## 3 Results and Discussion
### Prediction of Near-Wake Velocity at \(Re=60\)  
In the present study, we use an FCNN with instantaneous wall pressures on the cylinder surface as the input variables to predict the transverse velocity around a circular cylinder at \(Re=60\). As illustrated in Figure 1a, 44 locations of wall pressures are used to predict 8192 locations of transverse velocities around the cylinder. This FCNN is constructed by training it with the input and output database of flow fields obtained from the numerical simulation at \(Re=60\). The correlation coefficients \(R\) between the transverse velocity (\(v\)) from the numerical simulation and that (\(\vartheta\)) predicted by the FCNN constructed are found to be greater than 0.999 at all locations (Figure 1a) around the cylinder. Here, the correlation coefficient \(R\) is defined as follows:  
\[R=\frac{\sum_{i=1}^{M}(v_{i}-\langle v\rangle)(\hat{v}_{i}-\langle\hat{v} \rangle)}{\sqrt{\sum_{i=1}^{M}(v_{i}-\langle v\rangle)^{2}}\sqrt{\sum_{i=1}^{ M}(\hat{v}_{i}-\langle\phi\rangle)^{2}}}, \tag{4}\]  
where \(M\) is the number of testing data and \(\langle\cdot\rangle\) denotes averaged quantities.  
Figure 3 displays the time histories of the transverse velocity at a wake location (\(x/d=1.01\) and \(y/d=0\)) behind the cylinder predicted by the neural network together with that from the numerical simulation at \(Re=60\). In the figure, temporal behaviors of the transverse velocity, such as the peak amplitude and temporal periodicity, are well-predicted by the present neural network constructed. Figure 4 shows the contours of the instantaneous transverse velocity predicted by the present FCNN compared with those from the numerical simulation at various temporal instances. As expected from the correlation coefficient of 0.999, the transverse velocity contours agree very well with those from the numerical simulation at all instances.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**References** & \(Re\) & \(C_{D}\) & \(-C_{P_{\mathrm{a}}}\) & \(St\) \\ \hline Present study & 60 & 1.396 & 0.57 & 0.136 \\ [PERSON] et al. [38] & 60 & 1.39 & 0.59 & 0.136 \\ \hline Present study & 300 & 1.291 & 1.00 & 0.203 \\ [PERSON] et al. [39] & 300 & 1.28 & 1.01 & 0.203 \\ [PERSON] et al. [40] & 300 & 1.296 & 1.05 & 0.204 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Flow statistics for laminar flows over a circular cylinder at \(Re=60\) and 300 obtained from the present numerical simulations together with those from previous studies.  
Figure 3: Time histories of the transverse velocity at \(x/d=1.01\) and \(y/d=0\) (\(Re=60\)). \(\bullet\), \(v/u_{\infty}\) from the numerical simulation; green line, \(\vartheta/u_{\infty}\) predicted by FCNN.  
In the study by [PERSON] et al. [24], they showed that velocity fields in the two-dimensional laminar flow over a circular cylinder could be successfully predicted with neural networks having as their input the pressures on the cylinder surface. In their work, the input variables of the neural network included the pressures at the time \(t=t_{1}\) together with the pressures at multiple past instances at \(t<t_{1}\), and the velocity field at \(t=t_{1}\) was predicted using these input pressures. On the other hand, our study only employs the instantaneous pressures at the time \(t=t_{1}\) without past pressure information, resulting in a compact neural network structure. Nevertheless, the prediction quality is comparable to that of [PERSON] et al. [24] at the Reynolds number considered.