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

Let P be a linear differential operator over πββ^d and U = (U_x)_x βπ a second order stochastic process. In the first part of this article, we prove a new simple 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. The novelty of this result is that the equality T(U) = 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 Bayesian approach to machine learning. To do so, we put Gaussian process (GP) priors over the wave equation's initial conditions and propagate them through the wave equation. We obtain explicit formulas for the covariance kernel of the corresponding stochastic process; this kernel can then be used for GPR. We explore two particular cases : the radial symmetry and the 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 e.g. in GPS systems. Additionally, this Bayesian framework gives rise to a new answer for the ill-posed inverse problem of reconstructing initial conditions for the wave equation with finite dimensional data, and simultaneously provides a way of estimating physical parameters from this data as in [Raissi et al,2017]. We finish by showcasing this physically informed GPR on a number of practical examples.

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