用深度学习预测特征值分布，自动设计积分路径，加速大规模特征值求解。
Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers
- 结合深度神经网络与核密度估计，自动构建最优积分路径。
- 在多个科学数据集上实现最高5.63倍加速，精度严格达标。
- 适合需要快速求解大规模特征值问题的研究者使用。
求解大规模广义特征值问题（GEP）是科学与工程中的基础性任务，但计算成本极高。基于积分路径（CI）的方法虽具高效并行潜力，但其性能严重依赖于积分路径的设计——若缺乏对特征值分布的可靠先验知识，会导致显著计算开销并影响数值精度。为此，本文提出Deepcontour，一种融合深度学习谱预测与核密度估计（KDE）的混合框架。该框架利用专为特征值设计的神经算子（ENO）快速提供谱分布先验，驱动KDE模块自动生成优化的积分路径，引导CI求解器高效定位目标特征值。实验表明，Deepcontour在多种科学数据集上实现最高5.63倍加速，同时保持严格的数值精度。本工作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，建立了一种高效且鲁棒的大规模GEP求解新范式。
原文摘要 · Abstract (English)
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 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 (KDE) for principled contour design. Specifically, Deepcontour utilizes its specialized Eigen-Neural-Operator (ENO) to provide rapid spectral distribution priors, driving a KDE module to automatically construct the optimized integration contours, which guide the CI solver to efficiently find the desired eigenvalues. Deepcontour achieves up to a 5.63x speedup across diverse scientific datasets while maintaining strict numerical rigor. By merging the predictive power of deep learning with the numerical rigor of classical solvers, this work establishes an efficient and robust paradigm for solving large-scale GEPs.
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