A data-driven approximation of the koopman operator: extending dynamic mode decomposition

DOI10.1007/s00332-015-9258-5zbMath1329.65310arXiv1408.4408OpenAlexW3099674394MaRDI QIDQ897161

Clarence W. Rowley, Matthew O. Williams, Ioannis G. Kevrekidis

Publication date: 17 December 2015

Published in: Journal of Nonlinear Science (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1408.4408

zbMATH Keywords

algorithm numerical example eigenvalue eigenfunction data mining nonlinear dynamical systems spectral methods dynamic mode decomposition Kolmogorov backward equation Koopman spectral analysis reduced-rder models set oriented methods

Mathematics Subject Classification ID

Generation, random and stochastic difference and differential equations (37H10) Linear composition operators (47B33) Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) (37M25) Numerical problems in dynamical systems (65P99)

Related Items

Study of the thermo-magneto-hydrodynamic flow of micropolar-nanofluid in square enclosure using dynamic mode decomposition and proper orthogonal decomposition, Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism, Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems, Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques, Total-variation mode decomposition, Spectral and modal methods for studying stability and control of electric power systems, Koopman operator method for solution of generalized aggregate data inverse problems, A data-driven, physics-informed framework for forecasting the spatiotemporal evolution of chaotic dynamics with nonlinearities modeled as exogenous forcings, Extended dynamic mode decomposition for inhomogeneous problems, Deep learning nonlinear multiscale dynamic problems using Koopman operator, A reduced order method for nonlinear parameterized partial differential equations using dynamic mode decomposition coupled with $k$-nearest-neighbors regression, Stochastic embeddings of dynamical phenomena through variational autoencoders, A Koopman framework for rare event simulation in stochastic differential equations, Correcting noisy dynamic mode decomposition with Kalman filters, A general neural particle method for hydrodynamics modeling, Data Driven Modal Decompositions: Analysis and Enhancements, Tensor-based dynamic mode decomposition, Network resilience, A kernel-based method for data-driven Koopman spectral analysis, tgEDMD: approximation of the Kolmogorov operator in tensor train format, On the numerical approximation of the Perron-Frobenius and Koopman operator, Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks, Nonlinear observability via Koopman analysis: characterizing the role of symmetry, A few techniques to improve data-driven reduced-order simulations for unsteady flows, Lyapunov modal analysis and participation factors applied to small-signal stability of power systems, Operator inference and physics-informed learning of low-dimensional models for incompressible flows, Sparsity enabled cluster reduced-order models for control, Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables, Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference, Assessment of end-to-end and sequential data-driven learning for non-intrusive modeling of fluid flows, On data-driven computation of information transfer for causal inference in discrete-time dynamical systems, Extension of dynamic mode decomposition for dynamic systems with incomplete information based on t-model of optimal prediction, On the universal transformation of data-driven models to control systems, Evaluating the accuracy of the dynamic mode decomposition, Higher-order dynamic mode decomposition on-the-fly: a low-order algorithm for complex fluid flows, An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling, A tale of two vortices: how numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems, Time-resolved denoising using model order reduction, dynamic mode decomposition, and Kalman filter and smoother, Extended dynamic mode decomposition for two paradigms of non-linear dynamical systems, High-dimensional time series prediction using kernel-based koopman mode regression, Data-driven control of agent-based models: an equation/variable-free machine learning approach, Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning, Koopman operator-based driver-vehicle dynamic model for shared control systems, Spectral Identification of Networks Using Sparse Measurements, Dynamic mode decomposition for analytic maps, A parallel and streaming dynamic mode decomposition algorithm with finite precision error analysis for large data, Numerical convergence of the Lyapunov spectrum computed using low Mach number