A Data-Driven ROM Based on HODMD

A preliminary version of a data driven reduced order model (ROM) for dynamical systems is presented in this Chapter. This ROM synergically and adaptively combines a black-box full model (FM) of the system and extrapolate conveniently using a recent extension of standard dynamic mode decomposition called higher order dynamic mode decomposition (HODMD). These two are applied in interspersed time intervals, called the FM-intervals and the HODMD-intervals, respectively. The data for the each HODMD-interval is obtained from the application of the FM in the previous FM-interval. The main question is when extrapolation from HODMD is no longer valid and switching to a new FM-interval is necessary. This is made attending to two criteria, ensuring that an estimate of the extrapolation error and a measure of consistency are both conveniently small. In this sense, the present method is similar to a previous method called POD on the Fly, which was not a purely data driven method. Instead, POD on the Fly was based on a Galerkin projection of the governing equations that thus should be known. The new method presented in this paper is illustrated with several transient dynamics for the complex Ginzburg-Landau equation that converges to either periodic or quasi-periodic attractors. The resulting CPU accelerations factors (compared to the full model) are quite large.