Neural Networks and Chaos: Construction, Evaluation of Chaotic Networks, and Prediction of Chaos with Multilayer Feedforward Networks

TL;DR

This paper establishes a rigorous theoretical link between chaotic iterations and neural networks, providing methods to construct and verify chaos in neural networks, and examines the challenges of learning chaotic data with multilayer perceptrons.

Contribution

It introduces a formal framework connecting Devaney's chaos with neural networks and offers practical methods for constructing and identifying chaotic neural networks.

Findings

Chaotic iterations can be represented by specific neural networks.

Chaotic behaviors are more difficult for neural networks to learn.

The paper provides a method to verify chaos in neural networks.

Abstract

Many research works deal with chaotic neural networks for various fields of application. Unfortunately, up to now these networks are usually claimed to be chaotic without any mathematical proof. The purpose of this paper is to establish, based on a rigorous theoretical framework, an equivalence between chaotic iterations according to Devaney and a particular class of neural networks. On the one hand we show how to build such a network, on the other hand we provide a method to check if a neural network is a chaotic one. Finally, the ability of classical feedforward multilayer perceptrons to learn sets of data obtained from a dynamical system is regarded. Various Boolean functions are iterated on finite states. Iterations of some of them are proven to be chaotic as it is defined by Devaney. In that context, important differences occur in the training process, establishing with various neural networks that chaotic behaviors are far more difficult to learn.