Title: Rootfinding and approximation approaches through neural networks

Authors: Michael G. Epitropakis, Michael N. Vrahatis, University of Patras

Abstract:

Two important scientific problems that arise in a huge variety of
applications are the approximation of Multivariate High Order Polynomials,
as well as, the estimation of roots of High Order Univariate Polynomials.
Among the various proposed approaches for these problems Artificial Neural
Networks have recently demonstrated their ability to provide accurate and
fast approximations and root estimations. In this contribution we employ
High Order Neural Networks, and specifically Ridge Polynomial Networks for
the approximation of multivariate polynomial equations. We recommend the
use of stochastic global optimization techniques, like Differential
Evolution and Particle Swarm Optimization for the learning process to
obtain better solutions. We further propose a two step technique for the
estimation of a number of arbitrary roots of higher order polynomials.
This technique at a first step considers a trained Feedforward Neural
Network as an oracle, that returns the number of real roots of a
univariate polynomial. At the next step this information is used by a
known High Order Neural Network root-finder to provide the final roots
estimation.