Research map / Learning theory and algorithms / Matrix and tensor methods
STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes STNet, a neural-network eigensolver for differential operators that combines deflation projection and a rational filter transform inside a power-method-style iteration to compute multiple eigenpairs accurately.
Problem
Learning-based methods for operator eigenvalue problems are sensitive to the operator's spectral gaps, struggle to compute multiple eigenvalues, and can converge to already-found eigenfunctions.
Approach
During each iteration, use approximate eigenpairs to deflate the already-solved subspace and apply a rational filter (product of shifted operator factors around approximate eigenvalues ± xi) to amplify target eigenvalues and suppress others, reformulating the operator into an easier equivalent problem for MLP-based power-method iterations.
Key results
- On the 2D Harmonic problem STNet reaches absolute error 1.02e-5 for lambda_1, versus NeuralSVD and NeuralEF errors of several orders larger; for 5D Harmonic, STNet's residuals are about 1e-3 vs 1 to 30 for baselines.
- On 5D Harmonic with 9^5 points, STNet achieves relative error 2.32e-4 for lambda_4 with 1.14 GB memory, while FDM at 45^5 points (22.9 GB) reaches 1.27e-2 for lambda_4.
- For the Schrödinger oscillator, STNet gives relative error 7.24e-7 in 1D versus 2.34e-6 for PMNN and 1.25e-3 for NeuralSVD.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use STNet instead of NeuralSVD | Harmonic eigenvalue problem · Harmonic 5D λ1 relative error | relative error: 2.88e-2 -> 4.66e-6 | Helped |
| use STNet instead of PMNN | Schrödinger oscillator · 5D principal eigenvalue | relative error: 1.57e-1 -> 1.29e-1 | Helped |
| use STNet instead of NeuralSVD | Fokker-Planck operator · ci=0.5, 2D λ1 | absolute error: 3.33e-1 -> 5.26e-6 | Helped |
| use STNet instead of FDM | 5D Harmonic eigenvalue problem · λ1 relative error | relative error: 3.82e-3 -> 4.62e-5 | Helped |
| add filter transform (ablation) | STNet · 2D Harmonic λ1 | absolute error: 6.73e-5 -> 1.02e-5 | Helped |
| add filter transform (ablation) | STNet · 2D Harmonic λ2 | absolute error: 5.10e-2 -> 3.04e-2 | Helped |
| add deflation projection (ablation) | STNet · 2D Harmonic λ2 | absolute error: 2.96e+1 -> 3.04e-2 | Helped |
| add deflation projection (ablation) | STNet · 2D Harmonic λ3 | absolute error: 2.97e+1 -> 6.76e-1 | Helped |
| remove both filter transform and deflation projection (ablation) | STNet · 2D Harmonic λ2 | absolute error: 3.04e-2 -> 2.96e+1 | Hurt |
| increase model depth from 3 to 6 layers (ablation) | STNet · 2D Harmonic λ4 | absolute error: 1.00e-1 -> 3.81e+1 | Hurt |
| increase model width from 20 to 40 (ablation) | STNet · 2D Harmonic λ4 | absolute error: 2.00e+1 -> 1.50e+2 | Hurt |
| increase sampling points from 20000 to 50000 (ablation) | STNet · 2D Harmonic λ2 | absolute error: 1.25e+0 -> 2.53e-1 | Helped |
Surprising findings
- Removing the deflation projection causes the network to converge only to the first eigenfunction, with small residuals but large errors for lambda_2 and lambda_3.
- Compared with the basic method, STNet's full convergence takes longer in wall-clock time (about 3.1e+4 s vs 1.2e+4 s for baselines in the 5D runtime table) despite higher accuracy.
Limitations
- Only handles linear operators; nonlinear eigenvalue problems are left to future work.
- Does not investigate broader matrix preconditioning techniques.
- Total convergence time is longer than baseline neural methods due to the iterative filter transform.
Experimental setup
Benchmarks: Harmonic operator eigenvalues (1D, 2D, 5D) Schrödinger oscillator principal eigenvalue (1D, 2D, 5D) Fokker-Planck zero eigenvalues (1D, 2D, 5D)
Metrics: relative error absolute error residual error
Baselines: PMNN NeuralEF NeuralSVD finite difference method (FDM)
Builds on: power method neural network (PMNN) NeuralEF NeuralSVD shift-invert power method
Closest papers by meaning
- Learning to Discover Iterative Spectral Algorithms (NeurIPS 2026 · similarity 0.86)
- GIST: Gauge-Invariant Spectral Transformers for Scalable Graph Neural Operators (NeurIPS 2026 · similarity 0.85)
- Operator SVD with Neural Networks via Nested Low-Rank Approximation (ICML 2024 · similarity 0.84)
- Modulate Your Spectrum in Self-Supervised Learning (ICLR 2024 · similarity 0.84)
- Neural Spectral Methods: Self-supervised learning in the spectral domain (ICLR 2024 · similarity 0.84)
- Explicit Eigenvalue Regularization Improves Sharpness-Aware Minimization (NeurIPS 2024 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.