Adaptive Koopman Operator Learning via Iterative Projections: Time-Series Data Prediction Using Extended Dynamic Mode Decomposition

This paper presents a novel framework for adaptive learning of Koopman operator to predict the behavior of nonlinear time-varying dynamical systems based on the celebrated extended dynamic mode decomposition (EDMD). When applied to time-varying systems, the conventional EDMD framework has two limitations: degradation of prediction accuracy due to the use of predetermined dictionary and error accumulation caused by the changes of statistical properties. The proposed approach consists of three main components: (i) a data-driven dictionary selection algorithm using multiscale Gaussian functions as well as a novelty criterion, (ii) a “data-reusing” online learning algorithm based on the iterative orthogonal projection onto the linear variety depending on a fixed amount of most recent data, and (iii) a short-term prediction framework to prevent error accumulation. This leads to remarkable adaptivity as well as significant computational efficiency because the proposed algorithm is free from large matrix inversion. The simulation results show the efficacy of the proposed approach in time-varying environments.