On Dynamic Mode Decomposition: Theory and Applications

Abstract

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature, but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

Cited by in corpus (241)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Deep learning for universal linear embeddings of nonlinear dynamics
• Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
• Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
• Machine learning for molecular simulation
• Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
• Chaos as an Intermittently Forced Linear System
• VAMPnets: Deep learning of molecular kinetics
• Data-Driven Sparse Sensor Placement for Reconstruction
• Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
• Sparse identification of nonlinear dynamics for model predictive control in the low-data limit
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
• Spectral proper orthogonal decomposition
• On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator
• Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks
• De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis
• Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition
• Data-driven model reduction and transfer operator approximation
• Dynamic Mode Decomposition for Large and Streaming Datasets
• Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation
• Deep Neural Networks with Koopman Operators for Modeling and Control of Autonomous Vehicles
• Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows
• Koopman operator-based model reduction for switched-system control of PDEs
• Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels
• Deep learning of dynamics and signal-noise decomposition with time-stepping constraints
• Robust flow field reconstruction from limited measurements via sparse representation
• Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings
• Data-driven prediction of a multi-scale Lorenz 96 chaotic system using deep learning methods: Reservoir computing, ANN, and RNN-LSTM
• Construction of Reduced Order Models for Fluid Flows Using Deep Feedforward Neural Networks
• Multi-Scale Proper Orthogonal Decomposition of Complex Fluid Flows
• Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations
• Sparse Identification of Nonlinear Dynamics for Rapid Model Recovery
• On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues
• Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition
• Randomized Dynamic Mode Decomposition
• Toward compressed DMD: spectral analysis of fluid flows using sub-Nyquist-rate PIV data
• Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
• Study of dynamics in post-transient flows using Koopman mode decomposition
• Subspace dynamic mode decomposition for stochastic Koopman analysis
• Data-Driven Model Predictive Control using Interpolated Koopman Generators
• Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches
• Data-driven resolvent analysis
• Sparse sensing and DMD based identification of flow regimes and bifurcations in complex flows
• Residual Dynamic Mode Decomposition: Robust and verified Koopmanism
• Applied Koopman Operator Theory for Power Systems Technology
• Real-time forecasting of metro origin-destination matrices with high-order weighted dynamic mode decomposition
• On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations
• Special Topic: Markov Models of Molecular Kinetics
• Tensor-based dynamic mode decomposition
• Active Learning in Robotics: A Review of Control Principles
• Bagging, optimized dynamic mode decomposition (BOP-DMD) for robust, stable forecasting with spatial and temporal uncertainty-quantification
• Data-driven cardiovascular flow modeling: examples and opportunities
• Delay-coordinate maps and the spectra of Koopman operators
• Data-driven Experimental Modal Analysis by Dynamic Mode Decomposition
• On the Structure of Time-delay Embedding in Linear Models of Non-linear Dynamical Systems
• Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
• Data-driven modal decomposition methods as feature detection techniques for flow problems: a critical assessment
• Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)
• Characterizing Magnetized Plasmas with Dynamic Mode Decomposition
• Generating Random Earthquake Events for PTHA
• Dynamic Mode Decomposition for Plasma Diagnostics and Validation
• Dynamic mode decomposition using a Kalman filter for parameter estimation
• Extended-Kalman-filter-based dynamic mode decomposition for simultaneous system identification and denoising
• Koopman analysis of the long-term evolution in a turbulent convection cell
• Koopman spectra in reproducing kernel Hilbert spaces
• Reproducing kernel Hilbert space compactification of unitary evolution groups
• Memory embedded non-intrusive reduced order modeling of non-ergodic flows
• Optimized Sampling for Multiscale Dynamics
• On generalized residue network for deep learning of unknown dynamical systems
• On the role of nonlinear correlations in reduced-order modeling
• Koopman mode expansions between simple invariant solutions
