Learning Dynamics by Reservoir Computing (In Memory of Prof. Pavol Brunovský)

We study reservoir computing, a machine learning method, from the viewpoint of learning dynamics. We present numerical results of learning the dynamics of the logistic map, one of the typical examples of chaotic dynamical systems, using a 30-node reservoir and a three-node reservoir. When the learning is successful, an attractor that is smoothly conjugate to the logistic map to be learned is observed in the phase space of the reservoir. Inspired by this numerical result, we introduce a degenerate reservoir system and use it to mathematically confirm this observation. We also show that reservoir computing can learn information about dynamics not included in the training data, which we believe is a remarkable feature of reservoir computing compared to other machine learning methods. We discuss this feature in connection with the above observation that there is a smooth conjugacy between the attractor in the reservoir and the dynamics to be learned.