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课题之一。随着机器学习的兴起与发展，神经特征值方法引起了广泛关注，因为其推理前向传播所需计算时间仅为传统求解器的极小一部分。然而，该方法的关键局限性在于训练时需要大量包含算子及其特征值的标注数据。为克服这一局限，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之间的相似性——这一被现有方法忽视的因素——显著加速特征值数据生成。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，并构建切比雪夫子空间滤波器，利用先前求解问题的特征对辅助后续问题的求解，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实现了最高达3.5倍的加速比。