Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers

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 \textit{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.

Large scientific simulations often need to find key values that describe how a system behaves, such as vibration frequencies, energy levels, or stability patterns. Finding these values in very large problems can be slow, and existing fast solvers still depend heavily on choosing good search regions. We propose Deepcontour, a method that uses machine learning to quickly estimate where the important values are likely to be, then automatically builds efficient search regions for reliable numerical solvers. Across several science and engineering problems, Deepcontour speeds up computation while preserving accuracy, making large-scale simulations faster and easier to run.