Implementing polynomial expressions by means of reciprocal-function-based neural networks

In this paper, a new approach to the problem of implementing polynomial expressions by means of neural networks is proposed. In our method, the polynomial describing a given problem is regarded as an inverse of a particular fractional-rational function, where the function numerator is a real number equal to one. Since such a rational function can be decomposed into first order partial fractions (when operating with complex numbers), the neural implementation of the considered polynomial can be carried out without using activation functions of the exp(.) and ln(.) type and can be realized by applying weighted summation and inverting (reciprocal) operations only. This means that the proposed technique can lead to simpler neural implementations than that presented in the literature [1]-[10], where the implementation is based on utilizing the exp(.) and ln(.) operators. As a consequence, the proposed approach allows, among others, successful learning the network when our aim is to determine parameter values of the polynomial describing a given set of empirical data in order the discover laws governing the numerical data. Due to the lack of the exp(.) and ln(.) activation function in the proposed network structure, some problems related with calculating ln(.) for negative-number arguments can be eliminated or alleviated. The main advantages of our network are simplicity of implementing the given polynomials and high speed of the network training. Apart from theoretical considerations, simulation results concerning real-number operations are shown to be in a good agreement with the theory.