Dynamic mode decomposition for detecting 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 to overcome this limitation 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 and faithful representation of non-steady dynamics. We apply the proposed method to fluid flow data exhibiting transient behavior and demonstrate that it can capture the temporal structure of mode activations that are not accessible with the standard DMD method.