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

Abstract

Let P be a linear differential operator over D Ă R d and U " pU x q xPD 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 pUq " 0. This condition is formulated in terms of the covariance kernel of U. When compared to previous similar results [1], the novelty of this result is that the equality T pUq " 0 is understood in the sense of distributions, which is a functional analysis framework particularly adapted to the study of PDEs. This theorem provides precious insights during the second part of this article, which is dedicated to performing "physically informed" machine learning on data that is solution to the homogeneous 3 dimensional free space wave equation. We perform Gaussian Process Regression (GPR) on this data, which is a kernel based machine learning technique. To do so, we begin by modelling the solution of this PDE as a trajectory drawn from a Gaussian process (GP). We obtain explicit formulas for the covariance kernel of the corresponding stochastic process; this kernel can then be used for GPR. Our theorem states that this kernel, the trajectories of the corresponding GP and the predictions provided by GPR are all solutions to the wave equation in the sense of distributions. We explore two particular cases : the radial symmetry and the point source. In the case of radial symmetry, we derive "fast to compute" GPR formulas; in the case of the point source, we show a direct link between GPR and the classical triangulation method for point source localization used e.g. in GPS systems. We also show that this use of GPR can be interpreted as a new answer to the ill-posed inverse problem of reconstructing initial conditions for the wave equation with finite dimensional data, and also provides a way of estimating physical parameters from this data as in [2]. We finish by showcasing this physically informed GPR on a number of practical examples.