The paper introduces Spectrally Optimised Neural Discretisations (SpeND), a mesh‑free framework that learns discretisation weights from local stencil geometry on unstructured point clouds. By embedding discrete moment conditions into the network architecture, SpeND guarantees polynomial consistency and allows the weights to be optimised for spectral accuracy over a chosen wavenumber band, using an unsupervised Fourier‑mode loss. The resulting operators are PDE‑agnostic, perform well on Poisson, Burgers, and Navier–Stokes equations, and can reduce wall‑clock time by 3–20× compared to existing mesh‑free methods at the same accuracy.
By Lucas Gerken Starepravo, Henry Broadley, Steven Lind, Jack R. C. King
SPARCL introduces a spectral partitioned analytic continual learning method that addresses forgetting in analytic class‑incremental learning. By decomposing the running autocorrelation into a high‑energy core and a residual complement, SPARCL freezes core components for old classes and updates only the residual block, ensuring closed‑form updates with an invariance guarantee. Experiments on CIFAR‑100, CUB‑200, ImageNet‑R, and ImageNet‑A with a frozen ViT‑B/16 protocol show that SPARCL narrows the performance gap between classical analytic learners and strong representation matchers while complementing sparse feature‑decorrelation approaches.
By James Hartley, Zeropy Surio, Daniel Whitmore, Hannah Clarke, Thomas Reed
The paper introduces HermNet, a spectral graph neural network that uses Hermite polynomials for nodewise prediction and normalized propagation, avoiding eigendecomposition or learned bases. It compares HermNet to other complete polynomial bases, examining how coordinate choices affect optimization under limited training. Experiments on synthetic and real data show regimes where HermNet outperforms alternatives, and analyze the impact of calibration, regularization, and training duration on performance.
By Shuang Wu
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:2609. 23529v1 Announce Type: new Abstract: Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment.
By Hang-Cheng Dong, Pengcheng Cheng
arXiv:2606. 07289v1 Announce Type: new Abstract: Model merging combines several independently fine-tuned experts into a single multi-task model without any training data, reducing the storage, serving, and decentralized-development costs of large foundation models.
By Yongxian Wei, Runxi Cheng, Xingxuan Zhang, Li Shen, Chun Yuan, Peng Cui, Dacheng Tao
arXiv:2607. 22931v1 Announce Type: new Abstract: Analytic Continual Learning (ACL) offers a computationally efficient alternative to gradient-based approaches.
By Quyen Tran, Hai Nguyen, Quan Dao, Zhuowei Li, Nam Le, Trung Le, Dimitris Metaxas
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:2607. 02199v1 Announce Type: cross Abstract: Mutual information (MI)-inspired feature learning techniques are capable of generating low-dimensional embeddings that retain nonlinear dependence structures, but direct estimations of MI suffer from noisy probability distribution estimates in the low-data regime.
By Preston Pitzer, Anish Pradhan, Harpreet S. Dhillon
arXiv:2608.21234v1 Announce Type: cross
Abstract: These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classic...
By Victorita Dolean, Jemima Tabeart
arXiv:2505. 17868v2 Announce Type: replace Abstract: We present the first provable method for identifying symmetric linear dynamical systems (LDS) with accuracy guarantees that are independent of the systems' state dimension or effective memory.
By Devan Shah, Shlomo Fortgang, Sofiia Druchyna, Elad Hazan
arXiv:2604. 15613v4 Announce Type: replace-cross Abstract: We present Green-ELM, a non-iterative neural architecture that replaces gradient-based optimization of the output layer with a closed-form analytic solution over a fixed, high-dimensional random feature representation.
By Wladimir Silva