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
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
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. L2
maxij∣Kij−Kij∣/maxij∣Kij∣
N
Nc
ϱN
Lh
LϱN
Lh
LϱN
Degree p=2
500
150
1.1107†
1.43×10−1
3.11
4.32×10−2
4.92×10−1
1000
150
0.6113
4.14×10−2
3.93×10−1
8.26×10−2
2.07×10−1
2000
150
0.3064
9.99×10−2
6.29×10−2
1.59×10−1
5.04×10−2
4000
200
0.1536
8.68×10−1
1.44×10−2
3.64×10−1
5.87×10−3
Table 1: Advection–diffusion–reaction on S1 . 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 ϱN is set to 1 by the first fallback.
Figure 2: Advection–diffusion–reaction on S1 . Left column p=2 and right column p=3 . Top: relative L2 error against the exact solution. Bottom: maxi,j∣Kij−Kij∣ . The curves are the direct GFDM solve ( ), PMM with exact operator and basis ( ), PMM with Lh and the exact Matérn kernel ( ), PMM with Lh and the graph kernel ( ) and PMM with LϱN and the graph kernel ( ).
Sphere S2
Method
rel. L2
rel. L∞
cov. 68.3% (%)
cov. 95% (%)
med. post. sd
Known manifold, analytic tangent frames
GFDM Lh , direct solve
4.61×10−1
2.17
–
–
–
PMM, exact operator and basis (ours)
1.38×10−8
3.30×10−8
100.0
100.0
1.97×10−7
PMM, Lh + exact Matérn (ours)
5.48×10−3
5.95×10−3
51.2
72.8
3.84×10−5
Implicit manifold, estimated tangent frames
Table 2: Steady advection–diffusion–reaction on S2 and T2 . The first two columns are the relative L2 and L∞ 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 S2 . The top row reports the predictions for (a) the manufactured exact solution, (b) the direct GFDM solve with Lh , (c) PMM with the exact operator and basis, (d) PMM with Lh and the exact Matérn kernel, (e) PMM with Lh and the graph kernel, and (f) PMM with LϱN and the graph kernel. The bottom row reports the absolute error against the manufactured solution, that is ∣u−μ~∣ 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 T2 , with the panels drawn in the fundamental domain (ϑ,φ)∈[0,2π)2 . The top row reports the predictions for (a) the manufactured exact solution, (b) the direct GFDM solve with Lh , (c) PMM with the exact operator and basis, (d) PMM with Lh and the exact Matérn kernel, (e) PMM with Lh and the graph kernel, and (f) PMM with LϱN 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.
Method
rel. L2
rel. L∞
cov. 68.3% (%)
cov. 95% (%)
med. post. sd
t=0.002
PMM, Lh + graph Matérn (ours)
3.50×10−2
1.08×10−1
64.5
91.0
0.067
PMM, LNϱ + graph Matérn (ours)
3.66×10−2
1.10×10−1
63.3
90.4
0.069
Iterated kernel smoothing ( Chung et al., 2005 )
6.52×10−2
4.16×10−1
–
–
–
Explicit FEM diffusion ( Chung and Taylor, 2004 )
2.21×10−2
7.84×10−2
–
–
–
t=0.01
Table 3: Cortical thickness smoothing at three diffusion times t . Errors are relative L2 and L∞ against the heat semigroup etΔg , computed as the eigenfunction expansion of Seo et al. (2010) in the cotangent basis truncated at 1000 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 , 10242 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 Lh ; (d) PMM with LϱN ; (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 Nc=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 Lh ; (b) PMM with LϱN ; (c) the finite-element diffusion of Chung and Taylor (2004) ; (d) the iterated kernel smoothing of Chung et al. (2005) .
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,
(κ2−Δ)α/2u=W,κ∈R,α∈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.
We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications. Existing methods for generative modeling on manifolds rely on substantial geometric information such as geodesic curves or eigenfunctions of the Laplace-Beltrami operator and, as a result, they are limited to manifolds where such information is available. In contrast, our method, built on a projection scheme, can be applied to more general manifolds, as it only requires being able to evaluate the value and the first order derivatives of the function that defines the submanifold. We provide a theoretical analysis of our method in the continuous-time limit, which elucidates the connection between our RDDPMs and score-based generative models on manifolds. The capability of our method is demonstrated on datasets from previous studies and on new datasets sampled from two high-dimensional manifolds, i.e. SO(10) and the configuration space of molecular system alanine dipeptide with fixed dihedral angle.
Zichen Liu, Wei Zhang, Christof Schütte +1
Center for Data Science, Peking University, Beijing 100871, P.R. China · Zuse Institute Berlin, Takustrasse 7, Berlin 14195, Germany · Institute of Mathematics, Freie Universität Berlin and Zuse Institute Berlin, Takustrasse 7, Berlin 14195, Germany +1
This paper proposes a new formulation of functional Gaussian Process regression on manifolds, based on an Empirical Bayes approach, in the spatiotemporal random field context. We apply the machinery of tight Gaussian measures in separable Hilbert spaces, exploiting the invariance property of covariance kernels under the group of isometries of the manifold. The identification via characteristic function of these measures with the infinite product of one-dimensional Gaussian measures is then obtained, in terms of the eigenfunctions of the Laplace-Beltrami operator on the manifold. The involved time-varying angular spectrum constitutes the key tool for dimension reduction in the implementation of this regression approach, adopting a suitable truncation scheme depending on the functional sample size. The simulation study and synthetic data application illustrate the performance of the proposed functional regression predictor.