Probabilistic Meshless Methods on Manifolds

Federico Pavesi; Noémie Jaquier; Antonio Candelieri

stat.MLOct 29, 2026

Probabilistic Meshless Methods on Manifolds

Authors: Federico Pavesi, Noémie Jaquier, Antonio Candelieri

Organizations: Department of Economics, Management and Statistics University of Milano-Bicocca Piazza dell’Ateneo Nuovo 1, 20126 Milan, Italy · Department of Robotics, Perception and Learning KTH Royal Institute of Technology Lindstedtsvägen 24, 10044 Stockholm, Sweden

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.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of the method. The equation is posed on a manifold known only through a sample of points, from which we build both a Gaussian prior over the solution and a point cloud approximation of the differential operator. Conditioning on the values of the forcing term at the collocation points (orange) returns a posterior, which gives the solution at every node together with its uncertainty.

			rel. $L^{2}$		$\max_{ij}\|\widehat{K}_{ij}-K_{ij}\|\,/\,\max_{ij}\|K_{ij}\|$
$N$	$N_{c}$	$\varrho_{N}$	$L^{h}$	$L^{\varrho_{N}}$	$L^{h}$	$L^{\varrho_{N}}$
Degree $p=2$
$500$	$150$	$1.1107^{\dagger}$	$\mathbf{1.43{\times}10^{-1}}$	$3.11$	$\mathbf{4.32{\times}10^{-2}}$	$4.92{\times}10^{-1}$
$1\,000$	$150$	$0.6113$	$\mathbf{4.14{\times}10^{-2}}$	$3.93{\times}10^{-1}$	$\mathbf{8.26{\times}10^{-2}}$	$2.07{\times}10^{-1}$
$2\,000$	$150$	$0.3064$	$9.99{\times}10^{-2}$	$\mathbf{6.29{\times}10^{-2}}$	$1.59{\times}10^{-1}$	$\mathbf{5.04{\times}10^{-2}}$
$4\,000$	$200$	$0.1536$	$8.68{\times}10^{-1}$	$\mathbf{1.44{\times}10^{-2}}$	$3.64{\times}10^{-1}$	$\mathbf{5.87{\times}10^{-3}}$

Table 1: Advection–diffusion–reaction on

\mathbb{S}^{1}

. Bold marks the lower of the two errors in each metric at each cloud size. A dagger marks the clouds on which the admissibility window is empty. The reported value is the estimated ceiling and

\varrho_{N}

is set to

1

by the first fallback.

Figure 2: Advection–diffusion–reaction on

\mathbb{S}^{1}

. Left column

p=2

and right column

p=3

. Top: relative

L^{2}

error against the exact solution. Bottom:

\max_{i,j}|\widehat{K}_{ij}-K_{ij}|

. The curves are the direct GFDM solve ( ), PMM with exact operator and basis ( ), PMM with

L^{h}

and the exact Matérn kernel ( ), PMM with

L^{h}

and the graph kernel ( ) and PMM with

L^{\varrho_{N}}

and the graph kernel ( ).

Sphere $\mathbb{S}^{2}$
Method	rel. $L^{2}$	rel. $L^{\infty}$	cov. $68.3\%$ (%)	cov. $95\%$ (%)	med. post. sd
Known manifold, analytic tangent frames
GFDM $L^{h}$ , direct solve	$4.61{\times}10^{-1}$	$2.17$	–	–	–
PMM, exact operator and basis (ours)	$\mathbf{1.38{\times}10^{-8}}$	$\mathbf{3.30{\times}10^{-8}}$	$100.0$	$100.0$	$1.97{\times}10^{-7}$
PMM, $L^{h}$ + exact Matérn (ours)	$5.48{\times}10^{-3}$	$5.95{\times}10^{-3}$	$51.2$	$72.8$	$3.84{\times}10^{-5}$
Implicit manifold, estimated tangent frames

Table 2: Steady advection–diffusion–reaction on

\mathbb{S}^{2}

and

\mathbb{T}^{2}

. The first two columns are the relative

L^{2}

and

L^{\infty}

error against the ground truth. The remaining columns are the empirical coverage of the credible intervals at those nominal levels and the median posterior standard deviation, and exist only for the probabilistic methods. Bold marks the lowest error in each column within each block.

