Physics-informed, boundary-constrained Gaussian process regression for the reconstruction of fluid flow fields

Gaussian process regression techniques have been used in fluid mechanics for the reconstruction of flow fields from a reduction-of-dimension perspective. A main ingredient in this setting is the construction of adapted covariance functions, or kernels, to obtain such estimates. In this paper, we derive physics-informed kernels for simulating two-dimensional velocity fields of an incompressible (divergence-free) flow around aerodynamic profiles. These kernels allow to define Gaussian process priors satisfying the incompressibility condition and the prescribed boundary conditions along the profile in a continuous manner. Such physical and boundary constraints can be applied to any pre-defined scalar kernel in the proposed methodology, which is very general and can be implemented with high flexibility for a broad range of engineering applications. Its relevance and performances are illustrated by numerical simulations of flows around a cylinder and a NACA 0412 airfoil profile, for which no observation at the boundary is needed at all.