The paper introduces two multi-stage neural operator learning frameworks—Deep Collocation Neural Operator (DCNO) and Deep Galerkin Neural Operator (DGNO)—for efficiently computing convolution integrals. DCNO is a supervised method that iteratively refines operator approximations by learning residuals from data pairs, while DGNO is an unsupervised approach that uses the weak form of a PDE residual when the operator can be represented by a PDE. Both frameworks build basis operators across multiple training stages, yielding markedly higher accuracy than one-shot learning and achieving near machine‑precision results for convolution problems, with significant efficiency gains for repeated queries or parametric variations.
By Zhiping Mao, Zhenye Wen, Yong Zhang, Xiaofei Zhao
arXiv:2510. 14217v2 Announce Type: replace Abstract: The spectral properties of feature embeddings offer critical insights into model generalization and representation quality.
By Asma Jamali, Tin Sum Cheng, Rodrigo A. Vargas-Hern\'andez
arXiv:2603. 23647v2 Announce Type: replace-cross Abstract: In fluorescence microscopy, spectral unmixing aims to recover individual fluorophore concentrations from spectral images that capture mixed fluorophore emissions.
By Federico Carrara, Talley Lambert, Mehdi Seifi, Florian Jug
arXiv:2606. 00401v1 Announce Type: cross Abstract: Simulating large molecular systems comprising thousands of atoms requires highly scalable methodologies.
By Abhiram Badrinarayanan, Davor Davidovic, Edoardo Di Napoli, Jurica Novak, Luigi Genovese, Gustavo Ramirez-Hidalgo, Xinzhe Wu
arXiv:2601. 17090v2 Announce Type: replace-cross Abstract: Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps.
By Noam Koren, Rafael Moschopoulos, Kira Radinsky, Elad Hazan
arXiv:2607. 03692v1 Announce Type: new Abstract: Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling.
By Varvara Nazarenkko, Timur Lidzhiev, Alexander Tarakanov
arXiv:2604. 07421v3 Announce Type: replace Abstract: Full-waveform inversion (FWI) is pivotal for reconstructing high-resolution subsurface velocity models but remains computationally intensive and ill-posed.
By Zhenyu Wang, Peiyuan Li, Yongxiang Shi, Ruoyu Wu, Chenfei Liao, Lei Zhang
arXiv:2607. 12570v1 Announce Type: cross Abstract: Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations.
By Marc Haltmayer, Jaemin Seo, Yuseung Lee, Sungyeop Lee, Jaehoon Jeong, Jae Yong Lee
arXiv:2609.36047v1 Announce Type: cross
Abstract: Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional....
By Alexis de Villeroch\'e, Beniamin Bogosel, St\'ephane Breuils, Dorin Bucur, Jacques-Olivier Lachaud
arXiv:2407. 00809v4 Announce Type: replace Abstract: This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from functions to functions).
By Matthew Lowery, John Turnage, Zachary Morrow, John D. Jakeman, Akil Narayan, Shandian Zhe, Varun Shankar
The survey reviews frequency‑domain techniques for time‑series analysis, covering classical Fourier methods to modern neural operators. It identifies three main research challenges: preserving causal structure during spectral transformations, quantifying uncertainty in learned frequency representations, and performing topology‑aware analysis for non‑Euclidean data. By reviewing over 100 studies, the authors propose a unified taxonomy, establish standardized benchmarks, and highlight gaps in geometric deep learning and quantum‑enhanced spectral analysis.
By Qianru Zhang, Yuting Sun, Honggang Wen, Peng Yang, Xinzhu Li, Ming Li, Kwok-Yan Lam, Siu-Ming Yiu, Hongzhi Yin
arXiv:2608. 06894v1 Announce Type: new Abstract: Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations.
By Zhentao Tan, Ruijie Quan, Yi Yang