A unified perspective of Gaussian process approximation for differential equations

2026-07-07

AI Summary

This work addresses the fragmentation in existing Gaussian process–based approaches for approximating solutions to differential equations by proposing a unified Bayesian probabilistic framework. The framework embeds differential equation constraints into the likelihood function through derivative matching, thereby enabling simultaneous estimation of unknown parameters and quantification of uncertainty in the solution. It systematically integrates several established Gaussian process methods for the first time, elucidating their underlying connections. The generality and efficacy of the proposed approach are demonstrated across multiple benchmark problems, establishing a coherent foundation for future theoretical advancements and practical applications in this domain.

Abstract

The use of Gaussian processes for approximating differential equations has expanded rapidly, leading to a growing, diverse, and fragmented body of numerical methods. We present a unified Bayesian perspective that places these techniques within a common probabilistic framework, based on a derivative matching interpretation for incorporating differential equation constraints into likelihood. This unified perspective supports both parameter estimation and solution approximation, and shows how a range of existing methods can be understood within it. This work aims to consolidate current developments and provide a foundation for future research.

Problem

• Gaussian processes
• differential equations
• probabilistic framework
• numerical methods
• unified perspective

Innovation

• Gaussian processes
• differential equations
• Bayesian framework
• derivative matching
• probabilistic numerical methods