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:2608.29892v1 Announce Type: new
Abstract: Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scient...
By Abdolmehdi Behroozi, Chaopeng Shen
arXiv:2607. 20857v1 Announce Type: cross Abstract: Scientific machine learning methods such as neural operators and physics-informed neural networks have advanced engineering applications and inverse problems, but their training typically requires large volumes of simulated data.
By Amirhossein Nouranizadeh, Sarang Rajendra Patil, Alan John Varghese, Varsha Narayanan, Amit Chakraborty, Mengjia Xu
arXiv:2607. 19387v1 Announce Type: cross Abstract: Surrogate modeling for high-dimensional nonlinear dynamical systems that exhibit chaos requires mechanisms that preserve not only pointwise accuracy but also the scale-dependent structure of physical fields.
By Kanad Sen, Romit Maulik
arXiv:2604. 01802v2 Announce Type: replace Abstract: Real-time inference of inaccessible interior physical fields from sparse boundary observations is a fundamental but unresolved problem in scientific machine learning, with direct relevance to safety-critical monitoring across many engineering applications.
By William Howes, Jason Yoo, Kazuma Kobayashi, Subhankar Sarkar, Farid Ahmed, Souvik Chakraborty, Syed Bahauddin Alam
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)
TWIG (Time‑Causal Wavelet Operator for Irregular Graphs) is a graph‑native neural operator designed for autoregressive surrogate modeling on static irregular graphs. It transforms each node’s history into causal multiscale temporal features, separating recent changes from slower memory components, and propagates these through graph‑wavelet operator blocks with gated pointwise channel mixing. The architecture is causal by construction, enabling closed‑loop forecasting where predictions are recursively reused as future inputs, and it consistently outperforms non‑time‑causal baselines across three irregular‑domain forecasting problems.
By Subashree Venkatasubramanian, David A. Barajas-Solano, Chuyang Liu, Daniel M. Tartakovsky, Dipankar Dwivedi
arXiv:2608. 07161v1 Announce Type: cross Abstract: Simulating complex fluid flows requires capturing full equilibrium distributions rather than just mean trajectories, yet high-fidelity solvers remain computationally prohibitive.
By Shentong Mo, Guolin Ke
arXiv:2608. 13827v1 Announce Type: new Abstract: Machine-learned physical surrogate models have become promising alternatives to mesh-based numerical solvers.
By SiHun Lee, Dong-Hyuk Park, Taesoo Bang, Seung-Hoon Kang
arXiv:2602. 05352v3 Announce Type: replace Abstract: Modern neural networks have shown promise for solving partial differential equations over surfaces, often by discretizing the surface as a mesh and learning with a mesh-aware graph neural network.
By Edward Berman, Luisa Li, Jung Yeon Park, Robin Walters
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
arXiv:2607. 07718v1 Announce Type: cross Abstract: Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations.
By Oded Ovadia, Eli Turkel