Where to Compute and How to Interact: Operator-Readable Adaptation with Gauge-Aware Transport
Read the original on arXiv Machine Learning →The Flow has not summarised this story yet — read it at arXiv Machine Learning.
The Flow has not summarised this story yet — read it at arXiv Machine Learning.
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.
arXiv:2604. 07366v2 Announce Type: replace Abstract: Partial differential equations (PDEs) govern nearly every physical process in science and engineering, but solving them at scale remains prohibitively expensive.
The paper introduces meshless Latent Dynamics Models (LDMs) for model order reduction of time‑dependent PDEs, using conditioning in Neural Fields to represent spatially continuous dynamics. It presents a unified mathematical framework, proves a Universal Approximation Theorem, and analyzes stability through Lipschitz bounds for different conditioning mechanisms, revealing an expressivity–stability trade‑off. Experiments on Burgers’ equation, Navier–Stokes flow, and linear advection demonstrate that meshless LDMs achieve comparable or better predictive accuracy than mesh‑based models while using only 2% of the parameters and maintaining near‑constant error across spatial discretizations.
arXiv:2608. 09764v1 Announce Type: cross Abstract: Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces.
arXiv:2607. 07718v1 Announce Type: cross Abstract: Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations.
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.