## 4 Conclusions  
In the present study, to predict the transverse velocity field in the near-wake of laminar flow over a circular cylinder at the Reynolds numbers of 60 and 300, we constructed neural networks with instantaneous wall pressures on the cylinder surface as the input variables. For the two-dimensional unsteady flow at \(Re=60\), a fully connected neural network was considered. On the other hand, for the three-dimensional unsteady flow at \(Re=300\) having spanwise variations, we employed two different convolutional neural networks of CNN-F and CNN-D. Numerical simulations were carried out for both Reynolds numbers to obtain instantaneous flow fields, from which the input and output datasets were generated for training these neural networks. At the Reynolds numbers considered, the neural networks constructed accurately predicted the transverse velocity fields in the near-wake over the cylinder using the information of instantaneous wall pressures as the input variables. In addition, at \(Re=300\), it was observed that CNN-D showed a better prediction ability than CNN-F. The neural networks in this study were trained at a specific Reynolds number. Consequently, their performance at predicting other Reynolds numbers may not be as reliable. To improve the prediction performance for a wider range of Reynolds numbers, one could consider training the networks using flow fields from various Reynolds numbers. In addition, flow structures such as flow separation, shear layer, and wake are commonly observed in the flow over a cylinder across different Reynolds numbers. Designing neural networks to consider these features could therefore be a promising approach for future studies. The transverse velocity in the near-wake of the cylinder is a useful information indicating the status of the Karman vortex shedding. Thus, various feedback controls applied to flow over a circular cylinder measure the transverse velocity in the wake as sensors and aim to reduce its amplitude to weaken the strength of the Karman vortex shedding [27; 28; 29; 30; 31]. Figure 10: Contours of the instantaneous transverse velocity on the centerplane (\(y/d=0\)) at different temporal instances for \(Re=300\): (**a**) \(\sigma/u_{\infty}\) from the numerical simulation; (**b**) \(\partial/u_{\infty}\) predicted by CNN-D. Here, the four contour plots in (**a**,**b**) correspond to temporal instances arbitrarily chosen from the unsteady numerical simulation at \(Re=300\). However, it would not be practical to measure the transverse velocity in the wake in real applications of these control methods. Therefore, we believe that incorporating the present neural networks with feedback control methods would be an effective control strategy to improve its practical applicability because this approach does not need to measure the velocity and only measures the wall pressures as a sensor of feedback controller [27]. Conceptualization, methodology, investigation: J.Y. and J.L. ; validation, formal analysis, writing--original draft preparation, visualization: J.Y. and J.L. ; writing--review and editing, supervision, project administration, funding acquisition: J.L. All authors have read and agreed to the published version of the manuscript. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03932120). Additionally, this work was supported by Korea Ministry of Environment (MOE) as \"Chemical Accident Advancement Project\" (No. 2022003620005/1485018894). The data presented in this study are available from the corresponding author upon reasonable request. * [PERSON] et al. (2001) [PERSON]; [PERSON]; [PERSON] An immersed-boundary finite-volume method for simulations of flow in complex geometries. _J. Comput. Phys._**2001**, _171_, 132-150. [CrossRef]
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