Abstract

Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a data-driven hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators (infinite-dimensional operators that globally linearize nonlinear dynamical systems). However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which occur regularly in turbulent flows. Residual Dynamic Mode Decomposition (ResDMD) overcomes these challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator, allowing practitioners to gain confidence in the computed results. We apply ResDMD to several problems in fluid dynamics. For example, we compare ResDMD and DMD for particle image velocimetry data from turbulent wall-jet flow, prediction of the acoustic signature of laser-induced plasma, and the turbulent flow past a cascade of aerofoils. We also discuss the theory of ResDMD. ResDMD rigorously computes spectra of general Koopman operators with error control and spectral measures (including continuous spectra) with explicit high-order convergence theorems. Moreover, the error control provided by ResDMD allows a posteriori verification of learned basis functions. [1] Colbrook, Matthew J., and Alex Townsend. “Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems.” arXiv preprint arXiv:2111.14889 (2021). [2] Colbrook, Matthew J., Lorna J. Ayton, and Máté Szőke. “Residual dynamic mode decomposition: robust and verified Koopmanism.” Journal of Fluid Mechanics 955 (2023). [3] Colbrook, Matthew J. “The mpEDMD algorithm for data-driven computations of measure-preserving dynamical systems.” arXiv preprint arXiv:2209.02244 (2022).