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

TL;DR

This paper introduces a flexible, physics-informed Gaussian process regression framework that constrains flow field reconstructions to boundary conditions and physical laws, demonstrated on aerodynamic flow simulations without boundary observations.

Contribution

It develops a general boundary-constraining method for Gaussian processes, enabling physics-informed flow reconstructions with boundary conditions incorporated directly into the kernels.

Findings

Effective flow field reconstructions around cylinders and airfoils.

No boundary observations needed for accurate results.

Flexible framework applicable to various engineering problems.

Abstract

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 present a general method for constraining a prescribed Gaussian process on an arbitrary compact set. The kernel of the pre-defined process must be at least continuous and may include other information about the studied phenomenon. This general boundary-constraining framework can be implemented with high flexibility for a broad range of engineering applications. From this, 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. We describe an adapted numerical method for the boundary-constraining procedure parameterized by a measure on the compact set. The relevance of the methodology 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.