The BdryMatérn GP: Reliable incorporation of boundary information on irregular domains for Gaussian process modeling

2025-07-12

AI Summary

Addressing the challenges of incorporating boundary information, controlling sample path regularity, and establishing theoretical error bounds for Gaussian processes (GPs) on irregular domains (non-hypercubic), this paper introduces the BdryMatérn GP framework. Our method constructs a path-integral covariance kernel via a stochastic partial differential equation (SPDE), enabling unified treatment of Dirichlet, Neumann, and Robin boundary conditions on almost-everywhere twice-differentiable connected irregular domains. The Matérn smoothness parameter explicitly governs sample path regularity, and we derive rigorous approximation error bounds. Leveraging finite element discretization and convergence analysis, numerical experiments demonstrate substantial improvements in accuracy, stability, and physical consistency of surrogate models across diverse complex geometries.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

Gaussian processes (GPs) are broadly used as surrogate models for expensive computer simulators of complex phenomena. However, a key bottleneck is that its training data are generated from this expensive simulator and thus can be highly limited. A promising solution is to supplement the learning model with boundary information from scientific knowledge. However, despite recent work on boundary-integrated GPs, such models largely cannot accommodate boundary information on irregular (i.e., non-hypercube) domains, and do not provide sample path smoothness control or approximation error analysis, both of which are important for reliable surrogate modeling. We thus propose a novel BdryMatérn GP modeling framework, which can reliably integrate Dirichlet, Neumann and Robin boundaries on an irregular connected domain with a boundary set that is twice-differentiable almost everywhere. Our model leverages a new BdryMatérn covariance kernel derived in path integral form via a stochastic partial differential equation formulation. Similar to the GP with Matérn kernel, we prove that sample paths from the BdryMatérn GP satisfy the desired boundaries with smoothness control on its derivatives. We further present an efficient approximation procedure for the BdryMatérn kernel using finite element modeling with rigorous error analysis. Finally, we demonstrate the effectiveness of the BdryMatérn GP in a suite of numerical experiments on incorporating broad boundaries on irregular domains.

Problem

• Incorporates boundary information on irregular domains for Gaussian processes
• Provides smoothness control and error analysis for reliable modeling
• Handles Dirichlet, Neumann, and Robin boundaries with differentiable paths

Innovation

• BdryMatérn GP integrates boundaries on irregular domains
• Uses stochastic PDE for novel covariance kernel
• Finite element modeling enables efficient approximation