In this article, we propose two kinds of neural networks inspired by power method and inverse power method to solve linear eigenvalue problems. These neural networks share similar ideas with traditional methods, in which the differential operator is realized by automatic differentiation. The eigenfunction of the eigenvalue problem is learned by the neural network and the iterative algorithms are implemented by optimizing the specially defined loss function. The largest positive eigenvalue, smallest eigenvalue and interior eigenvalues with the given prior knowledge can be solved efficiently. We examine the applicability and accuracy of our methods in the numerical experiments in one dimension, two dimensions and higher dimensions. Numerical results show that accurate eigenvalue and eigenfunction approximations can be obtained by our methods.

翻译：本文提出两种受幂法和逆幂法启发的神经网络，用于求解线性特征值问题。这些神经网络与传统方法具有相似的思想，其中微分算子通过自动微分实现。特征值问题的特征函数由神经网络学习，迭代算法通过优化特定定义的损失函数来执行。基于给定的先验知识，可以高效求解最大正特征值、最小特征值以及内部特征值。我们通过一维、二维及更高维度的数值实验检验了方法的适用性和准确性。数值结果表明，我们的方法能够获得精确的特征值和特征函数近似。