Fractional Neural Network Systems

In the last two decades, artificial neural networks have been identified as efficient tools for forecasting porosity, permeability, fluid density and water, gas and oil saturation, design of tunnels, image processing, graph flow, data deconvolution, energy demand forecasting, ecosystem evaluation, scheduling optimization, targeted marketing, medical diagnosis, time series analysis, stock market, etc. This approach is advantageous because the implementation of artificial neural networks is straightforward in particular, it does not call for a thorough understanding of the phenomena. In turn, artificial neural networks offer a considerable degree of flexibility. An artificial neural network can easily address nonlinearities and multi-dimensional data; it is noise- and fault-tolerant, cost-efficient and boasts excellent pattern recognition capabilities. In this document, we propose an investigation of fractional-order neural network systems. Certain phenomena cannot be described in Newtonian terms, thus requiring a non-local term (memory) to account for the entire prehistory of states of the system. It is now widely recognized that fractional derivatives serve as an excellent tool for the description of memory and hereditary phenomena and fractal media. Moreover, noise is considerably reduced as derivative orders decrease and the contribution of an additional degree of freedom to the system facilitates information processing. This proposal is a continuation of some earlier works by the proponent. It is motivated by the notion of replacing the traditional capacitor in certain circuits with a generalized mechanism called fractance. We thereby plan to replace the first-order time derivative in a Hopfield (or Cohen-Grossberg) neural network system with fractional derivatives (between 0 and 1). Our objective is to study asymptotic behaviors (eventually, power type, Mittag-Leffler and possibly other forms of stability) of these systems under different types of fractional derivatives, nonlinearities and activation functions. We expect being able to find reasonable sufficient conditions ensuring quick convergence and stability therefore validate the considered models.