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Learning Chaotic Attractors by Neural Networks

DelftChemTech, Delft University of Technology, 2628 BL Delft, The Netherlands

Jaap C. Schouten

Chemical Reactor Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

C. Lee Giles

NEC Research Institute, Princeton, NJ 08540, U.S.A.

Floris Takens

Department of Mathematics, University of Groningen, 9700 AV Groningen, The Netherlands

Cor M. van den Bleek

DelftChemTech, Delft University of Technology, 5600 MB Eindhoven, The Netherlands

An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series. During training, the algorithm learns to short-term predict the time series. At the same time a criterion, developed by Diks, van Zwet, Takens, and de Goede (1996) is monitored that tests the hypothesis that the reconstructed attractors of model-generated and measured data are the same. Training is stopped when the prediction error is low and the model passes this test. Two other features of the algorithm are (1) the way the state of the system, consisting of delays from the time series, has its dimension reduced by weighted principal component analysis data reduction, and (2) the user-adjustable prediction horizon obtained by "error propagation"—partially propagating prediction errors to the next time step.

The algorithm is first applied to data from an experimental-driven chaotic pendulum, of which two of the three state variables are known. This is a comprehensive example that shows how well the Diks test can distinguish between slightly different attractors. Second, the algorithm is applied to the same problem, but now one of the two known state variables is ignored. Finally, we present a model for the laser data from the Santa Fe time-series competition (set A). It is the first model for these data that is not only useful for short-term predictions but also generates time series with similar chaotic characteristics as the measured data.