Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem

Hong Wang, Jiang Yixuan, Jie Wang, Xinyi Li, Jian Luo, Huanshuo Dong

TL;DR

This work tackles high-dimensional operator eigenvalue problems, where traditional discretization struggles under the curse of dimensionality. It introduces STNet, a mesh-free neural solver that iteratively updates neural approximations to eigenfunctions while applying spectral transformations—deflation projection and filter transform—to reshape the spectrum and guide convergence. The key contributions are the deflation-based subspace exclusion that preserves orthogonality across multiple eigenpairs and a rational-function filter that concentrates effort on target spectral regions, yielding state-of-the-art accuracy on Harmonic, Schrödinger, and Fokker-Planck tests. Empirical results show STNet achieves superior accuracy and memory efficiency compared to neural baselines and outperforms traditional grid-based methods in high dimensions, making it a scalable tool for high-dimensional PDE eigenproblems.

Abstract

Operator eigenvalue problems play a critical role in various scientific fields and engineering applications, yet numerical methods are hindered by the curse of dimensionality. Recent deep learning methods provide an efficient approach to address this challenge by iteratively updating neural networks. These methods' performance relies heavily on the spectral distribution of the given operator: larger gaps between the operator's eigenvalues will improve precision, thus tailored spectral transformations that leverage the spectral distribution can enhance their performance. Based on this observation, we propose the Spectral Transformation Network (STNet). During each iteration, STNet uses approximate eigenvalues and eigenfunctions to perform spectral transformations on the original operator, turning it into an equivalent but easier problem. Specifically, we employ deflation projection to exclude the subspace corresponding to already solved eigenfunctions, thereby reducing the search space and avoiding converging to existing eigenfunctions. Additionally, our filter transform magnifies eigenvalues in the desired region and suppresses those outside, further improving performance. Extensive experiments demonstrate that STNet consistently outperforms existing learning-based methods, achieving state-of-the-art performance in accuracy.

STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem

TL;DR

This work tackles high-dimensional operator eigenvalue problems, where traditional discretization struggles under the curse of dimensionality. It introduces STNet, a mesh-free neural solver that iteratively updates neural approximations to eigenfunctions while applying spectral transformations—deflation projection and filter transform—to reshape the spectrum and guide convergence. The key contributions are the deflation-based subspace exclusion that preserves orthogonality across multiple eigenpairs and a rational-function filter that concentrates effort on target spectral regions, yielding state-of-the-art accuracy on Harmonic, Schrödinger, and Fokker-Planck tests. Empirical results show STNet achieves superior accuracy and memory efficiency compared to neural baselines and outperforms traditional grid-based methods in high dimensions, making it a scalable tool for high-dimensional PDE eigenproblems.

Abstract

Operator eigenvalue problems play a critical role in various scientific fields and engineering applications, yet numerical methods are hindered by the curse of dimensionality. Recent deep learning methods provide an efficient approach to address this challenge by iteratively updating neural networks. These methods' performance relies heavily on the spectral distribution of the given operator: larger gaps between the operator's eigenvalues will improve precision, thus tailored spectral transformations that leverage the spectral distribution can enhance their performance. Based on this observation, we propose the Spectral Transformation Network (STNet). During each iteration, STNet uses approximate eigenvalues and eigenfunctions to perform spectral transformations on the original operator, turning it into an equivalent but easier problem. Specifically, we employ deflation projection to exclude the subspace corresponding to already solved eigenfunctions, thereby reducing the search space and avoiding converging to existing eigenfunctions. Additionally, our filter transform magnifies eigenvalues in the desired region and suppresses those outside, further improving performance. Extensive experiments demonstrate that STNet consistently outperforms existing learning-based methods, achieving state-of-the-art performance in accuracy.

Paper Structure

This paper contains 33 sections, 2 theorems, 29 equations, 3 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Operator Eigenvalue Problem
  4. Power Method
  5. Deflation Projection
  6. Filter Transform
  7. Method
  8. Problem Formulation
  9. Spectral Transformation Network
  10. Deflation Projection
  11. Filter Transform
  12. Experiments
  13. Harmonic Eigenvalue Problem
  14. Schrödinger Oscillator Equation
  15. Fokker-Planck Equation
  16. ...and 18 more sections

Key Result

Proposition B.1

Let $\bm{v}_1$ be an eigenvector of $\bm{A}$ of norm 1, associated with the eigenvalue $\lambda_1$ and let $\bm{A}_1 \equiv \bm{A} - \sigma \bm{v}_1 \bm{v}_1^H$. Then the eigenvalues of $\bm{A}_1$ are $\tilde{\lambda}_1 = \lambda_1 - \sigma$ and $\tilde{\lambda}_j = \lambda_j, j = 2, 3, \dots, n$. M

Figures (3)

  • Figure 1: Left. Absolute error results of zero eigenvalues for the Fokker-Planck operator computed using various algorithms, the $x$ axis represents the operator dimension. Right. Comparison of the eigenfunctions of the 2D Harmonic operator computed by STNet and the ground truth.
  • Figure 2: Overview of the STNet. (a) Introduction to the inputs and outputs. (b) STNet comprises multiple neural networks, each tasked with predicting distinct eigenvalues. If the accuracy of the solution reaches the expectation, then STNet will output the result.
  • Figure 3: Illustration of the modules' impact on the operator spectrum: (a) Deflation projection sets the solved eigenvalues to zero, (b) Filter transform enlarges the target eigenvalue region and suppresses others.

Theorems & Definitions (4)

  • Proposition B.1
  • proof
  • Proposition B.2
  • proof