A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

Abstract

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions.

References in corpus (3)

• On Dynamic Mode Decomposition: Theory and Applications
• Applied Koopmanism
• Geometry of the ergodic quotient reveals coherent structures in flows

Cited by in corpus (303)

• Deep learning for universal linear embeddings of nonlinear dynamics
• Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
• Data-driven discovery of coordinates and governing equations
• 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
• 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
• SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics
• Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
• Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems
• Data-driven model reduction and transfer operator approximation
• Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation
• Network resilience
• Machine learning force fields and coarse-grained variables in molecular dynamics: application to materials and biological systems
• 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
• On Explaining the Surprising Success of Reservoir Computing Forecaster of Chaos? The Universal Machine Learning Dynamical System with Contrasts to VAR and DMD
• Robust Tube-based Model Predictive Control with Koopman Operators--Extended Version
• On the numerical approximation of the Perron-Frobenius and Koopman operator
• 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
• Deep Learning in Deterministic Computational Mechanics
• Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations
• Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability
• On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues
• Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms
• Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
• Study of dynamics in post-transient flows using Koopman mode decomposition
• Data-Driven Model Predictive Control using Interpolated Koopman Generators
• Subspace dynamic mode decomposition for stochastic Koopman analysis
• Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems
• Galerkin Approximation of Dynamical Quantities using Trajectory Data
• 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
• Data-driven Discovery of Closure Models
• Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems
• Supervised learning from noisy observations: Combining machine-learning techniques with data assimilation
• A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise
• Special Topic: Markov Models of Molecular Kinetics
• Recovering missing CFD data for high-order discretizations using deep neural networks and dynamics learning
• Tensor-based dynamic mode decomposition
• Active Learning in Robotics: A Review of Control Principles
• Delay-coordinate maps and the spectra of Koopman operators
• Data-driven Experimental Modal Analysis by Dynamic Mode Decomposition
• Data-driven cardiovascular flow modeling: examples and opportunities
• Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
• Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques
• 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
• Kernel-based approximation of the Koopman generator and Schrödinger operator
• Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspaces
• Koopman analysis of the long-term evolution in a turbulent convection cell
• On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems
• Koopman spectra in reproducing kernel Hilbert spaces
• Reproducing kernel Hilbert space compactification of unitary evolution groups
• Optimized Sampling for Multiscale Dynamics
• Koopman mode expansions between simple invariant solutions
• A Geometric Approach to Dynamical Model-Order Reduction
• Data-driven modeling of the chaotic thermal convection in an annular thermosyphon
• Deep Learning Enhanced Dynamic Mode Decomposition
• Enhanced Sampling in the Age of Machine Learning: Algorithms and Applications
• Spectral analysis of climate dynamics with operator-theoretic approaches
• Deep Learning Models for Global Coordinate Transformations that Linearize PDEs
• Koopman-based feedback design with stability guarantees
• Data-Driven Participation Factors for Nonlinear Systems Based on Koopman Mode Decomposition
• Koopman Operator and its Approximations for Systems with Symmetries
• Nonlinear Model Reduction to Fractional and Mixed-Mode Spectral Submanifolds
• Searching turbulence for periodic orbits with dynamic mode decomposition
• Crisis of the Chaotic Attractor of a Climate Model: A Transfer Operator Approach
• Data-adaptive harmonic spectra and multilayer Stuart-Landau models
• Robust data-driven control for nonlinear systems using the Koopman operator
• Theoretical tools for understanding the climate crisis from Hasselmann's program and beyond
• 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
• Koopman analysis of Burgers equation
• Are extreme dissipation events predictable in turbulent fluid flows?
