Rayleigh-Based Physical Characterization of Multi-Resolution Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) represents complex, high-dimensionaldata as a combination of spatial modes evolving linearly in time. DMDmodes are characterized by their fixed exponential growth rates and oscillation frequencies but are not inherently associated with physical or spatialquantities which are often of interest. This paper introduces a frameworkthat applies generalized Rayleigh eigenproblems to reorganize DMD modalsubspaces into a new basis with respect to chosen physical quantities. Theresulting modes in this basis, which we call Rayleigh Modes, span the original modal subspaces while preserving the DMD reconstruction and temporal evolution. The framework is demonstrated using multi-resolution DMD(mrDMD) on motion video, CFD cylinder-flow, and seismic wave propagation datasets, with physical characterization by wavenumber, vorticity, orwavespeed.