## 3 Results and Discussion
### Prediction of Near-Wake Velocity at \(Re=300\)  
In laminar flow over a circular cylinder at \(Re<194\), it is well-known that the Karman vortex shedding maintains its two-dimensionality without spanwise variations in its vortical structure [1, 41]. Thus, we considered two-dimensional flow at \(Re=60\) in Section 3.2. However, the Karman vortex shedding behind a circular cylinder becomes three-dimensional for \(Re\geq 194\), where spanwise variations in vortical structures are observed owing to the occurrences of vortex loops and streamwise vortices [1, 41]. Therefore, predictions of the wake flow field over a circular cylinder at \(Re=300\) would be a more difficult task than those at \(Re=60\). In the present study, to predict the transverse velocity on the centerplane in the near-wake behind the cylinder at \(Re=300\), we construct two different neural network structures (CNN-F and CNN-D) as described in Figure 2b,c. We note that a neural network similar to CNN-F was used in the study of [PERSON] et al. [24]. Figure 5 shows loss functions versus the number of epochs for the training of both CNN-D and CNN-F structures at \(Re=300\). As shown, CNN-D exhibits lower values of loss function than CNN-F as the number of epochs increases, suggesting that CNN-D has better prediction accuracy for the hyperparameters and dataset considered in this study. Notably, during the learning process of CNN-F, the value of loss function for testing data increases after 100 epochs, which suggests that a mild overfitting occurs for CNN-F. We suspect that this overfitting occurs because the number of instantaneous flow fields is insufficient for the proper training of CNN-F. On the other hand, CNN-D does not exhibit an overfitting behavior. We believe that this could be attributed to the fact that the decoder structure of  
Figure 4: Contours of the instantaneous transverse velocity at \(Re=60\): (**a**) \(v/u_{\infty}\) from the numerical simulation; (**b**) \(\partial/u_{\infty}\) predicted by FCNN. The three contour plots in (**a**,**b**) correspond to temporal instances arbitrarily chosen from the unsteady numerical simulation at \(Re=60\). CNN-D has fewer parameters in hidden layers than the FCNN structure of CNN-F, making it less vulnerable to overfitting when dealing with a smaller dataset. Figure 6 shows the correlation function \(R\) according to the streamwise location \((x/d)\) at \(Re=300\), where \(R\) is obtained by Equation (4). Here, owing to the statistical homogeneity in the spanwise direction, we conduct an averaging of \(R\) in the spanwise direction. As shown, the prediction performance of CNN-D is superior to that of CNN-F at all streamwise locations. Compared to CNN-F with an average of \(R=0.974\), the prediction using CNN-D yields an average of \(R=0.996\) with variations of \(0.994<R<0.999\) depending on \(x\). This superiority of CNN-D over CNN-F is also evident in the root mean square (rms) of transverse velocity on the centerplane of the wake presented in Figure 7. The rms of transverse velocity from CNN-D agrees very well with that from the numerical simulation, while that from CNN-F shows a slight deviation. Figure 8 shows the scatter plots of \(v/u_{\infty}\) from the numerical simulation and \(\vartheta/u_{\infty}\) predicted by CNN-D at two different streamwise locations (\(x/d=0.71\) and \(x/d=1.01\)). For both locations, the transverse velocity predicted by CNN-D shows a good prediction performance, supporting the results of Figures 5-7. Figure 9 presents the time histories of the transverse velocity at the point of \(x/d=1.06\) and \(z/d=6\) in the wake for \(Re=300\). The velocity from CNN-D accurately follows the temporal behaviors of the numerical simulation, while that from CNN-F becomes less accurate in the vicinity of the peak locations of the velocity. Figure 10 displays the contours of the instantaneous transverse velocity on the \(x\)-\(z\) plane of \(y/d=0\) (centerplane) predicted by CNN-D, together with those from the numerical simulation at four different temporal instances.