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Deep learning of the Artificial Neural Networks (ANN) can be treated as aparticular class of interpolation problems. The goal is to find a neuralnetwork whose input-output map approximates well the desired map on a finite oran infinite training set. Our idea consists of taking as an approximant theinput-output map, which arises from a nonlinear continuous-time control system.In the limit such control system can be seen as a network with a continuum oflayers, each one labelled by the time variable. The values of the controls ateach instant of time are the parameters of the layer.

Control on the Manifolds of Mappings with a View to the Deep Learning