{"ID":2890228,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.20024","arxiv_id":"2507.20024","title":"Discrete Gaussian Vector Fields On Meshes","abstract":"Though the underlying fields associated with vector-valued environmental data are continuous, observations themselves are discrete. For example, climate models typically output grid-based representations of wind fields or ocean currents, and these are often downscaled to a discrete set of points. By treating the area of interest as a two-dimensional manifold that can be represented as a triangular mesh and embedded in Euclidean space, this work shows that discrete intrinsic Gaussian processes for vector-valued data can be developed from discrete differential operators defined with respect to a mesh. These Gaussian processes account for the geometry and curvature of the manifold whilst also providing a flexible and practical formulation that can be readily applied to any two-dimensional mesh. We show that these models can capture harmonic flows, incorporate boundary conditions, and model non-stationary data. Finally, we apply these models to downscaling stationary and non-stationary gridded wind data on the globe, and to inference of ocean currents from sparse observations in bounded domains.","short_abstract":"Though the underlying fields associated with vector-valued environmental data are continuous, observations themselves are discrete. For example, climate models typically output grid-based representations of wind fields or ocean currents, and these are often downscaled to a discrete set of points. By treating the area o...","url_abs":"https://arxiv.org/abs/2507.20024","url_pdf":"https://arxiv.org/pdf/2507.20024v1","authors":"[\"Michael Gillan\",\"Stefan Siegert\",\"Ben Youngman\"]","published":"2025-07-26T17:43:31Z","proceeding":"stat.ME","tasks":"[\"stat.ME\",\"math.ST\",\"stat.ML\"]","methods":"[]","has_code":false}