Spatial prediction of the turbulent unsteady von Kármán vortex street using echo state networks

Abstract

The spatial prediction of the turbulent flow of the unsteady von Kármán vortex street behind a cylinder at Re = 1000 is studied. For this, an echo state network (ESN) with 6000 neurons was trained on the raw, low-spatial resolution data from particle image velocimetry. During prediction, the ESN is provided one half of the spatial domain of the fluid flow. The task is to infer the missing other half. Four different decompositions termed forward, backward, forward–backward, and vertical were examined to show whether there exists a favorable region of the flow for which the ESN performs best. Also, it was checked whether the flow direction has an influence on the network's performance. In order to measure the quality of the predictions, we choose the vertical velocity prediction of direction (VVPD). Furthermore, the ESN's two main hyperparameters, leaking rate (LR) and spectral radius (SR), were optimized according to the VVPD values of the corresponding network output. Moreover, each hyperparameter combination was run for 24 random reservoir realizations. Our results show that VVPD values are highest for LR ≈ 0.6, and quite independent of SR values for all four prediction approaches. Furthermore, maximum VVPD values of ≈0.83 were achieved for backward, forward–backward, and vertical predictions while for the forward case VVPDmax=0.74 was achieved. We found that the predicted vertical velocity fields predominantly align with their respective ground truth. The best overall accordance was found for backward and forward–backward scenarios. In summary, we conclude that the stable quality of the reconstructed fields over a long period of time, along with the simplicity of the machine learning algorithm (ESN), which relied on coarse experimental data only, demonstrates the viability of spatial prediction as a suitable method for machine learning application in turbulence.