Discrete Gaussian Vector Fields On Meshes

2025-07-26

AI Summary

Modeling sparse vector field observations (e.g., wind, ocean currents) on two-dimensional manifolds—such as the Earth’s surface—poses challenges due to intrinsic curvature, boundary effects, and physical constraints (e.g., irrotationality or incompressibility). Method: We propose the first discrete intrinsic vector-valued Gaussian process (GP) model defined on arbitrary triangle meshes. By coupling discrete differential geometry—specifically covariant derivatives and Hodge decomposition—with intrinsic GPs, we construct a geometry-aware covariance operator that explicitly incorporates manifold curvature, boundary conditions, and physical priors. Efficient inference is achieved via variational approximation and sparse GP techniques. Results: Our model significantly outperforms Euclidean GPs and standard interpolation methods in global wind-field downscaling and sparse ocean-flow reconstruction. It accurately recovers complex circulation patterns, quantifies predictive uncertainty, and—uniquely—unifies geometric structure with physical constraints within a single probabilistic framework.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Develop discrete Gaussian processes for vector data on meshes
• Account for manifold geometry and curvature in modeling
• Apply models to downscale wind data and infer ocean currents

Innovation

• Discrete Gaussian processes for vector data
• Mesh-based differential operators for modeling
• Geometry-aware downscaling of environmental data