# Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems

arXiv: https://arxiv.org/abs/2209.11134 · HTML: https://mlsanity.com/paper/2209.11134/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 22 Sep 2022.

## Builds on (most cited first, 10 of 10)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Physics-Informed Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2203.00451/) (2022)
- [Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/2108.13054/) (2021)
- [Physics-informed Neural Networks for Solving Inverse Problems of Nonlinear Biot's Equations: Batch Training](https://mlsanity.com/paper/2005.09638/) (2020)
- [Solving the functional Eigen-Problem using Neural Networks](https://mlsanity.com/paper/2007.10205/) (2020)

## Built on by (most cited first, 8 of 8)

- [Operator SVD with Neural Networks via Nested Low-Rank Approximation](https://mlsanity.com/paper/2402.03655/) (2024)
- [Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient](https://mlsanity.com/paper/2506.04375/) (2025)
- [Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations](https://mlsanity.com/paper/2407.19707/) (2024)
- [STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem](https://mlsanity.com/paper/2510.23986/) (2025)
- [On the uncertainty analysis of the data-enabled physics-informed neural network for solving neutron diffusion eigenvalue problem](https://mlsanity.com/paper/2303.08455/) (2023)
- [Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDE](https://mlsanity.com/paper/2311.16167/) (2023)
- [Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers](https://mlsanity.com/paper/2511.01927/) (2025)
- [Low-Fidelity FDM Spectral Guidance for Neural Eigenvalue Solvers](https://mlsanity.com/paper/2610.04695/) (2026)

## Abstract (arXiv)

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.