• Efficient space-time reduced order model for linear dynamical systems in Python using less than 120 lines of code
• Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables
• SE(3) Koopman-MPC: Data-driven Learning and Control of Quadrotor UAVs
• On the Koopman operator of algorithms
• Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems
• On analytical construction of observable functions in extended dynamic mode decomposition for nonlinear estimation and prediction
• Optimal data-driven estimation of generalized Markov state models for non-equilibrium dynamics
• Detecting the maximum likelihood transition path from data of stochastic dynamic systems
• Resonances in a Chaotic Attractor Crisis of the Lorenz Flow
• Reduced-order Koopman modeling and predictive control of nonlinear processes
• Vehicular Applications of Koopman Operator Theory -- A Survey
• Discovering transition phenomena from data of stochastic dynamical systems with Lévy noise
• Scalable Extended Dynamic Mode Decomposition using Random Kernel Approximation
• Variational approach to closure of nonlinear dynamical systems: Autonomous case
• Hypergraph reconstruction from dynamics
• Time Series Forecasting Using Manifold Learning
• Identification of MIMO Wiener-type Koopman Models for Data-Driven Model Reduction using Deep Learning
• Extended Dynamic Mode Decomposition for Inhomogeneous Problems
• On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis
• Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems
• Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach
• The Linear-Time-Invariance Notion of the Koopman Analysis-Part 1: The Architecture, Practical Rendering on the Prism Wake, and Fluid-Structure Association
• Approximate Analytical Solution to the Zonal Harmonics Problem Using Koopman Operator Theory
• A kernel-based approach to molecular conformation analysis
• Discovering Conservation Laws using Optimal Transport and Manifold Learning
• Koopman operator learning using invertible neural networks
• Kernel methods for detecting coherent structures in dynamical data
• Deep Koopman Learning of Nonlinear Time-Varying Systems
• Predicting rare events using neural networks and short-trajectory data
• Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator
• Propagating Parameter Uncertainty in Power System Nonlinear Dynamic Simulations Using a Koopman Operator-Based Surrogate Model
• Informative and non-informative decomposition of turbulent flow fields
• Learning the temporal evolution of multivariate densities via normalizing flows
• 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 overview of Koopman-based control: From error bounds to closed-loop guarantees
• Dynamics of random pressure fields over bluff bodies: a dynamic mode decomposition perspective
• Symbolic Regression via Neural Networks
• Dynamic mode decomposition for analytic maps
• Anti-circulant dynamic mode decomposition with sparsity-promoting for highway traffic dynamics analysis
• Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds
• Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator
• 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
• Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain
• A Koopman framework for rare event simulation in stochastic differential equations
• The Dynamic-Mode Decomposition and Optimal Prediction
• Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks
• DDK: A Deep Koopman Approach for Dynamics Modeling and Trajectory Tracking of Autonomous Vehicles
• Efficient Approximation of Molecular Kinetics using Random Fourier Features
• Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas
• Deeptime: a Python library for machine learning dynamical models from time series data
• Koopman analysis of quantum systems
• Probabilistic forecast of nonlinear dynamical systems with uncertainty quantification
• Data-driven Nonlinear Model Reduction using Koopman Theory: Integrated Control Form and NMPC Case Study
• Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data
• Feedback control of nonlinear PDEs using data-efficient reduced order models based on the Koopman operator
• Manifold Learning with Contracting Observers for Data-driven Time-series Analysis
• Koopman and Perron-Frobenius Operators on reproducing kernel Banach spaces
• Two methods to approximate the Koopman operator with a reservoir computer
• An Efficient Forecasting Approach to Reduce Boundary Effects in Real-Time Time-Frequency Analysis
• Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization
• CD-ROM: Complemented Deep-Reduced Order Model
• Detecting regime transitions in time series using dynamic mode decomposition
• Deep Identification of Nonlinear Systems in Koopman Form
• Sparse Convolution-based Markov Models for Nonlinear Fluid Flows
• Data Driven Learning of Mori-Zwanzig Operators for Isotropic Turbulence
• Eigenvalues of Autocovariance Matrix: A Practical Method to Identify the Koopman Eigenfrequencies
• Combining Dynamic Mode Decomposition with Ensemble Kalman Filtering for Tracking and Forecasting
• Stochastic Deep Koopman Model for Quality Propagation Analysis in Multistage Manufacturing Systems
• Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning
• Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning
• Density matrix formulation of dynamical systems
• Consistent spectral approximation of Koopman operators using resolvent compactification
• Quantum mechanics and data assimilation
• Data-driven model reduction of agent-based systems using the Koopman generator
• Simultaneous Coherent Structure Coloring facilitates interpretable clustering of scientific data by amplifying dissimilarity
• System Norm Regularization Methods for Koopman Operator Approximation
