From Homogeneous Network to Neural Nets with Fractional Derivative Mechanism

The paper refers to ANNs of the feed-forward type, homogeneous within individual layers. It extends the idea of network modelling and design with the use of calculus of finite differences proposed by Dudek-Dyduch E. and then developed jointly with Tadeusiewicz R. and others. This kind of neural nets was applied mainly to different features extraction i.e. edges, ridges, maxima, extrema and many others that can be defined with the use of classic derivative of any order and their linear combinations. Authors extend this type ANNs modelling by using fractional derivative theory. The formulae for weight distribution functions expressed by means of fractional derivative and its discrete approximation are given. It is also shown that the application of discrete approximation of fractional derivative of some base functions allows for modelling the transfer function of a single neuron for various characteristics. In such an approach smooth control of a derivative order allows to model the neuron dynamics without direct modification of the source code in IT model. The new approach presented in the paper, universalizes the model of the considered type of ANNs.