We introduce a physics-informed Gaussian process regression (GPR) method for data assimilation and uncertainty quantification of particle tracking velocimetry data. Unlike traditional methods based on regression, our approach transparently incorporates statistical information and physics such as mass conservation, boundary conditions, and statistical symmetries directly into the regression model. Furthermore, GPR quantifies prediction uncertainty and provides physics-constrained estimates of the two-point velocity covariance, a quantity of primary interest in turbulent flows. The methodology is demonstrated using synthetic and experimental data from three canonical turbulent flows: homogeneous isotropic turbulence (HIT), turbulent channel flow (TCF), and the turbulent wake behind a square prism (SPW). In all cases, we make comparisons relative to the performance of the vortex in cell method, $\mathrm{VIC}+$. For HIT, the model leverages isotropy to learn the velocity correlation function from even very noisy and sparse data, achieves a factor of two improvement over $\mathrm{VIC}+$ in velocity prediction error, and accurately quantifies the prediction uncertainty. For TCF, we introduce a scalable approach to train a high-dimensional GP model that respects wall-bounded flow physics. GPR significantly outperforms $\mathrm{VIC}+$ in terms of accuracy, uncertainty estimation, and resolution in this case. In the SPW case, GPR demonstrates improved accuracy in velocity prediction and improved coherence of the vorticity field obtained from independent snapshots of tracers. Our approach lays the groundwork for extensions to time-resolved data, inclusion of acceleration measurements, and reduced-parameter models based on resolvent analysis.

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• Turbulence
• Lagrangian particle tracking
• Regression analysis
• Supervised learning
• Training models