Autoencoding for the "Good Dictionary" of eigenpairs of the Koopman operator

Abstract

<abstract><p>Reduced order modelling relies on representing complex dynamical systems using simplified modes, which can be achieved through the Koopman operator(KO) analysis. However, computing Koopman eigenpairs for high-dimensional observable data can be inefficient. This paper proposes using deep autoencoders(AE), a type of deep learning technique, to perform nonlinear geometric transformations on raw data before computing Koopman eigenvectors. The encoded data produced by the deep AE is diffeomorphic to a manifold of the dynamical system and has a significantly lower dimension than the raw data. To handle high-dimensional time series data, Takens' time delay embedding is presented as a preprocessing technique. The paper concludes by presenting examples of these techniques in action.</p></abstract>