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

DeepMartNet -- A Martingale based Deep Neural Network Learning Algorithm for Eigenvalue/BVP Problems and Optimal Stochastic Controls