Gaussian processes for data fulfilling linear differential equations

TL;DR

This paper introduces a Gaussian process regression method tailored to data satisfying linear differential equations, improving physical consistency and interpretability of the reconstructed fields and parameters.

Contribution

It develops a novel approach that restricts Gaussian process covariance functions to those satisfying linear differential equations, enhancing physical relevance and interpretability.

Findings

More reliable regression results with physically consistent solutions

Hyperparameters become directly interpretable in physical terms

Effective estimation of source strengths and unknown parameters

Abstract

A method to reconstruct fields, source strengths and physical parameters based on Gaussian process regression is presented for the case where data are known to fulfill a given linear differential equation with localized sources. The approach is applicable to a wide range of data from physical measurements and numerical simulations. It is based on the well-known invariance of the Gaussian under linear operators, in particular differentiation. Instead of using a generic covariance function to represent data from an unknown field, the space of possible covariance functions is restricted to allow only Gaussian random fields that fulfil the homogeneous differential equation. The resulting tailored kernel functions lead to more reliable regression compared to using a generic kernel and makes some hyperparameters directly interpretable. For differential equations representing laws of physics such a choice limits realizations of random fields to physically possible solutions. Source terms are added by superposition and their strength estimated in a probabilistic fashion, together with possibly unknown hyperparameters with physical meaning in the differential operator.