###### Abstract  
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 construct 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 (FCNN) is considered. On the other hand, for a three-dimensional unsteady flow at \(Re=300\) having spanwise variations, we employ two different convolutional neural networks based on an encoder-FCNN (CNN-F) or an encoder-decoder (CNN-D) structure. Numerical simulations are carried out for both Reynolds numbers to obtain instantaneous flow fields, from which the input and output datasets are generated for training these neural networks. At the Reynolds numbers considered, the neural networks constructed accurately predict 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 is observed that CNN-D shows a better prediction ability than CNN-F.  
flow over a circular cylinder; laminar flow; wake; neural network; instantaneous wall pressure; transverse velocity +
Footnote †: journal: Applied  
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Footnote †: journal: Applied  
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variables to output variables with a series of fully connected layers. The adoption of FCNN distinguishes these approaches from conventional turbulence modelings where the target stresses are approximated with an algebraic equation using the resolved flow quantities [9; 10; 18; 19]. In turbulence modeling with machine learning techniques, various input variables of the FCNN such as velocity, velocity gradient tensor, strain rate tensor, rotation rate tensor, or their combinations have been investigated to determine the stress terms of RANS and LES as the output variables of FCNN. For example, [PERSON] and [PERSON] [14] investigated the use of a fully connected neural network to develop an SGS model for LES of turbulent channel flow at \(Re_{\tau}=178\). In an a priori test, this study found that the NN-based SGS model with filtered strain rate or velocity gradient tensor at multiple points as the input variables showed high correlation coefficients between predicted and true SGS stresses but required special treatments for stable solutions. In contrast, the NN-based SGS model with filtered strain rate tensor at a single point as the input variable demonstrated an excellent prediction ability for mean velocity and Reynolds shear stress of turbulent channel flow in an a posteriori test. In turbulence models for RANS and LES with neural networks, the output variables of a neural network are the Reynolds and SGS stresses (or their components) in the Reynolds-averaged and filtered Navier-Stokes equations, respectively. On the other hand, without relying on the framework of turbulence modeling, there is another machine-learning-based approach that directly predicts or reconstructs fluid flow fields by setting the flow field information of interest, such as velocity, as the output variable of a neural network [20; 21; 22; 23]. For the prediction of laminar flow over a circular cylinder, [PERSON] et al. [21] adopted convolutional neural networks (CNNs) and their modified version that reconstruct a high-resolution velocity field in the wake with the input of a low-resolution velocity dataset. With these neural networks developed, they demonstrated that the two-dimensional laminar velocity field over a circular cylinder at \(Re=100\) could be successfully reconstructed. Later, they extended this approach to predict temporal evolutions of flow field from grossly under-resolved input data in both space and time and examined its performances in two-dimensional decaying homogeneous isotropic turbulence and three-dimensional turbulent channel flow.