Probabilistic Modelling of Vector Fields using Structure-Preserving Discretisation

We introduce a structure-preserving framework for probabilistic modelling of vector fields on Riemannian manifolds using finite element exterior calculus. A Gaussian process prior on vector fields is first constructed using the discrete Hodge Laplacian and then extended to non-Gaussian priors through flow matching. Our approach enables probabilistic reconstruction of vector fields while preserving, at the discrete level, the differential and geometric structure induced by the Hodge decomposition.