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This paper tackles the data-driven approximation of unknown dynamical systems using Koopman-operator methods. Given a dictionary of functions, these methods approximate the projection of the action of the operator on the finite-dimensional…
Nonlinear phenomena can be analyzed via linear techniques using operator-theoretic approaches. Data-driven method called the extended dynamic mode decomposition (EDMD) and its variants, which approximate the Koopman operator associated with…
Dynamic Mode Decomposition (DMD) is a technique to approximate generally non-linear dynamical systems using linear techniques, which are better understood and easier to analyze. Koopman theory extends DMD by transforming the original system…
The Koopman operator is a mathematical tool that allows for a linear description of non-linear systems, but working in infinite dimensional spaces. Dynamic Mode Decomposition and Extended Dynamic Mode Decomposition are amongst the most…
Extracting the latent underlying structures of complex nonlinear local and nonlocal flows is essential for their analysis and modeling. In this work, we attempt to provide a consistent framework through Koopman theory and its related…
A Koopman decomposition of a complex system leads to a representation in which nonlinear dynamics appear to be linear. The existence of a linear framework with which to analyse nonlinear dynamical systems brings new strategies for…
The Koopman operator is a linear operator that describes the evolution of scalar observables (i.e., measurement functions of the states) in an infinitedimensional Hilbert space. This operator theoretic point of view lifts the dynamics of a…
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…
Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in…
Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear,…
Koopman analysis provides a general framework from which to analyze a nonlinear dynamical system in terms of a linear operator acting on an infinite-dimensional observable space. This theoretical framework provides a rigorous underpinning…
Koopman operator theory shows how nonlinear dynamical systems can be represented as an infinite-dimensional, linear operator acting on a Hilbert space of observables of the system. However, determining the relevant modes and eigenvalues of…
Discovery of mathematical descriptors of physical phenomena from observational and simulated data, as opposed to from the first principles, is a rapidly evolving research area. Two factors, time-dependence of the inputs and hidden…
We present an approach to construct approximate Koopman-type decompositions for dynamical systems depending on static or time-varying parameters. Our method simultaneously constructs an invariant subspace and a parametric family of…
The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode…
Extended dynamic mode decomposition (EDMD) is a powerful tool to construct linear predictors of nonlinear dynamical systems by approximating the action of the Koopman operator on a subspace spanned by finitely many observable functions.…
Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of…
The long- and short-time behavior of solutions to dissipative evolution equations is studied by applying the concept of hypocoercivity. Aiming at partial differential equations that allow for a modal decomposition, we compute estimates that…
The localization transition and the critical properties of the Lorentz model in three dimensions are investigated by computer simulations. We give a coherent and quantitative explanation of the dynamics in terms of continuum percolation…
Dynamic mode decomposition (DMD) gives a practical means of extracting dynamic information from data, in the form of spatial modes and their associated frequencies and growth/decay rates. DMD can be considered as a numerical approximation…