A unified perspective of Gaussian process approximation for differential equations

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.