• An Optimal Control Formulation of Pulse-Based Control Using Koopman Operator
• A Parametric and Feasibility Study for Data Sampling of the Dynamic Mode Decomposition--Range, Resolution, and Universal Convergence States
• Dynamic mode decomposition for multiscale nonlinear physics
• Reservoir Computing as a Tool for Climate Predictability Studies
• Dynamic mode decomposition based analysis of flow over a sinusoidally pitching airfoil
• A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions
• Data-driven modeling of the chaotic thermal convection in an annular thermosyphon
• Linear and Nonlinear Dimensionality Reduction from Fluid Mechanics to Machine Learning
• Multiscale Proper Orthogonal Decomposition (mPOD) of TR-PIV data-- a Case Study on Stationary and Transient Cylinder Wake Flows
• A spectral characterization of nonlinear normal modes
• Data-driven nonlinear aeroelastic models of morphing wings for control
• Reduced-Order Models for Coupled Dynamical Systems: Data-driven Methods and the Koopman Operator
• Deep-learning assisted reduced order model for high-dimensional flow prediction from sparse data
• Preconditioned dynamic mode decomposition and mode selection algorithms for large datasets using incremental proper orthogonal decomposition
• Dimensionality Reduction and Reduced Order Modeling for Traveling Wave Physics
• Searching turbulence for periodic orbits with dynamic mode decomposition
• Modal decomposition of fluid-structure interaction with application to flag flapping
• Data-adaptive harmonic spectra and multilayer Stuart-Landau models
• Environment Identification in Flight using Sparse Approximation of Wing Strain
• Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond
• Calculating Time Eigenvalues of the Neutron Transport Equation with Dynamic Mode Decomposition
• Reduced-order modeling of left ventricular flow subject to aortic valve regurgitation
• Learning Latent Dynamics for Partially-Observed Chaotic Systems
• A Parametric and Feasibility Study for Data Sampling of the Dynamic Mode Decomposition: Spectral Insights and Further Explorations
• Data-driven Koopman Operators for Model-based Shared Control of Human-Machine Systems
• Identification of kinetic order parameters for non-equilibrium dynamics
• Koopman analysis of Burgers equation
• Event-based Background-Oriented Schlieren
• Structured Time-Delay Models for Dynamical Systems with Connections to Frenet-Serret Frame
• Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables
• Scalable Extended Dynamic Mode Decomposition using Random Kernel Approximation
• Variational approach to closure of nonlinear dynamical systems: Autonomous case
• On Some Aspects of the Response to Stochastic and Deterministic Forcings
• Bilinear dynamic mode decomposition for quantum control
• Cluster-based hierarchical network model of the fluidic pinball -- Cartographing transient and post-transient, multi-frequency, multi-attractor behaviour
• Identification of MIMO Wiener-type Koopman Models for Data-Driven Model Reduction using Deep Learning
• Enhancing CFD predictions in shape design problems by model and parameter space reduction
• Robust Dynamic Mode Decomposition
• Quantitative Evaluation of a Linear Reduced-order Model based on Particle-image-velocimetry Data of Separated Flow Field around Airfoil
• Establishing Direct Phenomenological Connections between Fluid and Structure by the Koopman-Linearly-Time-Invariant Analysis
• Sampling and resolution characteristics in reduced order models of shallow water equations: intrusive vs non-intrusive
• Data-Driven Acceleration of Thermal Radiation Transfer Calculations with the Dynamic Mode Decomposition and a Sequential Singular Value Decomposition
• Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence
• The Linear-Time-Invariance Notion of the Koopman Analysis-Part 1: The Architecture, Practical Rendering on the Prism Wake, and Fluid-Structure Association
• Data-Driven Correction Reduced Order Models for the Quasi-Geostrophic Equations: A Numerical Investigation
• Data-driven Modeling of Rotating Detonation Waves
• Koopman operator learning using invertible neural networks
• Probing non-Markovian quantum dynamics with data-driven analysis: Beyond "black-box" machine learning models
• A three-dimensional dynamic mode decomposition analysis of wind farm flow aerodynamics
• Kernel methods for detecting coherent structures in dynamical data
• Data-Driven Selection of Actuators for Optimal Control of Airfoil Separation
• Detection and Prediction of Equilibrium States in Kinetic Plasma Simulations via Mode Tracking using Reduced-Order Dynamic Mode Decomposition
• An efficient computational framework for naval shape design and optimization problems by means of data-driven reduced order modeling techniques
• Model and Data Reduction for Data Assimilation: Particle Filters Employing Projected Forecasts and Data with Application to a Shallow Water Model
• Phase-consistent dynamic mode decomposition from multiple overlapping spatial domains
