Graduate Student - 04, Iain Henderson (INSA Toulouse): Stochastic Processes Under Linear Differential Constraints : Application to Gaussian Process Regression for the 3 Dimensional Free Space Wave Equation

Enforcing linear differential constraints on the sample paths of stochastic processes requires special care because partial differential equations (PDEs) are usually understood in a weak (locally averaged) sense. Therefore, this weak formulation should be considered instead of a classical pointwise-defined one. Given a linear differential operator L, we state a general theorem that provides a simple necessary and sufficient condition for the sample paths of any centered second order stochastic process to verify the PDE Lu = 0 in the weak sense. This condition is formulated in terms of the covariance kernel of U. This theorem comes in handy when trying to perform physically informed Gaussian process regression (GPR) on the 3D wave equation. With reference to this equation, we very shorty present the covariance kernels for which our theorem applies and briefly showcase the potential of performing GPR with those kernels in an inverse problem framework. All of the presented material is a small extract from a more complete preprint available at https://arxiv.org/abs/2111.12035