A PINN Methodology for Temperature Field Reconstruction in the PIV Measurement Plane: Case of Rayleigh-B\'enard Convection

We present a method to infer temperature fields from stereo particle-image velocimetry (PIV) data in turbulent Rayleigh-B\'enard convection (RBC) using Physics-informed neural networks (PINNs). The physical setup is a cubic RBC cell with Rayleigh number $\text{Ra}=10^7$ and Prandtl number $\text{Pr}=0.7$. With data only available in a vertical plane $A:x=x_0$, the residuals of the governing partial differential equations are minimised in an enclosing 3D domain around $A$ with thickness $\delta_x$. Dynamic collocation point sampling strategies are used to overcome the lack of 3D labelled information and to optimize the overall convergence of the PINN. In particular, in the out-of-plane direction $x$, the collocation points are distributed according to a normal distribution, in order to emphasize the region where data is provided. Along the vertical direction, we leverage meshing information and sample points from a distribution designed based on the grid of a direct numerical simulation (DNS). This approach points greater attention to critical regions, particularly the areas with high temperature gradients within the thermal boundary layers. Using planar three-component velocity data from a DNS, we successfully validate the reconstruction of the temperature fields in the PIV plane. We evaluate the robustness of our method with respect to characteristics of the labelled data used for training: the data time span, the sampling frequency, some noisy data and boundary data omission, aiming to better accommodate the challenges associated with experimental data. Developing PINNs on controlled simulation data is a crucial step toward their effective deployment on experimental data. The key is to systematically introduce noise, gaps, and uncertainties in simulated data to mimic real-world conditions and ensure robust generalization.