Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter

TL;DR

This work tackles the data-generation bottleneck for neural eigenvalue methods by proposing Sorting Chebyshev Subspace Filter (SCSF), which first groups operators with similar spectral properties using a truncated FFT-based sorting and then solves sequential eigenvalue problems with a Chebyshev filtered subspace iteration that reuses eigenpairs from previous problems. By exploiting inter-operator similarities, SCSF reduces redundant computations and accelerates dataset generation, achieving up to $3.5\times$ speedup over state-of-the-art solvers across multiple operator types, with larger gains for higher-dimensional problems. The approach is validated on generalized Poisson, elliptic, Helmholtz, and vibration operators, showing substantial reductions in solve times and iterations, and the sorting component is shown to be cost-effective and beneficial especially when problem similarity is high. These results imply significant practical impact for producing large, high-fidelity eigenvalue datasets needed to train neural operators and related AI-for-science tools. The work also points to future extensions to nonlinear eigenproblems and enhancements to similarity metrics to further boost efficiency.

Abstract

Eigenvalue problems are among the most important topics in many scientific disciplines. With the recent surge and development of machine learning, neural eigenvalue methods have attracted significant attention as a forward pass of inference requires only a tiny fraction of the computation time compared to traditional solvers. However, a key limitation is the requirement for large amounts of labeled data in training, including operators and their eigenvalues. To tackle this limitation, we propose a novel method, named Sorting Chebyshev Subspace Filter (SCSF), which significantly accelerates eigenvalue data generation by leveraging similarities between operators -- a factor overlooked by existing methods. Specifically, SCSF employs truncated fast Fourier transform sorting to group operators with similar eigenvalue distributions and constructs a Chebyshev subspace filter that leverages eigenpairs from previously solved problems to assist in solving subsequent ones, reducing redundant computations. To the best of our knowledge, SCSF is the first method to accelerate eigenvalue data generation. Experimental results show that SCSF achieves up to a $3.5\times$ speedup compared to various numerical solvers.

Figures (5)

• Figure 1: Left. Generation process of the eigenvalue dataset: 1. Generate a set of random problem parameters. 2. Derive the corresponding operators based on these parameters. 3. Convert the operators into matrices using discretization methods. 4. Independently solve for the matrix eigenvalues using numerical solvers. 5. Obtain the matrix eigenpairs, converting them into the operator eigenpairs. 6. Assemble the dataset. Right. Results of average computation times across various algorithms based on the number of eigenvalues solved on the Helmholtz operator dataset.
• Figure 2: Algorithm Flow Diagram: a. Generation of operators to be solved. b. Discretization of operators into matrices. c. Apply SCSF algorithm to sort matrices, obtaining a sequence with strong correlations. d. Other algorithms independently solve eigenvalue problems. d1, d2, d3. SCSF algorithm utilizes Chebyshev subspace iterations to sequentially solve the eigenvalue problems. e. Assembly of eigenvalue pairs into a dataset. f. Amplification of the interval of interest through spectral transformation. g. Replacement of initial subspaces with previously solved invariant subspaces.
• Figure 3: The Truncated FFT Sorting Algorithm
• Figure 4: Chebyshev Filtered Subspace Iteration
• Figure 5: Plot of average computation time versus matrix dimension for solving 400 eigenvalues with a precision of 1e-12 on the generalized Poisson operator dataset.