Application of next-generation reservoir computing for predicting chaotic systems from partial observations.

Next-generation reservoir computing is a machine-learning approach that has been recently proposed as an effective method for predicting the dynamics of chaotic systems. So far, this approach has been applied mainly under the assumption that all components of the state vector of dynamical systems are observable. Here we study the effectiveness of this method when only a scalar time series is available for observation. As illustrations, we use the time series of Rössler and Lorenz systems, as well as the chaotic time series generated by an electronic circuit. We found that prediction is only effective if the feature vector of a nonlinear autoregression algorithm contains monomials of a sufficiently high degree. Moreover, the prediction can be improved by replacing monomials with Chebyshev polynomials. Next-generation models, built on the basis of partial observations, are suitable not only for short-term forecasting, but are also capable of reproducing the long-term climate of chaotic systems. We demonstrate the reconstruction of the bifurcation diagram of the Rössler system and the return maps of the Lorenz and electronic circuit systems.