solvers, Applied Koopman theory for partial differential equations and data-driven modeling of spatio-temporal systems, Recovering missing CFD data for high-order discretizations using deep neural networks and dynamics learning, A data-driven non-linear assimilation framework with neural networks, Delay-coordinate maps, coherence, and approximate spectra of evolution operators, A data-driven smoothed particle hydrodynamics method for fluids, Kernel embedding based variational approach for low-dimensional approximation of dynamical systems, On convergence of extended dynamic mode decomposition to the Koopman operator, Resonances in a chaotic attractor crisis of the Lorenz flow, On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis, Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator, Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms, Memory-based reduced modelling and data-based estimation of opinion spreading, Koopman operator framework for time series modeling and analysis, Koopman operator spectrum for random dynamical systems, On robust computation of Koopman operator and prediction in random dynamical systems, Go with the flow, on Jupiter and snow. Coherence from model-free video data without trajectories, On sample-based computations of invariant sets, Geometric considerations of a good dictionary for Koopman analysis of dynamical systems: cardinality, ``primary eigenfunction, and efficient representation, Data-driven model reduction and transfer operator approximation, Koopman operator-based model reduction for switched-system control of PDEs, Reproducing kernel Hilbert space compactification of unitary evolution groups, Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Data-driven spectral analysis of the Koopman operator, Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories, Spectral proper orthogonal decomposition, Variational approach for learning Markov processes from time series data, Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces, Modal decomposition methods for distributed excitation force field on tube bundle in cross flow, Dynamic mode decomposition for continuous time systems with the Liouville operator, Low-rank dynamic mode decomposition: an exact and tractable solution, Delay-coordinate maps and the spectra of Koopman operators, Robust tube-based model predictive control with Koopman operators, Extracting governing laws from sample path data of non-Gaussian stochastic dynamical systems, Tensor-based computation of metastable and coherent sets, Supervised learning from noisy observations: combining machine-learning techniques with data assimilation, Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits, Koopman spectra in reproducing kernel Hilbert spaces, Data-driven spectral decomposition and forecasting of ergodic dynamical systems, Intrusive and data-driven reduced order modelling of the rotating thermal shallow water equation, Spatiotemporal pattern extraction by spectral analysis of vector-valued observables, Detecting regime transitions in time series using dynamic mode decomposition, Variational approach to closure of nonlinear dynamical systems: autonomous case, Ruelle-Pollicott resonances of stochastic systems in reduced state space. Part III: Application to the Cane-Zebiak model of the El Niño-southern oscillation, Data-driven operator theoretic methods for phase space learning and analysis, Sparse feature map-based Markov models for nonlinear fluid flows, Koopman-based spectral clustering of directed and time-evolving graphs, Computation of invariant sets via immersion for discrete-time nonlinear systems, Finite-data error bounds for Koopman-based prediction and control, Data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media, Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems, Lift \& learn: physics-informed machine learning for large-scale nonlinear dynamical systems, Data-driven approximation of the Koopman generator: model reduction, system identification, and control, A data-driven approach for discovering stochastic dynamical systems with non-Gaussian Lévy noise, Combining dynamic mode decomposition with ensemble Kalman filtering for tracking and forecasting, Predicting rare events using neural networks and short-trajectory data, The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems, Generalizing dynamic mode decomposition: balancing accuracy and expressiveness in Koopman approximations, A real-time identification method of network structure in complex network systems, Active Operator Inference for Learning Low-Dimensional Dynamical-System Models from Noisy Data, Parsimony as the ultimate regularizer for physics-informed machine