A Mathematical Mechanism of Reservoir Computing for Dynamical Systems

Abstract. Reservoir computing (RC) is a machine learning framework based on recurrent neural networks, which can naturally be viewed as dynamical systems. We focus on the problem of learning a time series generated by an unknown dynamical system [Formula: see text]. As suggested by several numerical studies, once the reservoir has learned [Formula: see text], it appears to reproduce the dynamics of [Formula: see text]; however, the underlying mechanism behind this behavior has not yet been fully clarified. In this study, we prove that, under certain assumptions, a reservoir that has learned [Formula: see text] becomes topologically semiconjugate in a weak sense or topologically conjugate to [Formula: see text]. This theorem and its proof shed new light on the mathematical foundations of RC.