Mori-Zwanzig mode decomposition: Comparison with time-delay embeddings

We introduce the Mori-Zwanzig Mode Decomposition (MZMD), a novel data-driven technique for efficient modal analysis of and reduced-order modeling of large-scale spatio-temporal dynamical systems. MZMD represents an extension of Dynamic Mode Decomposition (DMD) by providing an approximate closure term with MZ memory kernels accounting for how the unresolved modes of DMD interact with the resolved modes, thus addressing limitations when the state-space observables do not form a Koopman-invariant subspace. Leveraging the Mori-Zwanzig (MZ) formalism, MZMD identifies the modes and spectrum of the discrete-time Generalized Langevin Equation (GLE); an integro-differential equation that governs the dynamics of selected observables and their memory-dependent coupling with the unresolved degrees of freedom. This feature fundamentally distinguishes MZMD from time-delay embedding methods, such as Higher-Order DMD (HODMD). In this work, we derive and analyze MZMD and compare it with DMD and HODMD, using two exemplary Direct Numerical Simulation (DNS) datasets: a 2D flow over a cylinder (as validation) and laminar-turbulent boundary-layer transition over a flared cone at Mach 6. We demonstrate that MZMD, via the addition of MZ memory terms, improves the resolution of spatio-temporal structures within the transitional/turbulent regime by the introduction of transient and periodic modes (not captured by DMD), which contain features that arise due to nonlinear mechanisms, such as the generation of the so-called hot streaks on the surface of the flared cone. Our results demonstrate that MZMD serves as an efficient generalization of DMD (reducing to DMD in the absence of memory), improves stability, and exhibits greater robustness and resistance to overfitting compared to HODMD.