Pressure reconstruction from error-embedded gradient measurements: a Gaussian-process generalization of Green's function integration

Reconstructing scalar fields from error-embedded gradient measurements is a fundamental linear inverse problem with broad applications in computational physics. Conventional approaches, such as Poisson-based solvers and the Green's Function Integration (GFI) method, require explicit boundary conditions extracted from the same error-embedded observations. In this study we assess the accuracy of a Gaussian Process Regression (GPR) framework for reconstructing pressure fields in turbulent flows from error-embedded pressure-gradient data derived from kinematic measurements. The probabilistic nature of GPR inherently provides tunable denoising, eliminates the need for boundary conditions, and produces a pointwise posterior-variance error estimate. A central theoretical result of the present work is that GFI is the noiseless limit of GPR, which on the unbounded plane reduces to the well-known logarithmic kernel and in three dimensions to the inverse-distance kernel. The framework is validated on two-dimensional slices and three-dimensional subdomains of a forced homogeneous isotropic turbulence from the Johns Hopkins Turbulence Database. With an empirical mixture-of-Gaussians (MoG-$3$) kernel fitted directly to the pressure correlation function, GPR performs at least as well as GFI. In situations with under-resolved data or high noise, GPR outperforms GFI, while delivering a calibrated pointwise posterior uncertainty whose standardized residuals satisfy $|z|<2$ over $95\%$ of grid points. The framework extends to three dimensions through a tensor-product Kronecker solver coupled to conjugate gradients with close to $\mathcal{O}(N^3\log N)$ cost. A closed-form error lower bound on a periodic cube is derived for the GPR operator, with the residual gap attributable to boundary contamination on non-periodic finite domains.