Dynamic Mode Decomposition
Dynamic mode decomposition extracts spatial modes and their growth and oscillation rates directly from time-series simulation data.
Modes with a time signature
Dynamic mode decomposition (DMD) analyzes a sequence of snapshots and finds spatial patterns, each tagged with a single complex frequency that sets how it grows, decays, and oscillates. Unlike proper orthogonal decomposition, whose modes are ranked only by energy, DMD modes each have a clean temporal behavior, making them a bridge between data and dynamical understanding.
The algorithm
DMD seeks the best linear operator that maps each snapshot to the next. It arranges snapshots into two matrices offset by one time step, then computes the operator that relates them, typically through a singular value decomposition for numerical stability. The eigenvalues of that operator give the modal frequencies and growth rates; its eigenvectors give the spatial modes.
Connection to Koopman
DMD is the leading practical approximation to the Koopman operator when the observables are just the raw state variables. Its modes approximate Koopman modes, and its eigenvalues approximate the Koopman spectrum. This link gives DMD a firm theoretical footing and motivates extensions that enrich the observables for stronger nonlinear systems.
Uses
- Identifying coherent structures and their frequencies in fluid flows
- Short-horizon forecasting from a purely linear model
- Detecting instability through growth rates of modes
- Background-foreground separation in image and signal data
Strengths and caveats
DMD is equation-free, fast, and interpretable, needing only snapshots and no knowledge of the governing equations. Its linear nature limits it to regimes where a linear model is adequate, and it is sensitive to noise and to how snapshots are sampled. Within those bounds it is a powerful diagnostic and a useful building block for reduced-order and control models.