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:2512. 07847v2 Announce Type: replace Abstract: Benchmarking has been the cornerstone of progress in computer vision, natural language processing, and the broader deep learning domain, driving algorithmic innovation through standardized datasets and reproducible evaluation protocols.
By Mohamed Elrefaie, Dule Shu, Matt Klenk, Faez Ahmed
arXiv:2607. 11974v1 Announce Type: cross Abstract: Most neural partial differential equation (PDE) surrogates learn how fields evolve after a grid has already been chosen.
By Zixuan Shen (Central South University), Bingchuan Wang (Central South University), Zhi Wang (Nanjing University), Yong Wang (Central South University)
arXiv:2609.36806v1 Announce Type: new
Abstract: Modern engineering systems, from automobiles to aircraft, are designed by using precise, continuous parametric computer-aided design (CAD) models. Eval...
By Daniel Leibovici, Nikola Borislavov Kovachki, Dawon Ahn, Ruben Ohana, Ira J. S. Shokar, Abouzar Ghasemi, Semih Akkurt, Rishikesh Ranade, Neil Ashton, Jan Kautz, Jean Kossaifi
arXiv:2602. 04940v2 Announce Type: replace Abstract: Deep learning has emerged as a transformative tool for the neural surrogate modeling of partial differential equations (PDEs), known as neural PDE solvers.
By Hang Zhou, Haixu Wu, Haonan Shangguan, Yuezhou Ma, Huikun Weng, Jianmin Wang, Mingsheng Long
arXiv:2609.36806v2 Announce Type: replace
Abstract: Modern engineering systems, from automobiles to aircraft, are designed by using precise, continuous parametric computer-aided design (CAD) models....
By Daniel Leibovici, Nikola Borislavov Kovachki, Dawon Ahn, Ruben Ohana, Ira J. S. Shokar, Abouzar Ghasemi, Semih Akkurt, Rishikesh Ranade, Neil Ashton, Jan Kautz, Jean Kossaifi
The paper introduces the Cluster Attention Neural Operator (CANO), a neural operator that uses a novel cross‑attention mechanism to dynamically cluster queries while keeping full‑resolution keys and values. This design eliminates the slice compression and weight‑sharing limitations of previous Transformer‑based operators, maintaining fast computation and global interactions. Experiments on a range of fluid and solid dynamics benchmarks—including Navier‑Stokes, Airfoil, Plasticity, Pipe Turbulence, and Composites—show that CANO achieves lower errors than existing baselines and demonstrates strong geometric adaptability and temporal consistency.
By Ming Zhong, Antonio Colanera, Gianluigi Rozza, Zhenya Yan
arXiv:2607. 18020v2 Announce Type: replace Abstract: Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators.
By Peiyu Zang, Bosen Xie, Ruoxiang Xu, Yongqiang Cai
arXiv:2607. 01128v1 Announce Type: new Abstract: Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation.
By Meenakshi Krishnan, Pranav Pulijala, Ke Chen, Haizhao Yang, Ramani Duraiswami
arXiv:2607. 18020v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) solve PDEs by incorporating physical constraints into neural-network training, but large-scale problems are limited by automatic-differentiation memory overhead and inefficient execution of grid-based PDE operators.
By Peiyu Zang, Bosen Xie, Ruoxiang Xu, Yongqiang Cai
arXiv:2609.06660v1 Announce Type: cross
Abstract: Accurate aerodynamic prediction is critical for designing fuel-efficient and safe transportation systems such as aircraft and automobiles, yet tradit...
By Wenxuan Jin, Jianguo Yao, Haibing Guan, Xijun Li
arXiv:2602. 02788v2 Announce Type: replace-cross Abstract: We aim to develop physics foundation models for science and engineering that provide real-time solutions to Partial Differential Equations (PDEs) which preserve structure and accuracy under adaptation to unseen geometries.
By Benjamin D. Shaffer, Shawn Koohy, Brooks Kinch, M. Ani Hsieh, Nathaniel Trask