A Dynamic Mode Decomposition Extension for the Forecasting of Parametric Dynamical Systems

Dynamic mode decomposition (DMD) has recently become a popular tool for the nonintrusive analysis of dynamical systems. Exploiting proper orthogonal decomposition (POD) as a dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of spatial bases evolving linearly in time, thus enabling a better understanding of the physical phenomena and forecasting of future time instants. In this work we propose an extension of DMD to parameterized dynamical systems, focusing on the future forecasting of the output of interest in a parametric context. Initially all the snapshots-for different parameters and different time instants-are projected to a reduced space; then DMD, or one of its variants, is employed to approximate reduced snapshots for future time instants. Exploiting the low dimension of the reduced space, we then combine the predicted reduced snapshots using regression techniques, thus enabling the possibility of approximating any untested parametric configuration in the future. This paper depicts in detail the algorithmic core of this method; we also present and discuss three test cases for our algorithm: a simple dynamical system with a linear parameter dependency, a heat problem with nonlinear parameter dependency, and a fluid dynamics problem with nonlinear parameter dependency.