Data reconstruction of homogeneous turbulence using Lagrangian Particle Tracking 

with Shake-The-Box and machine learning 

Dong Kim, Kyung Chun Kim 

School of Mechanical Engineering, Pusan National University, 46241 Busan, South Korea 

 

This paper proposes a data reconstruction of homogeneous turbulent flow combined machine learning 

(ML) approach using experimental Lagrangian Particle Tracking (LPT) data with Shake-The-Box 

(STB). The LPT with STB was adopted to measure a von Kármán flow with a homogeneous turbulent 

region in the center [1]. The STB results have been stored and a temporal filter using 3rd order B-splines 

has been applied with optimal weighting coefficients to be used as input for FlowFit data assimilation 

method [2]. FlowFit data was used as ground truth to train ML algorithm. The low-resolved data of the 

velocity and acceleration field was reconstructed using an Adaptive Neuro-Fuzzy Inference System 

(ANFIS) with the downsampled LPT data as an input to predict homogeneous turbulent flow [3]. The 

training process can be mathematically regarded as an optimization problem to determine the weighting 

factor. 

  

 

Fig. 1 Data reconstruction methods and ANFIS learning structure. 



Given the input data set 𝑥  (low resolved STB data) and the desired output data set 𝑦  (FlowFit 

results), we aim to find the optimal weight 𝑤 in a machine-learned model 𝐹 that acts as a nonlinear 

regression function such that 𝐹(𝑥; 𝑤) ≈ 𝑦. In the present case, 𝑥 and 𝐹(𝑥; 𝑤) represent the low-

resolution and reconstructed high-resolution data, respectively. The weight 𝑤 is optimized between 

the desired high-resolution output 𝑦 and the ML model output 𝐹(𝑥;𝑤) is minimized. The ANFIS-

based data assimilation was first trained with FlowFit data assimilation result as ground truth. Four 

ANFIS training inputs on the x, y, z coordinates, and time t of FlowFit results were assigned to 1st layer 

of ANFIS algorithm. The training targets on velocity, acceleration components were assigned to final 

layer of ANFIS algorithm to get weighting factor. The computations were performed on a computer 

with an Intel® Core™ i5-8250U CPU @ 1.60 GHz 1.80 GHz and 8.0 GB of RAM. With 300 epochs 

and 4 input membership functions, the ANFIS training satisfied the convergence criterion of RMSE < 

0.01. The training time takes 2 hours. 

Figure 2 shows the spatial data reconstruction of ANFIS model with different ratio of raw particle 

density, 𝜌𝑝. The particle density was downsampled by random reduction from raw STB data (~100,000 

particles to 75, 50). Compared to FlowFit, ANFIS model can be well reconstructed above 50% particle 

density. It revealed much more small-scale vortical structures. Machine learning based data assimilation 

provide a better understanding of the small turbulence structures and allow for more in-depth analysis 

by recovering data due to the resolution limit of the experiment. To provide a full turbulence spectrum, 

full time-series of STB results will later be trained and in-depth analyzed for ANFIS training. 

 

 

Fig. 2 (top) (a) Raw STB result (b, c) downsampled STB data as input. (bottom) Contour colored by x-

component of acceleration and iso-surface of Q-criterion, Q = 5,000 s-2 from (a) FlowFit and (b, c) ANFIS. 



Acknowledgements 

This research was supported by Basic Science Research Program through the National Research 

Foundation of Korea (NRF) funded by the Ministry of Education (2020R1A6A3A03038341) and the 

Korean government (MSIT) (2021R1C1C2011538). We acknowledge Daniel Schanz, Florian Huhn, 

Sebastian Gesemann and Andreras Schröder (German Aerospace Center) for providing the LPT and 

FlowFit data, Daniel Garaboa-Paz (University of Santiago de Compostella) and Eberhard Bodenschatz 

(Max-Planck Institute for Dynamics and Self-Organization) are acknowledged for their contributions 

to the experiment 

 

References 

[1] Schröder, A., et al. (2019) Measuring the full velocity gradient and dissipation rate tensor in 

homogeneous turbulence using Shake-The-Box and FlowFit. 17th ETC, Torino, Italy. 

[2] Gesemann, S., Huhn, F., Schanz, D., & Schröder, A. (2016) From noisy particle tracks to velocity, 

acceleration and pressure fields using B-splines and penalties. In 18th LXLASER, Lisbon, Portugal, 4-7. 

[3] Kim, D., Safdari, A., & Kim, K. C. (2021) Sound pressure level spectrum analysis by combination 

of 4D PTV and ANFIS method around automotive side-view mirror models. Scientific Reports, 11(1), 

1-15.