Dynamic Mode Decomposition Attribution

DMD is a data-driven approximation of the Koopman operator, which advances observables of a nonlinear dynamical system linearly in an infinite-dimensional space. This allows DMD to extract globally linear coherent structures from complex, nonlinear sequence model hidden states. The attribution is derived from the eigenvalues and eigenvectors of the approximated linear operator, where eigenvalues dictate temporal growth/decay and oscillation frequencies, and eigenvectors define spatial mode shapes.

DMD decomposes the hidden state trajectory into a set of DMD modes, each characterized by:

• Eigenvalue: Determines the mode's temporal behavior—exponential growth, decay, or oscillation at a specific frequency.
• Eigenvector (Mode Shape): Defines the spatial structure of the mode across the model's feature dimensions.
• Amplitude: Quantifies the mode's contribution to reconstructing the original dynamics. Attribution is performed by mapping these modes back to the input space, revealing which spatiotemporal patterns drive predictions.

The DMD eigenvalue spectrum provides a compact, interpretable summary of the system's dynamics. Attribution is frequency-resolved: modes with eigenvalues on the unit circle represent persistent oscillations, while those inside/outside indicate damping or growth. This allows engineers to attribute a model's forecast to specific physical frequencies—for example, linking a load forecast to a 24-hour diurnal mode or a 7-day weekly cycle—rather than arbitrary time steps.

Not all DMD modes are physically meaningful; many capture noise. Sparsity-promoting DMD uses an L1-regularized optimization to select a minimal subset of modes that optimally reconstruct the dynamics. The resulting sparse mode amplitudes serve directly as attribution scores, identifying the few dominant coherent structures responsible for a prediction. This provides a parsimonious explanation that aligns with Occam's razor.

DMD operates on the model's hidden state, but attribution must be delivered in the original input space. This is achieved by projecting DMD modes back through the model's encoder or embedding layers using a pseudoinverse or a learned decoder. The reconstructed input-space modes show exactly which spatiotemporal patterns in the raw data—such as a propagating voltage sag in a power grid or a seasonal temperature wave—are responsible for the model's output.

DMD Attribution excels in domains governed by underlying partial differential equations or coherent structures:

• Fluid dynamics: Attributing drag predictions to specific vortex shedding modes.
• Power grids: Linking instability forecasts to inter-area oscillation modes.
• Climate modeling: Decomposing temperature forecasts into seasonal and El Niño modes.
• Finance: Identifying dominant market regimes as coherent modes in high-dimensional order flow data.