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.
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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.
arXiv:2609.15620v1 Announce Type: new Abstract: Adaptive meshes enable neural operators for partial differential equations (PDEs) to allocate spatial samples and computation according to local physic...
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.
arXiv:2512. 19643v2 Announce Type: replace Abstract: Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings.
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 a new federated learning protocol for partial differential equations called solution-space PDE-Dirichlet, which transforms continuous supervised responses into reusable solution bins and measures client separation via optimal transport. It establishes an exact inverse relationship between population allocation heterogeneity and Dirichlet concentration, and shows how response heterogeneity can cause gradient disagreement, local-update dispersion, and parameter divergence. Experiments on seven PDE tasks, three neural-operator families, and five random seeds demonstrate that lower concentration consistently increases solution distance and optimization heterogeneity, with the most pronounced error increase observed in low-viscosity Burgers equations.