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We present an efficient deep learning technique for the model reduction ofthe Navier-Stokes equations for unsteady flow problems. The proposed techniquerelies on the Convolutional Neural Network (CNN) and the stochastic gradientdescent method. Of particular interest is to predict the unsteady fluid forcesfor different bluff body shapes at low Reynolds number. The discreteconvolution process with a nonlinear rectification is employed to approximatethe mapping between the bluff-body shape and the fluid forces. The deep neuralnetwork is fed by the Euclidean distance function as the input and the targetdata generated by the full-order Navier-Stokes computations for primitive bluffbody shapes. The convolutional networks are iteratively trained using thestochastic gradient descent method with the momentum term to predict the fluidforce coefficients of different geometries and the results are compared withthe full-order computations. We attempt to provide a physical analogy of thestochastic gradient method with the momentum term with the simplified form ofthe incompressible Navier-Stokes momentum equation. We also construct a directrelationship between the CNN-based deep learning and the Mori-Zwanzig formalismfor the model reduction of a fluid dynamical system. A systematic convergenceand sensitivity study is performed to identify the effective dimensions of thedeep-learned CNN process such as the convolution kernel size, the number ofkernels and the convolution layers. Within the error threshold, the predictionbased on our deep convolutional network has a speed-up nearly four orders ofmagnitude compared to the full-order results and consumes an insignificantfraction of computational resources. The proposed CNN-based approximationprocedure has a profound impact on the parametric design of bluff bodies andthe feedback control of separated flows.

An Efficient Deep Learning Technique for the Navier-Stokes Equations: Application to Unsteady Wake Flow Dynamics