arXiv Machine Learning
Probabilistic Meshless Methods on Manifolds
Read the original on arXiv Machine Learning →The Flow has not summarised this story yet — read it at arXiv Machine Learning.
Related stories
arXiv Machine Learning
Jun 15
Approximating Whittle-Matern Fields over Discretized Manifolds
arXiv:2606. 13827v1 Announce Type: cross Abstract: Markovian Whittle-Mat\'ern fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}.
arXiv Machine Learning
Jun 17
Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds
arXiv:2606. 13827v2 Announce Type: replace-cross Abstract: Markovian Whittle-Mat\'ern fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}.
arXiv Machine Learning
Sep 30
Learning Spectrally Optimised Mesh-Free Discretisations
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 Machine Learning
6d ago
PDE-Residual-Informed Kriging
The paper introduces PDE‑Residual‑Informed Kriging (PRIK), a Gaussian process framework that embeds global PDE residual information directly into its covariance structure via a PDE‑residual kernel. PRIK yields a posterior mean with minimal integrated squared PDE residual and offers statistical interpretations linking variance to residual energy. The authors prove quasi‑optimality and convergence rates, and demonstrate the method’s PDE‑solving capability through numerical experiments.
arXiv Machine Learning
Jul 14
Riemannian Denoising Diffusion Probabilistic Models
arXiv:2505. 04338v3 Announce Type: replace Abstract: We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications.
arXiv Statistics ML
Oct 27
Determinantal sampling on general spaces: fast approximation via diffusion geometry
arXiv:2607.06644v2 Announce Type: replace Abstract: Determinantal point processes (DPPs) have recently emerged as a kernel-based alternative to standard independent sampling for constructing efficien...