• Koopman-based spectral clustering of directed and time-evolving graphs
• Data-Driven Optimal Control of Tethered Space Robot Deployment with Learning Based Koopman Operator
• Extracting Non-Gaussian Governing Laws from Data on Mean Exit Time
• HAVOK Model Predictive Control for Time-Delay Systems with Applications to District Heating
• A Tailored Convolutional Neural Network for Nonlinear Manifold Learning of Computational Physics Data using Unstructured Spatial Discretizations
• Data-driven model construction for anisotropic dynamics of active matter
• Self-tuning moving horizon estimation of nonlinear systems via physics-informed machine learning Koopman modeling
• SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systems
• Spectral Discovery of Jointly Smooth Features for Multimodal Data
• tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format
• Kernel-based error bounds of bilinear Koopman surrogate models for nonlinear data-driven control
• Data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media
• Numerical convergence of the Lyapunov spectrum computed using low Mach number solvers
• Adaptive Koopman Embedding for Robust Control of Complex Nonlinear Dynamical Systems
• Data-Driven Inference of High-Accuracy Isostable-Based Dynamical Models in Response to External Inputs
• Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator
• Koopman Operator Dynamical Models: Learning, Analysis and Control
• Dynamical System Segmentation for Information Measures in Motion
• Data-Driven Linear Koopman Embedding for Networked Systems: Model-Predictive Grid Control
• Identifying efficient ensemble perturbations for initializing subseasonal-to-seasonal prediction
• Reduced-order modeling of fully turbulent buoyancy-driven flows using the Green's function method
• Data-Driven Koopman Controller Synthesis Based on the Extended ${\mathcal{H}}_2$ Norm Characterization
• Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems
• Data-Driven Models for Control Engineering Applications Using the Koopman Operator
• Data-driven MPC with terminal conditions in the Koopman framework
• On the Universal Transformation of Data-Driven Models to Control Systems
• Manifold Learning for Organizing Unstructured Sets of Process Observations
• A case study of spatiotemporal forecasting techniques for weather forecasting
• Error bounds for dynamical spectral estimation
• Koopman interpretation and analysis of a public-key cryptosystem: Diffie-Hellman key exchange
• Gaussian Process Koopman Mode Decomposition
• Koopman analysis in oscillator synchronization
• Sampling and Inference of Networked Dynamics using Log-Koopman Nonlinear Graph Fourier Transform
• Navigation with Probabilistic Safety Constraints: Convex Formulation
• Extended dynamic mode decomposition with dictionary learning using neural ordinary differential equations
• CKNet: A Convolutional Neural Network Based on Koopman Operator for Modeling Latent Dynamics from Pixels
• Generic Properties of Koopman Eigenfunctions for Stable Fixed Points and Periodic Orbits
• Decrypting Nonlinearity: Koopman Interpretation and Analysis of Cryptosystems
• Data-driven kinematics-consistent model order reduction of fluid-structure interaction problems: application to deformable microcapsules in a Stokes flow
• High-Fidelity Digital Twin Data Models by Randomized Dynamic Mode Decomposition and Deep Learning with Applications in Fluid Dynamics
• On Koopman Mode Decomposition and Tensor Component Analysis
• Dynamical systems and complex networks: A Koopman operator perspective
• Synthesizing Control Laws from Data using Sum-of-Squares Optimization
• A Reduced-Order Model for Nonlinear Radiative Transfer Problems Based on Moment Equations and POD-Petrov-Galerkin Projection of the Normalized Boltzmann Transport Equation
• Compressing and forecasting atomic material simulations with descriptors
• Mitigating Traffic Oscillations in Mixed Traffic Flow with Scalable Deep Koopman Predictive Control
• Memory-based reduced modelling and data-based estimation of opinion spreading
• Koopman-based control of nonlinear systems with closed-loop guarantees
• Approximating matrix eigenvalues by subspace iteration with repeated random sparsification
• Renormalization Group as a Koopman Operator
• Closed-Loop Koopman Operator Approximation
• Phase autoencoder for rapid data-driven synchronization of rhythmic spatiotemporal patterns
• Setting of the Poincaré section for accurately calculating the phase of rhythmic spatiotemporal dynamics
• Kernel-based parameter estimation of dynamical systems with unknown observation functions
• Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction
• Finite Dimensional Koopman Form of Polynomial Nonlinear Systems
• Detection of Functional Communities in Networks of Randomly Coupled Oscillators Using the Dynamic-Mode Decomposition
• Mean resolvent operator of statistically steady flows
• Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces
• Gaussian Process Phase Interpolation for estimating the asymptotic phase of a limit cycle oscillator from time series data
• End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data
• Koopman-based control using sum-of-squares optimization: Improved stability guarantees and data efficiency
• The spatiotemporal coupling in delay-coordinates dynamic mode decomposition
• A Supervised Machine-Learning Approach For Turboshaft Engine Dynamic Modeling Under Real Flight Conditions
• Data-driven discovery of quasiperiodically driven dynamics
• Sub-diffusive behavior in the Standard Map
• Physics-informed Split Koopman Operators for Data-efficient Soft Robotic Simulation
• Koopman Learning with Episodic Memory
• Numerical methods to evaluate Koopman matrix from system equations
