Inferring turbulent velocity and temperature fields and their statistics from Lagrangian velocity measurements using physics-informed Kolmogorov-Arnold Networks

TL;DR

This work introduces AIVT, a physics-informed, Chebyshev-based Kolmogorov-Arnold Network (cKAN) framework to infer continuous temperature fields from sparse Lagrangian velocity measurements in Rayleigh-Bénard convection. By formulating a velocity–vorticity representation and enforcing data, boundary, and PDE residuals within a sequential training regime augmented by Residual-Based Attention with resampling (RBA-R), the method achieves DNS-like fidelity for both velocity (relative $L^2$ errors ~9–12%) and temperature (relative $L^2$ error ~3.6%), including derived quantities such as convective heat transfer and dissipation rates. The approach yields high-resolution instantaneous fields and statistically consistent gradient-based quantities (e.g., $\omega$, $Q$-$R$ invariants) that align with DNS and experimental benchmarks, enabling detailed turbulence statistics from velocity data alone. This paradigm offers a practical path to DNS-level turbulence insights at high Reynolds numbers where DNS is infeasible, and it lays the groundwork for applying physics-informed Kan-based inference to broader turbulent flows and measurement modalities.

Abstract

We propose the Artificial Intelligence Velocimetry-Thermometry (AIVT) method to infer hidden temperature fields from experimental turbulent velocity data. This physics-informed machine learning method enables us to infer continuous temperature fields using only sparse velocity data, hence eliminating the need for direct temperature measurements. Specifically, AIVT is based on physics-informed Kolmogorov-Arnold Networks (not neural networks) and is trained by optimizing a combined loss function that minimizes the residuals of the velocity data, boundary conditions, and the governing equations. We apply AIVT to a unique set of experimental volumetric and simultaneous temperature and velocity data of Rayleigh-Bénard convection (RBC) that we acquired by combining Particle Image Thermometry and Lagrangian Particle Tracking. This allows us to compare AIVT predictions and measurements directly. We demonstrate that we can reconstruct and infer continuous and instantaneous velocity and temperature fields from sparse experimental data at a fidelity comparable to direct numerical simulations (DNS) of turbulence. This, in turn, enables us to compute important quantities for quantifying turbulence, such as fluctuations, viscous and thermal dissipation, and QR distribution. This paradigm shift in processing experimental data using AIVT to infer turbulent fields at DNS-level fidelity is a promising avenue in breaking the current deadlock of quantitative understanding of turbulence at high Reynolds numbers, where DNS is computationally infeasible.

Figures (10)

• Figure 1: Problem Setup. (A) The experimental setup includes a hexagonal RBC cell, cameras, and a light source. The illuminated vertical slice is indicated in yellow. Gravity $g$ (black arrow) acts in a negative $y$ direction. The experiment was performed at Rayleigh number Ra = $3.4\times10^7$ and Prandtl number Pr = 10.6. (B) Schematic view of the joint temperature and velocity measurement processing. The particle positions and velocity vectors are obtained using the Shake-the-Box method (DAVIS 10.2, LaVision GmbH) schanz_shake--box_2016. The particle temperature is derived from the individual particle color images using a multi-layer perceptron pre-trained on calibration data. (C) At each training iteration, the domain is sampled based on a PDF that identifies high-error regions. These coordinates are fed into a modified Kolmogorov Arnold network (cKAN) that predicts 3D velocities and temperature. Residuals for data, boundary conditions, and equations are calculated, and derivatives for equation constraints are obtained using automatic differentiation. Residual-based attention (RBA) multipliers are updated using an exponentially weighted moving average of the residuals, and a PDF is computed for the next iteration's sampling. Residuals are scaled with RBA, balancing the local contribution of each training point. The mean of scaled residuals to a power $q$ forms each loss subterm. The total loss updates cKAN parameters, resulting in continuous and differentiable temperature and velocity fields.
• Figure 2: Velocity reconstruction. (A) Exemplary instantaneous velocity vectors of the measured and reconstructed velocity at the particle positions in 3D space. The velocity magnitude is color-coded. (B) Streamline plot of the $u$ and $v$ velocity in the $x-y$ plane at $z=0$ and detailed view of the measured (blue) and inferred velocity (red).(C) PDFs of the measured (blue) and reconstructed (red) velocity components at particle locations. (D) Vertical profile of mean velocity fields in the $x$ ($\langle u \rangle _{A,t}$), $y$ ($\langle |v| \rangle _{A,t}$) and $z$ ($\langle w \rangle _{A,t}$) directions. Based on the profiles of horizontal velocity components, the orientation and rotation direction of the LSC can be determined.
• Figure 3: Velocity and temperature fluctuations profiles. (A) Normalized root-mean-squared ($\sigma/(\sigma)_{\max}$) velocity $u^\prime, v^\prime, w^\prime$ and (B) temperature $T^\prime$ fluctuations on the upper half domain. The vertical dashed lines indicate the viscous $\delta_{\nu}$ and thermal $\delta_T$ boundary layer thicknesses based on equation \ref{['eq:vicous_boundary_layer']} and equation \ref{['eq:T_boundary_layer']}, respectively. The profiles are consistent with results reported in zhang_statistics_2017lui_spatial_1998zhou_thermal_2013 and scaling theory grossmann_scaling_2000stevens2013unifying.
• Figure 4: Inferred temperature and heat transfer results. (A) Comparison of the scatter plot of the measured and inferred temperature fluctuations at particle positions for an exemplary snapshot. (B) PDFs of the measured (blue), inferred (red), and reconstructed (green) temperature fluctuations at the particle locations. Due to the nonuniform sensitivity of the TLCs, the measurable temperature range is not symmetric but shifted towards positive values. (C) Comparison of the scatter plot of the measured and inferred convective heat transfer at particle positions for an exemplary snapshot. (D) PDFs of the measured (blue), inferred (red), and reconstructed (green) convective heat transfer at the particle locations. PDFs are skewed towards positive values since overall heat is transferred from the bottom to the top. The tails of the measured convective heat transfer PDF are extended further. We attribute this to the measurement uncertainties, inevitably leading to wider PDFs and the tendency of the PIKANs to produce smoothed results. Including a few temperature observations in the training process shifts the results closer to the measurements. (E) Vertical profile of mean temperature $\langle T \rangle _{A,t}$. The red line denotes the inferred profile, the blue markers the profile obtained from the binned Lagrangian measurement data, and the black line the temperature profile as proposed in reference shishkina_mean_2009. Since measured and inferred profile collapse in the bulk region, the offset of the bulk temperature from the theoretical profile is likely caused by slight deviations from the idealized Boussinesq approximation case, as shown in reference horn_rotating_2014. Close to the heating and especially the cooling plate, temperature measurements become unreliable due to the limited temperature range of the TLC particles. (F) View of the temperature field in the top and bottom boundary region covering the distance of 5 $\delta_T$ from the respective plate. The thermal boundary layer thickness, $\delta_T$, is indicated as a black dashed line. The color bar for both plots is presented in the middle of the figure.
• Figure 5: Inferred turbulent fields. Three snapshots of the inferred temperature fluctuations (A), vorticity magnitude (B), viscous dissipation rate (C), and thermal dissipation rate (D). The snapshots show the development of the flow over 15 free-fall time units. Snapshots like this are usually only available from DNS. Note the correlation between features among the different quantities.