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.

翻译：特征值问题是众多科学领域中最重要的话题之一。随着机器学习近年来的兴起与发展，神经特征值方法因其前向推理过程仅需传统求解器极小部分的计算时间而备受关注。然而，一个关键限制在于训练时需要大量标注数据，包括算子及其特征值。为应对这一限制，我们提出了一种名为排序切比雪夫子空间滤波（SCSF）的新方法，该方法通过利用算子间的相似性——这一现有方法忽视的因素——显著加速了特征值数据生成。具体而言，SCSF采用截断快速傅里叶变换排序对具有相似特征值分布的算子进行分组，并构建一个切比雪夫子空间滤波器，该滤波器利用已求解问题的特征对来辅助求解后续问题，从而减少冗余计算。据我们所知，SCSF是首个加速特征值数据生成的方法。实验结果表明，与多种数值求解器相比，SCSF最高可实现$3.5\times$的加速比。