Mori--Zwanzig Mode Decomposition: a comparison to DMD and HODMD

We present the Mori–Zwanzig Mode Decomposition (MZMD), which extends Dynamic Mode Decomposition by embedding memory kernels derived from the Mori–Zwanzig formalism, providing a data-driven approximate closure for the dynamics that standard DMD leaves unresolved. By casting the evolution of chosen observables as a discrete-time Generalized Langevin Equation, MZMD extracts both modes and spectrum while explicitly modeling their memory-based coupling to unresolved degrees of freedom, a capability that distinguishes it from time-delay methods such as Higher-Order DMD. When applied to a Mach-6 flared-cone boundary-layer transition, MZMD resolves the leading coherent structures, especially the hypersonic "hot-streaks", with more fidelity than DMD, and less prone to overfitting than HODMD. Its memory kernels stabilize the reduced model, reveal higher-harmonic content that truncated DMD misses, and reduces short-time prediction error. Together, these benefits establish MZMD as a scalable, physically transparent extension of Koopman-based techniques for modal diagnosis, forecasting, and analysis.