Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations

2024-11-25

arXiv.org

AI Summary

This work addresses boundary value problems (BVPs) for linear partial differential equations (PDEs) with constant coefficients. Method: We propose the Boundary Ehrenpreis–Palamodov Gaussian Processes (B-EPGPs) framework, the first to construct analytically exact Gaussian process priors that rigorously satisfy both the PDE system and general linear boundary conditions. Grounded in the fundamental Ehrenpreis–Palamodov theorem from algebraic analysis, our approach integrates analytic kernel design with explicit boundary constraint embedding, enabling exact, approximation-free conditioning on finite observational data. Contribution/Results: B-EPGPs yield probabilistic solutions with zero discretization error and strict adherence to physical constraints—surpassing neural operators and other data-driven solvers. We provide a formal correctness proof and demonstrate unified applicability across diverse linear PDE BVPs, achieving substantial advances in both theoretical rigor and solution accuracy.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Solving systems of partial differential equations (PDEs) is a fundamental task in computational science, traditionally addressed by numerical solvers. Recent advancements have introduced neural operators and physics-informed neural networks (PINNs) to tackle PDEs, achieving reduced computational costs at the expense of solution quality and accuracy. Gaussian processes (GPs) have also been applied to linear PDEs, with the advantage of always yielding precise solutions. In this work, we propose Boundary Ehrenpreis-Palamodov Gaussian Processes (B-EPGPs), a novel framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions. We explicitly construct GP priors for representative PDE systems with practical boundary conditions. Formal proofs of correctness are provided and empirical results demonstrating significant accuracy improvements over state-of-the-art neural operator approaches.

Problem

• Solving linear PDEs with constant coefficients
• Incorporating boundary conditions into Gaussian processes
• Improving computational efficiency for PDE systems

Innovation

• Novel probabilistic framework for linear PDE boundary problems
• Gaussian process priors satisfy PDEs and boundary conditions
• Explicit construction for representative systems with proofs