# Residual Dynamic Mode Decomposition: Robust and verified Koopmanism arXiv: https://arxiv.org/abs/2205.09779 ยท HTML: https://mlsanity.com/paper/2205.09779/ ## Key facts - Cited by 36 papers in the map (data through 14 Oct 2026). - Builds on 10 of its 124 references that are in the map. - First posted on arXiv 19 May 2022. ## Builds on (most cited first, 10 of 10) - [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021) - [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014) - [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019) - [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017) - [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014) - [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021) - [Sparsity-promoting dynamic mode decomposition](https://mlsanity.com/paper/1309.4165/) (2013) - [Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control](https://mlsanity.com/paper/1902.02827/) (2019) - [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021) - [Physics-informed dynamic mode decomposition (piDMD)](https://mlsanity.com/paper/2112.04307/) (2021) ## Built on by (most cited first, 30 of 36) - [Koopman Kernel Regression](https://mlsanity.com/paper/2305.16215/) (2023) - [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022) - [Koopman operator learning using invertible neural networks](https://mlsanity.com/paper/2306.17396/) (2023) - [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023) - [Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems](https://mlsanity.com/paper/2308.10697/) (2023) - [Koopman-Assisted Reinforcement Learning](https://mlsanity.com/paper/2403.02290/) (2024) - [PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator](https://mlsanity.com/paper/2306.12962/) (2023) - [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024) - [A Control Theory of Predictability in Latent World Models](https://mlsanity.com/paper/2607.10362/) (2026) - [Koopman-informed recurrent neural networks](https://mlsanity.com/paper/2410.23467/) (2024) - [Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman Operators](https://mlsanity.com/paper/2405.00782/) (2024) - [Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories](https://mlsanity.com/paper/2209.09977/) (2022) - [ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals](https://mlsanity.com/paper/2501.00701/) (2025) - [An Introductory Guide to Koopman Learning](https://mlsanity.com/paper/2510.22002/) (2025) - [InputDSA: Demixing then Comparing Recurrent and Externally Driven Dynamics](https://mlsanity.com/paper/2510.25943/) (2025) - [Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches](https://mlsanity.com/paper/2310.06790/) (2023) - [Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems](https://mlsanity.com/paper/2507.07222/) (2025) - [Orthogonal polynomial approximation and Extended Dynamic Mode Decomposition in chaos](https://mlsanity.com/paper/2305.08074/) (2023) - [Multiplicative Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05334/) (2024) - [On the Convergence of Hermitian Dynamic Mode Decomposition](https://mlsanity.com/paper/2401.03192/) (2024) - [Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine](https://mlsanity.com/paper/2505.10438/) (2025) - [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025) - [Data-driven modelling of autonomous and forced dynamical systems](https://mlsanity.com/paper/2512.12432/) (2025) - [Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation](https://mlsanity.com/paper/2606.29083/) (2026) - [Learning dynamical systems from noisy data with Weak-form Kernel Ridge Regression](https://mlsanity.com/paper/2607.00257/) (2026) - [Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors](https://mlsanity.com/paper/2310.12156/) (2023) - [Real-Time Motion Detection Using Dynamic Mode Decomposition](https://mlsanity.com/paper/2405.05057/) (2024) - [Global Description of Flutter Dynamics via Koopman Theory](https://mlsanity.com/paper/2505.14697/) (2025) - [Avoiding spectral pollution for transfer operators using residuals](https://mlsanity.com/paper/2507.16915/) (2025) - [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025) ## Abstract (arXiv) Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a major challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes, and dealing with continuous spectra, both of which occur regularly in turbulent flows. Residual Dynamic Mode Decomposition (ResDMD), introduced by (Colbrook & Townsend 2021), overcomes some of these challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control, and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in a variety of fluid dynamic situations, at varying Reynolds numbers, arising from both numerical and experimental data. Examples include: vortex shedding behind a cylinder; hot-wire data acquired in a turbulent boundary layer; particle image velocimetry data focusing on a wall-jet flow; and acoustic pressure signals of laser-induced plasma. We present some advantages of ResDMD, namely, the ability to verifiably resolve non-linear, transient modes, and spectral calculation with reduced broadening effects. We also discuss how a new modal ordering based on residuals enables greater accuracy with a smaller dictionary than the traditional modulus ordering. This paves the way for greater dynamic compression of large datasets without sacrificing accuracy.