learning, Data‐assisted control: A framework development by exploiting NASA Generic Transport platform, Constrained optimized dynamic mode decomposition with control for physically stable systems with exogeneous inputs, Probabilistic forecast of nonlinear dynamical systems with uncertainty quantification, Data-driven inference of low order representations of observable dynamics for an airfoil model, Koopman operator learning using invertible neural networks, Operator inference with roll outs for learning reduced models from scarce and low-quality data, Quantum Mechanics for Closure of Dynamical Systems, Stochastic dynamics and data science, Data-driven method to extract mean exit time and escape probability for dynamical systems driven by Lévy noises, Reduced Order Characterization of Nonlinear Oscillations Using an Adaptive Phase-Amplitude Coordinate Framework, Data driven discovery of systems of ordinary differential equations using nonconvex multitask learning, Data-driven feedback stabilisation of nonlinear systems: Koopman-based model predictive control, Mean resolvent operator of a statistically steady flow, Neural dynamic mode decomposition for end-to-end modeling of nonlinear dynamics, The Adaptive Spectral Koopman Method for Dynamical Systems, Learning to Forecast Dynamical Systems from Streaming Data, A Reduced Order Modeling Framework for Strongly Perturbed Nonlinear Dynamical Systems Near Arbitrary Trajectory Sets, Koopman Operator Inspired Nonlinear System Identification, Rigorous data‐driven computation of spectral properties of Koopman operators for dynamical systems, Data-driven model identification using forcing-induced limit cycles, Ensemble forecasts in reproducing kernel Hilbert space family, Piecewise DMD for oscillatory and Turing spatio-temporal dynamics, Efficiency of randomised dynamic mode decomposition for reduced order modelling, Sparse Sensing and DMD-Based Identification of Flow Regimes and Bifurcations in Complex Flows, Approximating Matrix Eigenvalues by Subspace Iteration with Repeated Random Sparsification, Numerical methods to evaluate Koopman matrix from system equations*, Koopman analysis of quantum systems*, Extracting stochastic dynamical systems with α-stable Lévy noise from data, Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks, Spectral Discovery of Jointly Smooth Features for Multimodal Data, Discriminant Dynamic Mode Decomposition for Labeled Spatiotemporal Data Collections, Extracting Sparse High-Dimensional Dynamics from Limited Data, Data-Driven Discovery of Closure Models, Koopman Operator Family Spectrum for Nonautonomous Systems, Modern Koopman Theory for Dynamical Systems, Sparse reduced-order modelling: sensor-based dynamics to full-state estimation, Crisis of the chaotic attractor of a climate model: a transfer operator approach, Generalizing Koopman Theory to Allow for Inputs and Control, Reduced Operator Inference for Nonlinear Partial Differential Equations, Dynamically Orthogonal Numerical Schemes for Efficient Stochastic Advection and Lagrangian Transport, Koopman analysis of the long-term evolution in a turbulent convection cell, Robust Approximation of the Stochastic Koopman Operator, Insights into the dynamics of conical breakdown modes in coaxial swirling flow field, Data-driven reduced modelling of turbulent Rayleigh–Bénard convection using DMD-enhanced fluctuation–dissipation theorem, Generative Stochastic Modeling of Strongly Nonlinear Flows with Non-Gaussian Statistics, Higher Order Extended Dynamic Mode Decomposition Based on the Structured Total Least Squares, Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain, Koopman analysis of nonlinear systems with a neural network representation, Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning, DRIPS: a framework for dimension reduction and interpolation in parameter space, Data-driven probability density forecast for stochastic dynamical systems, Identifying finite-time coherent sets from limited quantities of Lagrangian data, Auxiliary functions as Koopman observables: data-driven analysis of dynamical systems via polynomial optimization, Regression-Based Projection for Learning Mori–Zwanzig Operators, Dynamic reconstruction and data reconstruction for subsampled or irregularly sampled data, Data‐driven identification of the spatiotemporal structure of turbulent flows by streaming dynamic mode decomposition, Deep Koopman model predictive control for enhancing transient stability in power grids, CD-ROM: complemented deep -- reduced order model, Data-Driven Modeling and Control of Complex Dynamical Systems Arising in Renal Anemia Therapy, Human-centered driving authority allocation for driver-automation