## 2 Numerical Details  
### Numerical Details for Simulation of Flow over a Circular Cylinder  
In the present study, we perform numerical simulations of laminar flows over a circular cylinder at \(Re=60\) and \(300\) to generate the input and output datasets for training neural networks. The governing equations are the non-dimensional continuity and Navier-Stokes equations for unsteady and incompressible flow, as shown in the following equations:  
\[\frac{\partial u_{i}}{\partial x_{i}}-q=0, \tag{1}\]  
and  
\[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}= -\frac{\partial p}{\partial x_{i}}+\frac{1}{Re}\frac{\partial^{2}u_{i}}{ \partial x_{j}\partial x_{j}}+f_{i}, \tag{2}\]  
where \(t\) is the time, \(x_{i}=(x,y,z)\) are the Cartesian coordinates, \(u_{i}=(u,v,w)\) are the corresponding velocity components, and \(p\) is the pressure. Here, \(x\), \(y\), and \(z\) denote the streamwise, transverse, and spanwise directions, respectively (see Figure 1). To satisfy the no-slip condition on the cylinder surface, we adopt an immersed boundary method by [PERSON] et al. [7]. Owing to the use of the immersed boundary method, Equations (1) and (2) include the mass source/sink term \(q\) and the momentum forcing term \(f_{i}\), respectively. We adopt the fully implicit fractional step method [5; 25] to numerically decouple the pressure and velocity in the Navier-Stokes equation using the Crank-Nicolson scheme. The second-order central difference scheme is used for spatial discretizations in each term of Equations (1) and (2). We note that the numerical details described above have been implemented through our own in-house code, and its accuracy has been proven in predicting various complex flow configurations [7; 9; 26; 27].  
Figure 1a,b shows the computational domains for \(Re=60\) and \(300\), respectively. The two-dimensional computational domain for \(Re=60\) is \(-40\leq x/d\leq 50\) and \(-50\leq y/d\leq 50\) (see Figure 1a). The computational domain for \(Re=300\) is three-dimensi onal with the same \(x\)-\(y\) domain as that for \(Re=60\), and its domain for the spanwise direction is set to be \(0\leq z/d\leq 12\) (see Figure 1b). The center axis of a circular cylinder is located at \((x,y)=(0,0)\). The numbers of grid points used for numerical simulations are \(385(x)\times 217(y)\) and \(385(x)\times 217(y)\times 128(z)\) for \(Re=60\) and \(300\), respectively. At the inflow boundary, a Dirichlet boundary condition (\(u=u_{\infty}\) and \(v=w=0\)) is employed. At the top and bottom boundaries, \(\partial u/\partial y=v=\partial w/\partial y=0\) is given. At the outflow boundary, a convective boundary condition is adopted as \(\partial u_{i}/\partial t+c\partial u_{i}/\partial x=0\), where \(c\) is the plane-averaged streamwise velocity on the outflow plane. A periodic boundary condition  
Figure 1: Computational domains for the numerical simulation of flow over a circular cylinder together with learning dataset domains for machine learning: (**a**) \(Re=60\); (**b**) \(Re=300\). Blue area: domains for wall pressures as the input variables of neural networks; red area: domains for the transverse velocity prediction as the output variables of neural networks.  
is applied in the spanwise direction for the three-dimensional computational domain of \(Re=300\).