mlsanityData to 14 Oct 2026

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

arXiv 2209.11134 ↗First posted 22 Sep 2022math.NA cs.AI cs.NAcited by 8

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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.

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

10 of 41 references.

References · 10

  1. 2015Automatic differentiation in machine learning: a surveycited by 855
  2. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  3. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  4. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  5. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 87
  6. 2020Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approachcited by 35
  7. 2022Physics-Informed Neural Networks for Quantum Eigenvalue Problemscited by 18
  8. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  9. 2020Physics-informed Neural Networks for Solving Inverse Problems of Nonlinear Biot's Equations: Batch Trainingcited by 5
  10. 2020Solving the functional Eigen-Problem using Neural Networkscited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 2; 2025: 3; 2026: 1.

8 papers.

  1. 2024Operator SVD with Neural Networks via Nested Low-Rank Approximationcited by 13
  2. 2025Solving engineering eigenvalue problems with neural networks using the Rayleigh quotientcited by 8
  3. 2024Neural networks for bifurcation and linear stability analysis of steady states in partial differential equationscited by 5
  4. 2025STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problemcited by 3
  5. 2023On the uncertainty analysis of the data-enabled physics-informed neural network for solving neutron diffusion eigenvalue problemcited by 2
  6. 2023Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDEcited by 2
  7. 2025Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolverscited by 0
  8. 2026Low-Fidelity FDM Spectral Guidance for Neural Eigenvalue Solverscited by 0