DeepContour: A Hybrid Deep Learning Framework for Accelerating Generalized Eigenvalue Problem Solving via Efficient Contour Design

TL;DR

This work proposes DeepContour, a novel hybrid framework that integrates a deep learning-based spectral predictor with Kernel Density Estimation for principled contour design and pioneers an efficient and robust paradigm for tackling difficult generalized eigenvalue involving matrices of high dimension.

Abstract

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.

Figures (6)

• Figure 1: The variation in tolerance for DeepContour compared to scouting methods. Each line shows experimental results using a contour selection strategy and a CI-based solver. Notably, DeepContour substantially enhances the efficiency of solving Generalized Eigenvalue Problem, achieving a speed-up of up to 5.63 times. A comparison with traditional iterative eigensolvers is provided in Appendix \ref{['subsec:appendix_iterative']}. Their significantly slower solving performance also highlights motivation for accelerating contour integral methods.
• Figure 2: Experiments demonstrate the CIRR's high sensitivity to contour selection, as evidenced by the large performance variance observed under a random contour strategy.
• Figure 3: Overall architecture of DeepContour: (a) Construct contour $\Gamma$ for CI solver to solve given matrices $A$ and $B$. (b) Traditional Scout-based Method: An iterative solver (e.g., Arnoldi) is run for a fixed number of steps to obtain a rough spectral distribution, and a safety margin is then applied to define the integration contour. (c1)DeepContour Eigenvalue Prediction Module: Utilizing a specialized eigenvalue neural operator for estimating target eigenvalues. (c2)DeepContour Adaptive Contour Construction Module: Leveraging Kernel Density Estimation (KDE) to automatically construct tight-fitting, optimized integration contours. (d) Final Solve Stage: The contours is passed to a CI-based Eigensolver (e.g., CIRR) for final solution.
• Figure 4: Experiments on the Kirchhoff-Love Plate and EGFR Electronic problems with varying matrix sizes. The results indicate that as the matrix size increases, both time speedup and iteration speedup increase.
• Figure 5: Training curves of our ENO model across five datasets.