  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.
