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Predicting chaotic time series using optoelectronic feedback laser

We study delay-based photonic reservoir computing using a semiconductor laser with an optoelectronic feedback. A rate-equation model for a laser with an optoelectronic filtered feedback is used. The filter allows only high-frequency signals to pass through the feedback loop. The delay-differential equation model consists of three equations for the normalized electric field intensity I(t), the carrier density N(t); and the filtered intensity signal IF (t). The stability boundaries which correspond to the Hopf bifurcation condition are determined analytically, showing multiple Hopf bifurcation branches in the dynamics, and the parity asymmetry with relation to the feedback sign. We use the Santa Fe time-series prediction task to evaluate the performance of reservoir computing. Our objective is to determine location of the optimal operating point defined as corresponding to minimal normalized means square error (NMSE) and relate it to the stability properties of the system. We use 3000 points for training and 1000 for testing, number of virtual nodes is chosen in regard to the relaxation oscillation frequency. Single-point prediction of the chaotic data is performed. Input signal is determined by the chaotic waveform having n sampling points, and three cases are investigated: prediction of n + 1 ,n + 2 or n + 3 sampling point. The best NMSE value order of 10^7 for n + 1 point prediction task is obtained in the absence of feedback and the rapid increase in NMSE is observed in the vicinity of Hopf bifurcation without regard to the feedback sign. On the contrary, the minimum values of NMSE for n + 2 and n + 3 point prediction task correspond to the Hopf bifurcation, and only for the positive feedback. We discuss whether the parity asymmetry can explain strongly asymmetric reservoir computing results.