SReNet: Spectral Refined Network for Solving Operator Eigenvalue Problem

Hong Wang, Jiang Yixuan, Jie Wang, Xinyi Li, Jian Luo, huanshuo dong

Keywords: AI for Science, Operator Eigenvalue Problem, Scientific Computing

Abstract:

Solving operator eigenvalue problems helps analyze intrinsic data structures and relationships, yielding substantial influence on scientific research and engineering applications.
Recently, novel approaches based on deep learning have been proposed to obtain eigenvalues and eigenfunctions from the given operator, which address the efficiency challenge arising from traditional numerical methods.
However, when solving top-$L$ eigenvalues problems, these learning-based methods ignore the information that could be inherited from other known eigenvectors, thus resulting in a less-than-ideal performance.
To address the challenge, we propose the **S**pectral **Re**fined **Net**work (**SReNet**).
Our novel approach incorporates the power method to approximate the top-$L$ eigenvalues and their corresponding eigenfunctions.
To effectively prevent convergence to previous eigenfunctions, we introduce the Deflation Projection that significantly improves the orthogonality of the computed eigenfunctions and enables more precise prediction of multiple eigenfunctions simultaneously.
Furthermore, we develop the adaptive filtering method that dynamically leverages intermediate approximate eigenvalues to construct rational filters that filter out predicted eigenvalues, when predicting the successive eigenvalue of the given problem.
During the iterative solving, the spectral transformation is performed based on the filter function, converting the original eigenvalue problem into an equivalent problem that is easier to converge.
Extensive experiments demonstrate that our approach consistently outperforms existing learning-based methods, achieving state-of-the-art performance in accuracy.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 8928