shared control: a two-layer game-theoretic approach, On the Koopman Operator of Algorithms, Time-Delay Observables for Koopman: Theory and Applications, Physics-Informed Probabilistic Learning of Linear Embeddings of Nonlinear Dynamics with Guaranteed Stability, Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh–Bénard convection, Prediction Accuracy of Dynamic Mode Decomposition, Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator, Time Series Source Separation Using Dynamic Mode Decomposition, A Geometric Approach to Dynamical Model Order Reduction, A Weak Characterization of Slow Variables in Stochastic Dynamical Systems, Dynamical Systems Theory and Algorithms for NP-hard Problems, On Least Squares Problems with Certain Vandermonde--Khatri--Rao Structure with Applications to DMD, Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings, Linearly Recurrent Autoencoder Networks for Learning Dynamics, Data-Driven Identification of Parametric Partial Differential Equations, Optimized Sampling for Multiscale Dynamics, Discovering transition phenomena from data of stochastic dynamical systems with Lévy noise, Extracting non-Gaussian governing laws from data on mean exit time, Detecting the maximum likelihood transition path from data of stochastic dynamical systems, Sampling Low-Dimensional Markovian Dynamics for Preasymptotically Recovering Reduced Models from Data with Operator Inference, Manifold learning for organizing unstructured sets of process observations, Discovery of Dynamics Using Linear Multistep Methods, Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems, Data-adaptive harmonic spectra and multilayer Stuart-Landau models, Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator, Geometric-control formulation and averaging analysis of the unsteady aerodynamics of a wing with oscillatory controls, SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics, On Matching, and Even Rectifying, Dynamical Systems through Koopman Operator Eigenfunctions, Single- and double-helix vortex breakdown as two dominant global modes in turbulent swirling jet flow, Kernel methods for detecting coherent structures in dynamical data, Searching turbulence for periodic orbits with dynamic mode decomposition, A data-driven phase and isostable reduced modeling framework for oscillatory dynamical systems, A Review on Reduced Order Modeling using DMD-Based Methods, Constrained sparse Galerkin regression, Scalable Extended Dynamic Mode Decomposition Using Random Kernel Approximation, Centering Data Improves the Dynamic Mode Decomposition, Data-Driven Model Predictive Control using Interpolated Koopman Generators, On explaining the surprising success of reservoir computing forecaster of chaos? The universal machine learning dynamical system with contrast to VAR and DMD, Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces, Koopman mode expansions between simple invariant solutions, Data-driven resolvent analysis, Kernel-based parameter estimation of dynamical systems with unknown observation functions, On the Approximation of Koopman Spectra for Measure Preserving Transformations, Online Dynamic Mode Decomposition for Time-Varying Systems, 9 From the POD-Galerkin method to sparse manifold models, Data Driven Koopman Spectral Analysis in Vandermonde--Cauchy Form via the DFT: Numerical Method and Theoretical Insights, On Koopman mode decomposition and tensor component analysis, Koopman operator and its approximations for systems with symmetries, Sparse identification of nonlinear dynamics for model predictive control in the low-data limit, Error Bounds for Dynamical Spectral Estimation, Data-driven inference of high-accuracy isostable-based dynamical models in response to external inputs, A Tailored Convolutional Neural Network for Nonlinear Manifold Learning of Computational Physics Data Using Unstructured Spatial Discretizations, Two methods to approximate the Koopman operator with a reservoir computer, Koopman Analysis of Isolated Fronts and Solitons, Modes of Homogeneous Gradient Flows, Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems, Deep learning models for global coordinate transformations that linearise PDEs, Higher Order Dynamic Mode Decomposition, Data-Driven Learning for the Mori--Zwanzig Formalism: A Generalization of the Koopman Learning Framework, An Adaptive Phase-Amplitude Reduction Framework without $\mathcal{O}(\epsilon)$ Constraints on Inputs, Data-driven kinematics-consistent model-order reduction of fluid–structure interaction problems: application to deformable microcapsules in a Stokes flow, Residual dynamic mode decomposition: robust and verified Koopmanism

Uses Software

Cites Work