Stochastic Processes Under Linear Differential Constraints : Application to Gaussian Process Regression for the 3 Dimensional Free Space Wave Equation

TL;DR

This paper develops a new theoretical framework for Gaussian process regression constrained by PDEs, specifically the wave equation, and demonstrates its application to inverse problems and source localization in 3D space.

Contribution

It introduces a distributional PDE constraint condition for stochastic processes and applies it to physically informed GPR for the wave equation, including explicit covariance formulas.

Findings

Derived explicit covariance kernels for wave equation GPR

Linked GPR with classical GPS source localization methods

Provided a Bayesian approach for inverse initial condition reconstruction

Abstract

Let $P$ be a linear differential operator over $\mathcal{D} \subset \mathbb{R}^d$ and $U = (U_x)_{x \in \mathcal{D}}$ a second order stochastic process. In the first part of this article, we prove a new necessary and sufficient condition for all the trajectories of $U$ to verify the partial differential equation (PDE) $T(U) = 0$. This condition is formulated in terms of the covariance kernel of $U$. When compared to previous similar results, the novelty lies in that the equality $T(U) = 0$ is understood in the \textit{sense of distributions}, which is a relevant framework for PDEs. This theorem provides precious insights during the second part of this article, devoted to performing "physically informed" machine learning for the homogeneous 3 dimensional free space wave equation. We perform Gaussian process regression (GPR) on pointwise observations of a solution of this PDE. To do so, we propagate Gaussian processes (GP) priors over its initial conditions through the wave equation. We obtain explicit formulas for the covariance kernel of the propagated GP, which can then be used for GPR. We then explore the particular cases of radial symmetry and point source. For the former, we derive convolution-free GPR formulas; for the latter, we show a direct link between GPR and the classical triangulation method for point source localization used in GPS systems. Additionally, this Bayesian framework provides a new answer for the ill-posed inverse problem of reconstructing initial conditions for the wave equation with a limited number of sensors, and simultaneously enables the inference of physical parameters from these data. Finally, we illustrate this physically informed GPR on a number of practical examples.