DeepMartNet -- A Martingale based Deep Neural Network learning algorithm for Eigenvalue Problems in High Dimensions

In this paper, we propose a neural network learning algorithm for finding eigenvalue and eigenfunction for elliptic operators in high dimensions using the Martingale property in the stochastic representation for the eigenvalue problem. A loss function based on the Martingale property can be used for efficient optimization by sampling the stochastic processes associated with the elliptic operators. The proposed algorithm can be used for Dirichlet, Neumann, and Robin eigenvalue problems in bounded or unbounded domains.