Functional Gaussian processes for regression with linear PDE models

TL;DR

This paper introduces a novel functional Gaussian process regression method that integrates linear PDE models with observational data, enabling improved physical system predictions and uncertainty quantification.

Contribution

It develops a functional Gaussian process framework for PDE-based regression that incorporates model and observational data, handling linear functionals and providing uncertainty estimates.

Findings

The method effectively combines PDE models with observations.

It handles linear functional observations of the field variable.

Numerical results show improved prediction accuracy and uncertainty quantification.

Abstract

In this paper, we present a new statistical approach to the problem of incorporating experimental observations into a mathematical model described by linear partial differential equations (PDEs) to improve the prediction of the state of a physical system. We augment the linear PDE with a functional that accounts for the uncertainty in the mathematical model and is modeled as a {\em Gaussian process}. This gives rise to a stochastic PDE which is characterized by the Gaussian functional. We develop a {\em functional Gaussian process regression} method to determine the posterior mean and covariance of the Gaussian functional, thereby solving the stochastic PDE to obtain the posterior distribution for our prediction of the physical state. Our method has the following features which distinguish itself from other regression methods. First, it incorporates both the mathematical model and the observations into the regression procedure. Second, it can handle the observations given in the form of linear functionals of the field variable. Third, the method is non-parametric in the sense that it provides a systematic way to optimally determine the prior covariance operator of the Gaussian functional based on the observations. Fourth, it provides the posterior distribution quantifying the magnitude of uncertainty in our prediction of the physical state. We present numerical results to illustrate these features of the method and compare its performance to that of the standard Gaussian process regression.