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

TL;DR

The paper introduces a dynamic mode decomposition extension for forecasting parametric dynamical systems by combining Proper Orthogonal Decomposition with DMD in a reduced space, followed by regression to map parameters to reduced states. It presents two offline strategies (monolithic and partitioned) and an online phase, along with a stabilization scheme to remove spurious modes. The approach is validated on three test cases—a linear-parameter system, a nonlinear heat problem, and a Navier–Stokes cylinder—demonstrating accurate predictions for unseen parameters in many scenarios while highlighting limitations when the POD space cannot capture essential dynamics. This work advances non-intrusive, data-driven parametric forecasting and supports fast, real-time evaluation in applications like CFD and related computational sciences.

Abstract

Dynamic mode decomposition (DMD) has recently become a popular tool for the non-intrusive 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 basis 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 the predicted reduced snapshots are then combined using regression techniques, thus enabling the possibility to approximate any untested parametric configuration in 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.

Figures (22)

• Figure 1: Scheme of the numerical pipeline to perform the offline stage, isolating the differences between monolithic and partitioned.
• Figure 2: Scheme of the numerical pipeline to perform the online stage.
• Figure 3: Graphical example of stabilization: on left, the eigenvalues (red) of the DMD operator are sketched together with the unitary circle (solid black line) in the complex plane. On right, we show the same entities after the stabilization: the eigenvalues distant from the unit circle have been discarded, whereas the closest are scaled in order to have the modulo equal to one.
• Figure 4: On the left, the two simple functions which we are using for the composition. We consider 10 values of $\mu$ to train our algorithm, the realizations corresponding to $\mu \in \{0.1,0.3,0.5,0.7\}$ are shown on the right.
• Figure 5: Looking at singular values of the system, on the left, we deduce that the most contribution is given by the first two POD modes. We drew those modes on the right.