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.

翻译：本文提出一种神经网络学习算法，利用特征值问题随机表示中的鞅性质，求解高维空间中椭圆算子的特征值与特征函数。基于鞅性质构建的损失函数，可通过采样与椭圆算子相关的随机过程实现高效优化。所提算法适用于有界域或无界域中的Dirichlet、Neumann及Robin特征值问题。