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Vector-Valued Gaussian Processes for Approximating Divergence- or Rotation-free Vector Fields

Quoc Thong Le Gia, Ian Hugh Sloan, Holger Wendland

TL;DR

This work develops a rigorous framework for vector-valued Gaussian processes focused on divergence-free or curl-free vector fields. It extends GP theory to Hilbert-space valued functions, introduces divergence-free and curl-free matrix-valued kernels, and connects these to vector-valued reproducing kernel Hilbert spaces and Sobolev spaces, including extension/restriction results. The authors provide comprehensive Sobolev-space error analyses for the predictive mean under both interpolation (noise-free) and approximation (noisy) regimes, with explicit rates governed by fill distance, mesh ratio, and smoothness indices. The results supply concrete convergence guarantees and kernel-design guidance for physically constrained vector-field reconstruction in applications such as incompressible fluid dynamics and electromagnetic modeling.

Abstract

In this paper, we discuss vector-valued Gaussian processes for the approximation of divergence- or rotation-free functions. We establish the theory for such Gaussian processes, then link the theory to multivariate approximation theory, and finally give error estimates for the predictive mean in various situations.

Vector-Valued Gaussian Processes for Approximating Divergence- or Rotation-free Vector Fields

TL;DR

This work develops a rigorous framework for vector-valued Gaussian processes focused on divergence-free or curl-free vector fields. It extends GP theory to Hilbert-space valued functions, introduces divergence-free and curl-free matrix-valued kernels, and connects these to vector-valued reproducing kernel Hilbert spaces and Sobolev spaces, including extension/restriction results. The authors provide comprehensive Sobolev-space error analyses for the predictive mean under both interpolation (noise-free) and approximation (noisy) regimes, with explicit rates governed by fill distance, mesh ratio, and smoothness indices. The results supply concrete convergence guarantees and kernel-design guidance for physically constrained vector-field reconstruction in applications such as incompressible fluid dynamics and electromagnetic modeling.

Abstract

In this paper, we discuss vector-valued Gaussian processes for the approximation of divergence- or rotation-free functions. We establish the theory for such Gaussian processes, then link the theory to multivariate approximation theory, and finally give error estimates for the predictive mean in various situations.

Paper Structure

This paper contains 14 sections, 25 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Gaussian Processes and Approximation
  3. Scalar-Valued Gaussian Processes
  4. Connection to Multivariate Approximation Theory
  5. Connection to Functional Analysis
  6. Vector-Valued Function Approximation
  7. Generalized Gaussian Processes
  8. Divergence- and Rotation-free Gaussian Processes
  9. Vector-Valued Reproducing Kernel Hilbert Spaces
  10. Sobolev Spaces
  11. Divergence- and Rotation-Free Kernels
  12. Error Analysis
  13. Error Analysis for the Predictive Mean using Interpolation
  14. Error Analysis for the Predictive Mean using Approximation

Key Result

Theorem 2.2

Let $K:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ be a positive definite kernel and let $X=\{\boldsymbol{x}_1,\ldots,\boldsymbol{x}_N\}\subseteq\mathcal{D}$ consist of pairwise distinct points. Define the $N$-dimensional approximation space $V_X=\mathop{\mathrm{span}}\limits\{K(\cdot,\boldsymbol{x}_

Theorems & Definitions (51)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5: Mercer
  • Theorem 2.6: Karhunen-Loève
  • Definition 3.1
  • Definition 3.2
  • ...and 41 more