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

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.