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Subspace method based on neural networks for eigenvalue problems

Xiaoying Dai, Yunying Fan, Zhiqiang Sheng

TL;DR

This work introduces a subspace method based on neural networks (SNN-EP) for eigenvalue problems by first training neural-network-based functions to form a subspace and then applying POD to obtain an orthogonal, low-dimensional basis. The eigenproblem is solved via Galerkin projection onto this neural-network-derived subspace, yielding very accurate eigenvalues and eigenfunctions at a fraction of the training effort required by traditional neural approaches. Key contributions include integrating a POD-based dimensionality reduction with a decoupled training-and-solving workflow, enforcing boundary conditions within the network, and demonstrating substantial accuracy and speed advantages on Laplace, harmonic oscillator, and Schrödinger-type problems. The approach promises efficient handling of high-dimensional or quantum-mechanical eigenproblems by combining machine learning with classical variational discretization.

Abstract

In this paper, we propose a subspace method based on neural networks for eigenvalue problems with high accuracy and low cost. We first construct a neural network-based orthogonal basis by some deep learning method and dimensionality reduction technique, and then calculate the Galerkin projection of the eigenvalue problem onto the subspace spanned by the orthogonal basis and obtain an approximate solution. Numerical experiments show that we can obtain approximate eigenvalues and eigenfunctions with very high accuracy but low cost.

Subspace method based on neural networks for eigenvalue problems

TL;DR

This work introduces a subspace method based on neural networks (SNN-EP) for eigenvalue problems by first training neural-network-based functions to form a subspace and then applying POD to obtain an orthogonal, low-dimensional basis. The eigenproblem is solved via Galerkin projection onto this neural-network-derived subspace, yielding very accurate eigenvalues and eigenfunctions at a fraction of the training effort required by traditional neural approaches. Key contributions include integrating a POD-based dimensionality reduction with a decoupled training-and-solving workflow, enforcing boundary conditions within the network, and demonstrating substantial accuracy and speed advantages on Laplace, harmonic oscillator, and Schrödinger-type problems. The approach promises efficient handling of high-dimensional or quantum-mechanical eigenproblems by combining machine learning with classical variational discretization.

Abstract

In this paper, we propose a subspace method based on neural networks for eigenvalue problems with high accuracy and low cost. We first construct a neural network-based orthogonal basis by some deep learning method and dimensionality reduction technique, and then calculate the Galerkin projection of the eigenvalue problem onto the subspace spanned by the orthogonal basis and obtain an approximate solution. Numerical experiments show that we can obtain approximate eigenvalues and eigenfunctions with very high accuracy but low cost.

Paper Structure

This paper contains 17 sections, 53 equations, 6 figures, 11 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A linear elliptic eigenvalue problem and its finite dimensional discretization
  4. Subspace method based on neural networks
  5. Subspace method based on neural networks for eigenvalue problems
  6. Constructing a neural network-based orthogonal basis
  7. Neural networks architecture
  8. Handling the boundary condition
  9. The loss function and the training
  10. Dimensionality reduction technique
  11. Algorithm
  12. Galerkin discretization in the neural network-based subspace
  13. Numerical experiments
  14. Laplace eigenvalue problem
  15. Harmonic oscillator problems
  16. ...and 2 more sections

Figures (6)

  • Figure 1: The dimension of the reduced subspace and the relative errors of the smallest 15 eigenvalues for 2D Laplace eigenvalue problem obtained by SNN-EP with different $M$.
  • Figure 2: The dimension of the reduced subspace and the relative errors for the 2D decoupled harmonic oscillator obtained by SNN-EP with different $M$.
  • Figure 3: The dimension of the reduced subspace and the relative errors of eigenvalues of the 2D coupled harmonic oscillator obtained by SNN-EP with different $M$.
  • Figure 4: The dimension of the reduced subspace and the relative error of eigenvalues of the Schrodinger equation for Hydrogen atom obtained by SNN-EP with different $M$.
  • Figure 5: The dimension of the reduced subspace and the relative error of eigenvalues of the Schrodinger equation for Hydrogen atom obtained by SNN-EP (10 random $\beta_i$) with different $M$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2