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

arXiv: https://arxiv.org/abs/2511.01927 · HTML: https://mlsanity.com/paper/2511.01927/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 71 references that are in the map.
- First posted on arXiv 2 Nov 2025.

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Spectral Inference Networks: Unifying Deep and Spectral Learning](https://mlsanity.com/paper/1806.02215/) (2018)
- [NeuralEF: Deconstructing Kernels by Deep Neural Networks](https://mlsanity.com/paper/2205.00165/) (2022)
- [Physics-Informed Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2203.00451/) (2022)
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- [Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2510.25803/) (2025)
- [Operator SVD with Neural Networks via Nested Low-Rank Approximation](https://mlsanity.com/paper/2402.03655/) (2024)
- [Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling](https://mlsanity.com/paper/2401.09516/) (2024)
- [Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problems](https://mlsanity.com/paper/2209.11134/) (2022)
- [Self-Attention to Operator Learning-based 3D-IC Thermal Simulation](https://mlsanity.com/paper/2510.15968/) (2025)
- [SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning](https://mlsanity.com/paper/2510.24170/) (2025)
- [Solving the functional Eigen-Problem using Neural Networks](https://mlsanity.com/paper/2007.10205/) (2020)
- [STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem](https://mlsanity.com/paper/2510.23986/) (2025)
- [A Shift Selection Strategy for Parallel Shift-Invert Spectrum Slicing in Symmetric Self-Consistent Eigenvalue Computation](https://mlsanity.com/paper/1908.06043/) (2019)
- [Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter](https://mlsanity.com/paper/2510.23215/) (2025)

## Abstract (arXiv)

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.