Using artificial neural networks to forecast chaotic time series

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Abstract

Two-layer feedforward neural network was used in this work to forecast chaotic time series with very promising results, especially for the Lorenz system, as in comparison to others that had been previously published elsewhere. It was observed that the architecture m:2m:m:1, where m is the embedding dimension of the attractor of the dynamical system in consideration, is a very good initial guess for the process of finding the ideal architecture for the neural network, which is usually hard to achieve. The results we obtained with this particular type to series, and also with some others like Henon and Logistic maps, clearly indicate that there is an interplay between the architecture of a multilayer network and the embedding dimension m of the time series used. From the very good forecasting results we obtained, it can be concluded that neural networks can be considered to be an important tool for making predictions of the time evolution of nonlinear systems.

Introduction

In many situations in science and technology we usually face the necessity of predicting the future evolution of a system from past measurements of it. Mathematical models of physical systems are generally investigated by writing down the equations of motion and by trying to integrate them, forward in time, to predict the future state of the system. Mathematically speaking, this dynamics is described by the motion of a point v, which represents the state of the system in a multi-dimensional space Γ. However, in nonlinear systems with many degrees of freedom, it is just impraticable to solve all the equations without making some sort of assumptions and simplifications. Dissipations can reduce the number of the effectively relevant degrees of freedom in apparently chaotic dynamical systems. Thus, the motion of the system becomes confined to a subspace ΓA, of Γ, known as attractor with lower dimension d [1], [2]. Within this scenario, neural networks may be considered an important forecasting tool to be used in such situations.

As it is already well known, the human brain is superior to even the most powerful digital computer in many tasks and a good example is the prediction and controlling of complex systems without the necessity of having an underlying knowledge of the physics of the system: any child can ride a bicycle without knowing Newtonian Mechanics, for instance.

Thus, this is the real motivation for studying neural computation, which is based on the knowledge of neuroscience apart from attempting to be biologically realistic in all details. The main point to be considered is the essential property of the biological neurons from the viewpoint of information processing [3]. In this respect, artificial neural network models can be constructed to imitate some aspects of the structure of the brain and the nervous system, and have been extensively used in a variety of areas such as Robotics, Medicine, Economics, Astronomy and, of course, in physics. In this work, we have concentrated our attention on predicting chaotic time series obtained from the solutions of Lorenz equations and by iterations from Henon and Logistic maps. However, the same procedure here described could be used for other rather complicated dynamical systems, such as turbulence in fluids [4], lasers [5], chemical reactions [6], and plasma physics [7], only to name a few.

In Section 2 a simplified description of the basics of neural networks will be given [1], [2]. In Section 3 the forecasting of chaotic time series will be reported. Although some works on Lorenz system predictions have already been reported by using different approachs [2], [9], we will be showing that artificial neural networks with the particular architecture m:2m:m:1, where m is the embedding dimension of the attractor of the dynamics in focus, can yield much better results. We have already been using neural network mainly to forecast disruption instabilities in magnetically confined plasma with relative success [10], and much more promising results have also been obtained from this new architeture [11]. Finally, in Section 4, the conclusions of this work will be presented.

Section snippets

Neural networks

Artificial neural networks are computer algorithms which simulate in a very simplified form the ability of brain neurons to process information. In general, a neural net is typically composed of interconnected units organized in layers which serve as model neurons. The function of the synapse, the structure responsable for storing information in the brain, is modelled by a modifiable weight which is associated to each connection between neural units located in adjacents layers (Fig. 1). Within

Prediction of chaotic time series

According to Takens [13], there exists a smooth function of at most 2d+1 past measurements of a temporal series that allows the correct prediction of its future value, and the prediction is just as good as the one it would be obtained if we had been able to solve the complete system with all its degrees of freedom [2], [13]. What the theorem of Takens does not provide is the explicit form of the function which would contain the desired extrapolation and it is in this context that neural

Conclusions

It has been shown in this work that neural network can be successfully used to predict the future state of nonlinear systems as the ones described by the Lorenz equations and Henon and Logistic maps. For all these cases, it has been verified that the architecture m:2m:m:1 is an initial good guess to construct the neural network architecture for predicting the future state of chaotic temporal series. The reasonably good results obtained here clearly suggests that there is an interplay between

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