  This paper introduces a novel algorithmic framework for a deep neural network
(DNN), which in a mathematically rigorous manner, allows us to incorporate
history (or memory) into the network -- it ensures all layers are connected to
one another. This DNN, called Fractional-DNN, can be viewed as a
time-discretization of a fractional in time nonlinear ordinary differential
equation (ODE). The learning problem then is a minimization problem subject to
that fractional ODE as constraints. We emphasize that an analogy between the
existing DNN and ODEs, with standard time derivative, is well-known by now. The
focus of our work is the Fractional-DNN. Using the Lagrangian approach, we
provide a derivation of the backward propagation and the design equations. We
test our network on several datasets for classification problems.
Fractional-DNN offers various advantages over the existing DNN. The key
benefits are a significant improvement to the vanishing gradient issue due to
the memory effect, and better handling of nonsmooth data due to the network's
ability to approximate non-smooth functions.
