A dynamic mode decomposition extension for the forecasting of parametric dynamical systems

Dynamic mode decomposition (DMD) has recently become a popular tool for the non-intrusive analysis of dynamical systems. Exploiting the proper orthogonal decomposition as dimensionality reduction technique, DMD is able to approximate a dynamical system as a sum of (spatial) basis evolving linearly in time, allowing for a better understanding of the physical phenomena or for a future forecasting. We propose in this contribution an extension of the DMD to parametrized 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 the reduced space, employing the DMD (or one of its variants) to approximate the reduced snapshots for a future instants. Still exploiting the low dimension of the reduced space, the predicted reduced snapshots are then combined using a regression technique, enabling the possibility to approximate any untested parametric configuration in any future instant. We are going to present here the algorithmic core of the aforementioned method, presenting at the end three different test cases with incremental complexity: 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.