Learning chaotic attractors by neural networks

An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series. During training, the algorith m learns to short-term predict the time series. At the same time a criterio n, developed by Diks, van Zwet, Takens, and de Goede (1996) is monitored th at tests the hypothesis that the reconstructed attractors of model-generate d and measured data are the same. Training is stopped when the prediction e rror is low and the model passes this test. Two other features of the algor ithm are (1) the way the state of the system, consisting of delays from the time series, has its dimension reduced by weighted principal component ana lysis data reduction, and (2) the user-adjustable prediction horizon obtain ed by "error propagation"-partially propagating prediction errors to the ne xt 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 co mprehensive example that shows how well the Diks test can distinguish betwe en slightly different attractors. Second, the algorithm is applied to the s ame problem, but now one of the two known state variables is ignored. Final ly, we present a model for the laser data from the Santa Fe time-series com petition (set A). It is the first model for these data that is not only use ful for short-term predictions but also generates time series with similar chaotic characteristics as the measured data.