Dynamic mode decomposition for detecting oscillatory transient activity via sparsity and smoothness regularization

TL;DR

This paper introduces an extension to Dynamic Mode Decomposition that incorporates sparsity and smoothness regularization to better identify and interpret oscillatory transient activities in complex dynamical systems.

Contribution

The proposed method enhances DMD by enabling extraction of transient oscillatory modes with time-varying amplitudes, improving interpretability of non-steady dynamics.

Findings

Successfully captures transient mode activations in fluid flow data.

Provides more interpretable representations of non-steady dynamics.

Demonstrates effectiveness on a laminar airfoil wake example.

Abstract

Dynamic Mode Decomposition (DMD) is a data-driven modal decomposition technique that extracts coherent spatio-temporal structures from high-dimensional time-series data. By decomposing the dynamics into a set of modes, each associated with a single frequency and a growth rate, DMD enables a natural modal decomposition and dimensionality reduction of complex dynamical systems. However, when DMD is applied to transient dynamics, even if a large number of modes are used, it remains difficult to interpret how these modes contribute to the transient behavior. In this study, we propose a simple extension of DMD that facilitates extraction of oscillatory transient activity by introducing time-varying amplitudes for the DMD modes based on sparsity and smoothness regularization. This approach enables identification of dynamically significant modes and extraction of their transient activities, providing a more interpretable representation of non-steady dynamics. We illustrate the validity of the proposed method using a simple example and then apply it to fluid flow data of a laminar airfoil wake exhibiting transient behavior. We demonstrate that it can capture the temporal structure of mode activations that are not accessible with the standard DMD method.