• Stable Invariant Models via Koopman Spectra
• Online learning of Koopman operator using streaming data from different dynamical regimes
• Learned Lifted Linearization Applied to Unstable Dynamic Systems Enabled by Koopman Direct Encoding
• Machine Learning in Viscoelastic Fluids via Energy-Based Kernel Embedding
• Data-Augmented Predictive Deep Neural Network: Enhancing the extrapolation capabilities of non-intrusive surrogate models
• Extraction of nonlinearity in neural networks with Koopman operator
• Data-driven Discovery of Invariant Measures
• Star Log-extended eMulation: a method for efficient computation of the Tolman-Oppenheimer-Volkoff equations
• Group-Convolutional Extended Dynamic Mode Decomposition
• Connection among stochastic Hamilton-Jacobi-Bellman equation, path-integral, and Koopman operator on nonlinear stochastic optimal control
• Transformer-based Koopman Autoencoder for Linearizing Fisher's Equation
• Assessment of End-to-End and Sequential Data-driven Learning of Fluid Flows
• Non-intrusive model combination for learning dynamical systems
• Total-Variation -- Fast Gradient Flow and Relations to Koopman Theory
• Data-driven system identification using quadratic embeddings of nonlinear dynamics
• Identifying Large-Scale Linear Parameter Varying Systems with Dynamic Mode Decomposition Methods
• Learning Data-Driven PCHD Models for Control Engineering Applications
• Data-driven identification of the spatio-temporal structure of turbulent flows by streaming Dynamic Mode Decomposition
• Dynamic Mode Decomposition of High Reynolds Number Supersonic Jet Flows
• A mode-in-state contribution factor based on Koopman operator and its application to power system analysis
• From reductionism to realism: Holistic mathematical modelling for complex biological systems
• Sketching Methods for Dynamic Mode Decomposition in Spherical Shallow Water Equations
• Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems
• A tutorial overview of model predictive control for continuous crystallization: current possibilities and future perspectives
• Adaptive Koopman-Based Models for Holistic Controller and Observer Design
• A General Framework for Linking Free and Forced Fluctuations via Koopmanism
• Bridging the Gap between Koopmanism and Response Theory: Using Natural Variability to Predict Forced Response
• A multichannel generalization of the HAVOK method for the analysis of nonlinear dynamical systems
• Learning dynamical systems from data: Gradient-based dictionary optimization
• Model predictive traction control system based on the Koopman operator
• Definition and data-driven reconstruction of asymptotic phase and amplitudes of stochastic oscillators via Koopman operator theory
• Improvement of system identification of stochastic systems via Koopman generator and locally weighted expectation
• Optimal Synthesis of LTI Koopman Models for Nonlinear Systems with Inputs
• Statistics for stochastic differential equations and approximations of resolvent
• Learning to Optimize with Dynamic Mode Decomposition
• Control of Oscillatory Temperature Field in a Building via Damping Assignment to Nonlinear Koopman Mode
• Kolmogorov Modes and Linear Response of Jump-Diffusion Models
• An MCMC Method for Uncertainty Set Generation via Operator-Theoretic Metrics
• Koopman analysis of combinatorial optimization problems with replica exchange Monte Carlo method
• Reconstruction of phase-amplitude dynamics from electrophysiological signals
• Signature of glassy dynamics in dynamic modes decompositions
• System stabilization with policy optimization on unstable latent manifolds
• $Q$-functions, synchronization, and Arnold tongues for coupled stochastic oscillators
• Participation Factors for Nonlinear Autonomous Dynamical Systems in the Koopman Operator Framework
• Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces
• Robust Control Design and Analysis Based on Lifting Linearization of Nonlinear Systems Under Uncertain Initial Conditions
• Statistical analysis and method to quantify the impact of measurement uncertainty on dynamic mode decomposition
• Data-driven network analysis using local delay embeddings
• Efficient pseudometrics for data-driven comparisons of nonlinear dynamical systems
• Digital Twins Meet the Koopman Operator: Data-Driven Learning for Robust Autonomy
• Atoms to Processes: The Role of Artificial Intelligence and Machine Learning in Chemical Engineering
• Koopman Resolvent: A Laplace-Domain Analysis of Nonlinear Autonomous Dynamical Systems
• Improving Estimation of the Koopman Operator with Kolmogorov-Smirnov Indicator Functions
• Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?
• Data-Assisted Control -- A Framework Development by Exploiting NASA GTM Platform
• Fast Subspace Fluid Simulation with a Temporally-Aware Basis
• SVD-based Causal Emergence for Gaussian Iterative Systems
• Data-Driven Contact-Aware Control Method for Real-Time Deformable Tool Manipulation: A Case Study in the Environmental Swabbing
• Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine
• Koopman operator-based discussion on partial observation in stochastic systems
• Dynamic mode decomposition for detecting oscillatory transient activity via sparsity and smoothness regularization
• Distributed Koopman Operator Learning from Sequential Observations
• Cause-and-effect approach to turbulence forecasting
• Deep Koopman Economic Model Predictive Control of a Pasteurisation Unit
• Consistent Projection of Langevin Dynamics: Preserving Thermodynamics and Kinetics in Coarse-Grained Models
• Reduced-Order Surrogates for Forced Flexible Mesh Coastal-Ocean Models
• Integrated utilization of equations and small dataset in the Koopman operator: applications to forward and inverse problems