Discrete Gaussian Vector Fields On Meshes

• URL: http://arxiv.org/abs/2507.20024v1
• Date: Sat, 26 Jul 2025 17:43:31 GMT
• Title: Discrete Gaussian Vector Fields On Meshes
• Authors: Michael Gillan, Stefan Siegert, Ben Youngman,
• Abstract summary: 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.<n>We show that these models can capture harmonic flows, incorporate boundary conditions, and model non-stationary data.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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