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

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.