  In this paper, we propose a neural network learning algorithm for solving
eigenvalue problems and boundary value problems (BVPs) for elliptic operators
and initial BVPs (IBVPs) of quasi-linear parabolic equations in high dimensions
as well as optimal stochastic controls. The method is based on the Martingale
property in the stochastic representation for the eigenvalue/BVP/IBVP problems
and martingale principle for optimal stochastic controls. A loss function based
on the Martingale property can be used for efficient optimization by sampling
the stochastic processes associated with the elliptic operators or value
process for stochastic controls. The proposed algorithm can be used for
eigenvalue problems and BVPs and IBVPs with Dirichlet, Neumann, and Robin
boundaries in bounded or unbounded domains and some feedback stochastic control
problems.
