Kolmogorov Artificial Intelligence Velocimetry infers hidden temperature from turbulent experimental velocity data*

We propose Kolmogorov Artificial Intelligence Velocimetry (KAIV) to infer hidden temperature fields from turbulent experimental velocity data. This scientific machine-learning approach allows temperature prediction using only velocity data, eliminating the need for direct temperature measurements. Our models are based on physics-informed Kolmogorov Arnold Networks (PIKANs) and are trained by optimizing a combined loss function that minimizes the residuals of the velocity data, boundary conditions, and the governing equations. To manage local imbalances in the optimization process, we propose a residual-based attention method with resampling (RBA-R) that enhances stability and efficiency by utilizing historical residual data for sampling and local multipliers to balance the point-wise errors. To ensure exact constraint enforcement, we use approximate distance functions for temperature boundary conditions and redesign the base model to predict divergence-free fields directly. We apply KAIV to experimental volumetric and simultaneous temperature and velocity data of Rayleigh-Bénard convection obtained from combined Particle Image Thermometry (PIT) and Lagrangian Particle Tracking (LPT), which allows us to compare KAIV predictions and measurements directly. Furthermore, we demonstrate its efficacy by accurately calculating convective heat transfer, analyzing the QR distribution, and viscous and thermal dissipation rates from the resulting velocity and temperature fields.