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.

翻译：本文提出一种神经网络学习算法，用于求解高维椭圆算子的特征值问题与边值问题（BVPs）、拟线性抛物型方程的初边值问题（IBVPs），以及最优随机控制。该方法基于特征值/边值/初边值问题随机表示中的鞅性质，以及最优随机控制的鞅原理。基于鞅性质的损失函数可通过采样与椭圆算子相关的随机过程或随机控制的值过程实现高效优化。所提算法适用于有界或无界区域中具有Dirichlet、Neumann、Robin边界条件的特征值问题、边值问题与初边值问题，以及部分反馈随机控制问题。