• Learning developmental mode dynamics from single-cell trajectories
• Data-driven reduced modelling of turbulent Rayleigh-Benard convection using DMD-enhanced Fluctuation-Dissipation Theorem
• A parallel and streaming Dynamic Mode Decomposition algorithm with finite precision error analysis for large data
• Linear stability analysis of detonations via numerical computation and dynamic mode decomposition
• An Operator Inference Oriented Approach for Mechanical Systems
• Energy preserving model order reduction of the nonlinear Schrödinger equation
• Dynamics of random pressure fields over bluff bodies: a dynamic mode decomposition perspective
• A coherent structure approach for parameter estimation in Lagrangian Data Assimilation
• Dynamic Mode Decomposition for Extrapolating Non-equilibrium Green's Functions Dynamics
• Magnetohydrodynamic wave mode identification in circular and elliptical sunspot umbrae: evidence for high order modes
• Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method
• Nonequilibrium Statistical Mechanics and Optimal Prediction of Partially-Observed Complex Systems
• Eigenfunctions of Galactic Phase Space Spirals from Dynamic Mode Decomposition
• Koopman analysis of quantum systems
• Deeptime: a Python library for machine learning dynamical models from time series data
• Physically-interpretable classification of biological network dynamics for complex collective motions
• Data-driven model order reduction for structures with piecewise linear nonlinearity using dynamic mode decomposition
• Probabilistic forecast of nonlinear dynamical systems with uncertainty quantification
• Near-Wake Dynamics of a Vertical-Axis Turbine
• Jupiter's cloud-level variability triggered by torsional oscillations in the interior
• Feedback control of nonlinear PDEs using data-efficient reduced order models based on the Koopman operator
• Learning of Causal Observable Functions for Koopman-DFL Lifting Linearization of Nonlinear Controlled Systems and Its Application to Excavation Automation
• Stable Dynamic Mode Decomposition Algorithm for Noisy Pressure-Sensitive Paint Measurement Data
• Go with the FLOW: Visualizing spatiotemporal dynamics in optical widefield calcium imaging
• Two methods to approximate the Koopman operator with a reservoir computer
• Detecting regime transitions in time series using dynamic mode decomposition
• Combining Dynamic Mode Decomposition with Ensemble Kalman Filtering for Tracking and Forecasting
• Forecasting long-time dynamics in quantum many-body systems by dynamic mode decomposition
• Robust orbital-angular-momentum-based underwater acoustic communication with dynamic modal decomposition method
• Dynamic mode decomposition of magnetohydrodynamic bubble chain flow in a rectangular vessel
• Consistent spectral approximation of Koopman operators using resolvent compactification
• Domain decomposition for data-driven reduced modeling of large-scale systems
• Data-driven Discovery of Delay Differential Equations with Discrete Delays
• HAVOK Model Predictive Control for Time-Delay Systems with Applications to District Heating
• Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data
• An Optimized Dynamic Mode Decomposition Model Robust to Multiplicative Noise
• Data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media
• Colored-LIM: A Data-Driven Method for Studying Dynamical Systems with Temporally Correlated Stochasticity
• A Data-driven Approach for Rapid Detection of Aeroelastic Modes from Flutter Flight Test Based on Limited Sensor Measurements
• Distributed computing for physics-based data-driven reduced modeling at scale: Application to a rotating detonation rocket engine
• Reduced-order modeling of fully turbulent buoyancy-driven flows using the Green's function method
• Koopman Operator Dynamical Models: Learning, Analysis and Control
• Data-Driven Koopman Controller Synthesis Based on the Extended ${\mathcal{H}}_2$ Norm Characterization
• Identifying efficient ensemble perturbations for initializing subseasonal-to-seasonal prediction
• On the sample complexity of stabilizing linear dynamical systems from data
• Dynamical System Segmentation for Information Measures in Motion
• Connections between propulsive efficiency and wake structure via modal decomposition
• Data-Driven Models for Control Engineering Applications Using the Koopman Operator
• A case study of spatiotemporal forecasting techniques for weather forecasting
• A minimal hyperbolic system for unstable shock waves
• Entangled gene regulatory networks with cooperative expression endow robust adaptive responses to unforeseen environmental changes
• Extended dynamic mode decomposition with dictionary learning using neural ordinary differential equations
• Understanding Mass Transfer Directions via Data-Driven Models with Application to Mobile Phone Data
• High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics
• Dynamic mode decomposition as an analysis tool for time-dependent partial differential equations
