Probabilistic Meshless Methods on Manifolds

Abstract

Probabilistic meshless methods solve partial differential equations (PDEs) by placing a Gaussian process prior on the solution and conditioning it on the PDE at a finite set of collocation nodes. The posterior provides both an approximation of the solution and a Bayesian quantification of its error. While these methods are well established on Euclidean domains, their extension to manifolds lacks, to our knowledge, any convergence analysis. We extend them to compact manifolds and study their convergence in two settings. First, for known manifolds whose Laplace-Beltrami eigenpairs are available, the prior can be evaluated exactly, but the PDE operator generally cannot be applied to it in closed form. We replace this operator by a standard generalized finite difference discretization on a point cloud sampled from the manifold, and show that the resulting method is consistent at the points of the cloud, at a rate governed by their fill distance. Second, for implicit manifolds, known only through a point cloud, both the prior and the operator must be approximated. We propose a new generalized finite difference scheme with a free stencil radius, and show that it retains the consistency of the standard construction while controlling the growth of its coefficients, and that an appropriate radius yields uniform convergence of the discrete Gram matrix. In both settings, we derive error bounds for the posterior mean which separate the collocation error intrinsic to the method from the error due to approximating the operator and, for implicit manifolds, the prior. Experiments on the circle, the sphere and the flat torus confirm our theoretical claims. On real-world cortical thickness data, our approach attains the accuracy of established discretizations, while conditioning on fewer than a tenth of the observations and providing a credible interval at every vertex.