Related papers: Sharp Spectral Rates for Koopman Operator Learning
The Koopman operator provides a powerful framework for representing the dynamics of general nonlinear dynamical systems. Data-driven techniques to learn the Koopman operator typically assume that the chosen function space is closed under…
Learning tractable linear representations of nonlinear dynamical systems via Koopman operator theory is often hindered by dictionary selection, temporal memory encoding, and numerical ill-conditioning. Inspired by Reservoir Computing (RC)…
The Koopman operator and extended dynamic mode decomposition (EDMD) as a data-driven technique for its approximation have attracted considerable attention as a key tool for modeling, analysis, and control of complex dynamical systems.…
In this article, we propose a new error bound for Koopman operator approximation using Kernel Extended Dynamic Mode Decomposition. The new estimate is $O(N^{-1/2})$, with a constant related to the probability of success of the bound, given…
We provide an overview of the Koopman operator analysis for a class of partial differential equations describing relaxation of the field variable to a stable stationary state. We introduce Koopman eigenfunctionals of the system and use the…
A dynamical system is a transformation of a phase space, and the transformation law is the primary means of defining as well as identifying the dynamical system. It is the object of focus of many learning techniques. Yet there are many…
The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches…
Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the $L^2$ space of observables associated with the invariant measure. Koopman eigenfunctions represent…
Asymptotic phase and amplitudes are fundamental concepts in the analysis of limit-cycle oscillators. In this paper, we briefly review the definition of these quantities, particularly a generalization to stochastic oscillatory systems from…
We present a novel data-driven approach for learning linear representations of a class of stable nonlinear systems using Koopman eigenfunctions. By learning the conjugacy map between a nonlinear system and its Jacobian linearization through…
The mathematical properties and data-driven learning of the Koopman operator, which represents nonlinear dynamics as a linear mapping on a properly defined functional spaces, have become key problems in nonlinear system identification and…
We present a low-rank Koopman operator formulation for accelerating deformable subspace simulation. Using a Dynamic Mode Decomposition (DMD) parameterization of the Koopman operator, our method learns the temporal evolution of deformable…
Extended dynamic mode decomposition (EDMD) is a data-driven algorithm for approximating spectral data of the Koopman operator associated to a dynamical system, combining a Galerkin method of order N and collocation method of order M.…
The Koopman operator framework can be used to identify a data-driven model of a nonlinear system. Unfortunately, when the data is corrupted by noise, the identified model can be biased. Additionally, depending on the choice of lifting…
A learning method is proposed for Koopman operator-based models with the goal of improving closed-loop control behavior. A neural network-based approach is used to discover a space of observables in which nonlinear dynamics is linearly…
The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with…
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…
An outstanding challenge in nonlinear systems theory is identification or learning of a given nonlinear system's Koopman operator directly from data or models. Advances in extended dynamic mode decomposition approaches and machine learning…
This paper proposes an original methodology to compute the regions of attraction in hyperbolic and polynomial nonlinear dynamical systems using the eigenfunctions of the discrete-time approximation of the Koopman operator given by the…
We develop a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman and Perron-Frobenius groups of…