• Accelerating Particle-in-Cell Kinetic Plasma Simulations via Reduced-Order Modeling of Space-Charge Dynamics using Dynamic Mode Decomposition
• On Koopman Mode Decomposition and Tensor Component Analysis
• A Reduced-Order Model for Nonlinear Radiative Transfer Problems Based on Moment Equations and POD-Petrov-Galerkin Projection of the Normalized Boltzmann Transport Equation
• Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation
• Data-driven kinematics-consistent model order reduction of fluid-structure interaction problems: application to deformable microcapsules in a Stokes flow
• Schrodinger dynamics and Berry phase of undulatory locomotion
• Reduced-order modeling using Dynamic Mode Decomposition and Least Angle Regression
• Characterization of elastic topological states using dynamic mode decomposition
• Effective equations for reaction coordinates in polymer transport
• A Dynamic Mode Decomposition Approach for Decentralized Spectral Clustering of Graphs
• A Quantum Algorithm for Dynamic Mode Decomposition Integrated with a Quantum Differential Equation Solver
• Real-Time Motion Detection Using Dynamic Mode Decomposition
• Memory-based reduced modelling and data-based estimation of opinion spreading
• Effects of confinement, impinging shock deflection angle, and Mach number on the flow field of a supersonic open cavity
• Schrödinger PCA: On the Duality between Principal Component Analysis and Schrödinger Equation
• Advanced modeling for the HIT-SI Experiment
• Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems
• Correlating Time Series with Interpretable Convolutional Kernels
• The detection matrix as a model-agnostic tool to estimate the number of degrees of freedom in mechanical systems and engineering structures
• Online learning of Koopman operator using streaming data from different dynamical regimes
• Inferring Time-Dependent Distribution Functions from Kinematic Snapshots
• Context-aware controller inference for stabilizing dynamical systems from scarce data
• Physics-informed Split Koopman Operators for Data-efficient Soft Robotic Simulation
• A Comparison of Parametric Dynamic Mode Decomposition Algorithms for Thermal-Hydraulics Applications
• On the Cyclostationary Linear Inverse Models: A Mathematical Insight and Implication
• Using Covariant Lyapunov Vectors to Quantify High Dimensional Chaos with a Conservation Law
• The spatiotemporal coupling in delay-coordinates dynamic mode decomposition
• Koopman Learning with Episodic Memory
• Transition to chaos and modal structure of magnetized Taylor-Couette flow
• Dynamic Mode Decomposition of Control-Affine Nonlinear Systems using Discrete Control Liouville Operators
• Data-Augmented Predictive Deep Neural Network: Enhancing the extrapolation capabilities of non-intrusive surrogate models
• Star Log-extended eMulation: a method for efficient computation of the Tolman-Oppenheimer-Volkoff equations
• Identifying Large-Scale Linear Parameter Varying Systems with Dynamic Mode Decomposition Methods
• Fast prediction of plasma instabilities with sparse-grid-accelerated optimized dynamic mode decomposition
• Reduced-Order Modeling for Heston Stochastic Volatility Model
• Sketching Methods for Dynamic Mode Decomposition in Spherical Shallow Water Equations
• Parametric Landau damping of space charge modes
• On the Predictive Capability of Dynamic Mode Decomposition for Nonlinear Periodic Systems with Focus on Orbital Mechanics
• System stabilization with policy optimization on unstable latent manifolds
• Signature of glassy dynamics in dynamic modes decompositions
• Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces
• Block-structured Operator Inference for coupled multiphysics model reduction
• A parallel implementation of reduced-order modeling of large-scale systems
• Learning to Optimize with Dynamic Mode Decomposition
• Reconstruction of phase-amplitude dynamics from electrophysiological signals
• Frequency-dependent stress response under thermal cycle: A thermal-crystal plasticity and dynamic mode decomposition study
• Uncertainty-Aware Flow Field Reconstruction Using SVGP Kolmogorov-Arnold Networks
• An Incremental Multi-Level, Multi-Scale Approach to Assessment of Multifidelity HPC Systems
• Dynamic mode decomposition for detecting oscillatory transient activity via sparsity and smoothness regularization
• Data-Driven Transient Growth Analysis
• Randomized Projection Learning Method forDynamic Mode Decomposition
• Data-Driven Reconstruction and Characterization of Stochastic Dynamics via Dynamical Mode Decomposition
• Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine
• Koopman spectral analysis of elementary cellular automata
• Efficient pseudometrics for data-driven comparisons of nonlinear dynamical systems
• Statistical analysis and method to quantify the impact of measurement uncertainty on dynamic mode decomposition
• Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems
• Integrated utilization of equations and small dataset in the Koopman operator: applications to forward and inverse problems
• Stabilizing RED using the Koopman Operator