Solving large-scale Generalized Eigenvalue Problems (GEPs) is a fundamental yet computationally prohibitive task in science and engineering. As a promising direction, contour integral (CI) methods, such as the CIRR algorithm, offer an efficient and parallelizable framework. However, their performance is critically dependent on the selection of integration contours -- improper selection without reliable prior knowledge of eigenvalue distribution can incur significant computational overhead and compromise numerical accuracy. To address this challenge, we propose DeepContour, a novel hybrid framework that integrates a deep learning-based spectral predictor with Kernel Density Estimation for principled contour design. Specifically, DeepContour first employs a Fourier Neural Operator (FNO) to rapidly predict the spectral distribution of a given GEP. Subsequently, Kernel Density Estimation (KDE) is applied to the predicted spectrum to automatically and systematically determine proper integration contours. Finally, these optimized contours guide the CI solver to efficiently find the desired eigenvalues. We demonstrate the effectiveness of our method on diverse challenging scientific problems. In our main experiments, DeepContour accelerates GEP solving across multiple datasets, achieving up to a 5.63$\times$ speedup. By combining the predictive power of deep learning with the numerical rigor of classical solvers, this work pioneers an efficient and robust paradigm for tackling difficult generalized eigenvalue involving matrices of high dimension.

翻译：求解大规模广义特征值问题是科学与工程领域的一项基础性但计算代价高昂的任务。作为一种有前景的方向，轮廓积分方法（例如CIRR算法）提供了一个高效且可并行化的框架。然而，其性能关键依赖于积分轮廓的选择——若缺乏特征值分布的先验知识而进行不当选择，会导致显著的计算开销并损害数值精度。为应对这一挑战，我们提出了DeepContour，一种新颖的混合框架，它将基于深度学习的谱预测器与核密度估计相结合，以实现有原则的轮廓设计。具体而言，DeepContour首先采用傅里叶神经算子快速预测给定广义特征值问题的谱分布。随后，对预测的谱应用核密度估计，以自动且系统地确定合适的积分轮廓。最后，这些优化后的轮廓指导轮廓积分求解器高效地找到所需的特征值。我们在多种具有挑战性的科学问题上验证了该方法的有效性。在主要实验中，DeepContour在多个数据集上加速了广义特征值问题的求解，实现了最高5.63倍的加速比。通过将深度学习的预测能力与经典求解器的数值严谨性相结合，这项工作开创了一种高效且鲁棒的范式，用于处理涉及高维矩阵的困难广义特征值问题。