Linear Latent Force Models using Gaussian Processes

TL;DR

This paper introduces a hybrid modeling approach combining Gaussian processes with differential equations to effectively incorporate physical knowledge into data-driven models, especially useful when data is limited.

Contribution

It develops physically-inspired kernel functions within Gaussian processes, enabling hybrid models that integrate mechanistic understanding with data-driven learning.

Findings

Versatile approach demonstrated across three case studies

Improved modeling with limited data

Seamless integration of physical models and data-driven methods

Abstract

Purely data driven approaches for machine learning present difficulties when data is scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data driven modelling with a physical model of the system. We show how different, physically-inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology and geostatistics.