Figure 3: Steady advection–diffusion–reaction on $\mathbb{S}^{2}$ . The top row reports the predictions for (a) the manufactured exact solution, (b) the direct GFDM solve with $L^{h}$ , (c) PMM with the exact operator and basis, (d) PMM with $L^{h}$ and the exact Matérn kernel, (e) PMM with $L^{h}$ and the graph kernel, and (f) PMM with $L^{\varrho_{N}}$ and the graph kernel. The bottom row reports the absolute error against the manufactured solution, that is $|u-\tilde{\mu}|$ at every node, of the panel directly above it; the cell under (a) is empty, the exact solution having no error against itself. Panels (a)–(d) use the analytic tangent frames, (e) and (f) the PCA estimates.

Figure 4: Steady advection–diffusion–reaction on $\mathbb{T}^{2}$ , with the panels drawn in the fundamental domain $(\vartheta,\varphi)\in[0,2\pi)^{2}$ . The top row reports the predictions for (a) the manufactured exact solution, (b) the direct GFDM solve with $L^{h}$ , (c) PMM with the exact operator and basis, (d) PMM with $L^{h}$ and the exact Matérn kernel, (e) PMM with $L^{h}$ and the graph kernel, and (f) PMM with $L^{\varrho_{N}}$ and the graph kernel. The bottom row reports the absolute error against the manufactured solution of the panel directly above it; the cell under (a) is empty, the exact solution having no error against itself. Panels (a)–(d) use the analytic tangent frames, (e) and (f) the PCA estimates.

$t=0.002$
Method	rel. $L^{2}$	rel. $L^{\infty}$	cov. $68.3\%$ (%)	cov. $95\%$ (%)	med. post. sd
PMM, $L^{h}$ + graph Matérn (ours)	$3.50{\times}10^{-2}$	$1.08{\times}10^{-1}$	$64.5$	$91.0$	$0.067$
PMM, $L^{\varrho}_{N}$ + graph Matérn (ours)	$3.66{\times}10^{-2}$	$1.10{\times}10^{-1}$	$63.3$	$90.4$	$0.069$
Iterated kernel smoothing ( Chung et al., 2005 )	$6.52{\times}10^{-2}$	$4.16{\times}10^{-1}$	–	–	–
Explicit FEM diffusion ( Chung and Taylor, 2004 )	$\mathbf{2.21{\times}10^{-2}}$	$\mathbf{7.84{\times}10^{-2}}$	–	–	–
$t=0.01$

Table 3: Cortical thickness smoothing at three diffusion times

t

. Errors are relative

L^{2}

and

L^{\infty}

against the heat semigroup

e^{t\Delta}g

, computed as the eigenfunction expansion of Seo et al. (2010) in the cotangent basis truncated at

1\,000

modes. The last three columns are the empirical coverage of the credible intervals at the two nominal levels and the median posterior standard deviation. Bold marks the lowest error in each column within each block.

Figure 5: Cortical thickness smoothing on the inflated left hemisphere of fsaverage5 , $10\,242$ vertices drawn on the triangulation for (a) the measured thickness $g$ , which is the datum being smoothed and is therefore identical in every row; (b) the reference, the eigenfunction expansion of Seo et al. (2010) in the cotangent basis at $1000$ modes; (c) PMM with $L^{h}$ ; (d) PMM with $L^{\varrho_{N}}$ ; (e) the finite-element diffusion of Chung and Taylor (2004) ; (f) the iterated kernel smoothing of Chung et al. (2005) . Columns (c) and (d) condition on $N_{c}=799$ vertices, $7.8\%$ of the mesh, while (b), (e) and (f) read all of them.

Figure 6: Absolute error of each compared method against the reference of Figure 5 (b) for (a) PMM with $L^{h}$ ; (b) PMM with $L^{\varrho_{N}}$ ; (c) the finite-element diffusion of Chung and Taylor (2004) ; (d) the iterated kernel smoothing of Chung et al. (2005) .

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Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds

Srinivas Nambirajan

Markov Random Fields Riemannian Geometry

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Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family,

(κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}.

of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to

α, κ

and thus allows a universal approximation scheme for the precision and covariance matrices of the entire

(α, κ)

-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

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