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.

翻译：高斯过程回归技术已从降维视角应用于流体力学中的流场重构。该方法的核心在于构建适应的协方差函数（即核函数）以获得此类估计。本文提出了一种在任意紧集上约束预设高斯过程的通用方法。预设过程的核函数至少需连续，并可包含所研究现象的其他信息。这一通用的边界约束框架具有高度灵活性，可广泛应用于各类工程问题。基于此，我们推导出用于模拟空气动力学剖面周围不可压缩（无散度）流动二维速度场的物理信息核函数。这些核函数能够以连续方式定义满足不可压缩条件及剖面处预设边界条件的高斯过程先验。我们描述了一种适用于边界约束过程的数值方法，该方法通过紧集上的测度进行参数化。通过对圆柱体及NACA 0412翼型剖面绕流的数值模拟，展示了该方法的相关性与性能优势，该过程完全不需要边界观测数据。