arXiv:2602. 11626v3 Announce Type: replace-cross Abstract: Learning solution operators on arbitrary geometries remains a central challenge in scientific machine learning, especially for many-query simulation, physics-informed learning, and evolving geometries requiring accurate, geometry-aware predictions at arbitrary spatial locations.
By Wenqian Chen, Zhi-Feng Wei, Yucheng Fu, Michael Penwarden, Pratanu Roy, Panos Stinis
arXiv:2505. 19809v3 Announce Type: replace-cross Abstract: In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency.
By Daniel Ordo\~nez-Apraez, Vladimir Kosti\'c, Alek Fr\"ohlich, Vivien Brandt, Karim Lounici, Massimiliano Pontil
arXiv:2606. 29440v1 Announce Type: new Abstract: Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training.
By Zirui Deng, Jingbo Sun, Deyu Meng, Fei Wang
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
arXiv:2609.38623v1 Announce Type: new
Abstract: Learning partial differential equation (PDE) dynamics across varying domains is central to predictive modelling and data-driven discovery of governing...
By Yinghao Cheng, Gengxiang Chen, Xu Liu, Qinglu Meng, Yixin Jing, Xiangguo Tang, Wenping Mou, Lihui Wang, Yingguang Li
arXiv:2606. 01427v1 Announce Type: cross Abstract: Foundation models (FMs) have achieved substantial success in generalizing across tasks without problemspecific training or fine-tuning.
By Tyler R. Johnson, Kian Ben-Jacob, Nima Negarandeh, Oriol Vendrell-Gallart, Ramin Bostanabad