Tensor train based higher order Dynamic Mode Decomposition for dynamical systems

Higher order dynamic mode decomposition (HODMD) has proved to be an efficient tool for the analysis and prediction of complex dynamical systems described by data-driven models. In the present paper, we propose a realization of HODMD that is based on low-rank tensor decomposition of potentially high-dimensional data sets. It is used to compute the HODMD modes and eigenvalues to effectively reduce the computational complexity of the problem. The proposed extension also provides a more efficient realization of the ordinary Dynamic Mode Decomposition with the use of the tensor-train decomposition. High efficiency of the tensor-train based HODMD (TT-HODMD) is illustrated by a few examples, including forecasting the load of a power system, which provide comparisons between TT-HODMD and HODMD with respect to the computing time and accuracy. The developed algorithm can be effectively used for the prediction of high-dimensional dynamical systems.