New efficient data-driven reduced order models for oscillatory dynamics

A new methodology is presented to construct data-driven reduced order models able to efficiently approximate periodic attractors of dynamical systems depending on one or more parameters. In an offline stage, some sets of temporally equidistant snapshots are computed for a limited number of parameter values. Such computation is performed using a standard numerical solver when the problem is governed by a low-dimensional system of ordinary differential equations. If, instead, the underlying dynamical system is high-dimensional (i.e., governed by partial differential equations), the snapshots are computed relying on a physics-based reduced order model obtained via proper orthogonal decomposition plus Galerkin projection. The snapshot sets are then treated using a recent, very robust data processing method, the higher order dynamic mode decomposition, which permits describing the periodic attractors as synchronized expansions in terms of spatial modes and associated temporal frequencies. Modes and frequencies are effectively interpolated at new parameter values, different from those involved in the offline stage. The online operation of the resulting data-driven reduced order model is very fast, since it requires carrying out only a small number of algebraic computations. The performance of the new method is tested for two representative dynamical systems, namely the three-dimensional Lorenz system and a high-dimensional system describing the electron transport in a semiconductor superlattice.