  We present an efficient deep learning technique for the model reduction of
the Navier-Stokes equations for unsteady flow problems. The proposed technique
relies on the Convolutional Neural Network (CNN) and the stochastic gradient
descent method. Of particular interest is to predict the unsteady fluid forces
for different bluff body shapes at low Reynolds number. The discrete
convolution process with a nonlinear rectification is employed to approximate
the mapping between the bluff-body shape and the fluid forces. The deep neural
network is fed by the Euclidean distance function as the input and the target
data generated by the full-order Navier-Stokes computations for primitive bluff
body shapes. The convolutional networks are iteratively trained using the
stochastic gradient descent method with the momentum term to predict the fluid
force coefficients of different geometries and the results are compared with
the full-order computations. We attempt to provide a physical analogy of the
stochastic gradient method with the momentum term with the simplified form of
the incompressible Navier-Stokes momentum equation. We also construct a direct
relationship between the CNN-based deep learning and the Mori-Zwanzig formalism
for the model reduction of a fluid dynamical system. A systematic convergence
and sensitivity study is performed to identify the effective dimensions of the
deep-learned CNN process such as the convolution kernel size, the number of
kernels and the convolution layers. Within the error threshold, the prediction
based on our deep convolutional network has a speed-up nearly four orders of
magnitude compared to the full-order results and consumes an insignificant
fraction of computational resources. The proposed CNN-based approximation
procedure has a profound impact on the parametric design of bluff bodies and
the feedback control of separated flows.
