics (SIAM)
- Volume
- 48
- Issue
- 1
- Pages
- C25–C50
- Publication date
- 2 January 2026
- Language
- EN