Operator eigenvalue problems play a critical role in various scientific fields and engineering applications, yet numerical methods are hindered by the curse of dimensionality. Recent deep learning methods provide an efficient approach to address this challenge by iteratively updating neural networks. These methods' performance relies heavily on the spectral distribution of the given operator: larger gaps between the operator's eigenvalues will improve precision, thus tailored spectral transformations that leverage the spectral distribution can enhance their performance. Based on this observation, we propose the Spectral Transformation Network (STNet). During each iteration, STNet uses approximate eigenvalues and eigenfunctions to perform spectral transformations on the original operator, turning it into an equivalent but easier problem. Specifically, we employ deflation projection to exclude the subspace corresponding to already solved eigenfunctions, thereby reducing the search space and avoiding converging to existing eigenfunctions. Additionally, our filter transform magnifies eigenvalues in the desired region and suppresses those outside, further improving performance. Extensive experiments demonstrate that STNet consistently outperforms existing learning-based methods, achieving state-of-the-art performance in accuracy.

翻译：算子特征值问题在众多科学领域与工程应用中扮演着关键角色，然而数值方法常受维度灾难的制约。近年来，深度学习通过迭代更新神经网络为应对这一挑战提供了高效途径。此类方法的性能在很大程度上依赖于给定算子的谱分布：算子特征值之间的间隔越大，精度通常越高，因此利用谱分布特性定制的谱变换能够有效提升其性能。基于这一观察，我们提出了谱变换网络（STNet）。在每次迭代中，STNet利用近似特征值与特征函数对原始算子进行谱变换，将其转化为等价但更易求解的问题。具体而言，我们采用收缩投影排除已求解特征函数对应的子空间，从而缩减搜索空间并避免收敛至已有特征函数。此外，我们的滤波器变换能够放大目标区域内的特征值并抑制区域外的特征值，进一步提升了算法性能。大量实验表明，STNet在精度上持续超越现有基于学习的方法，取得了最先进的性能。