Related papers: Tensor network approximation of Koopman operators
This paper presents a novel Koopman composition operator representation framework for control systems in reproducing kernel Hilbert spaces (RKHSs) that is free of explicit dictionary or input parametrizations. By establishing fundamental…
Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained…
Nonlinearity plays a crucial role in deep neural networks. In this paper, we investigate the degree to which the nonlinearity of the neural network is essential. For this purpose, we employ the Koopman operator, extended dynamic mode…
The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in…
We present a tensor train (TT) based algorithm designed for sampling from a target distribution and employ TT approximation to capture the high-dimensional probability density evolution of overdamped Langevin dynamics. This involves…
The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct…
We propose Koopman Spectral Wasserstein Gradient Descent (KSWGD), a particle-based generative modeling framework that learns the Langevin generator via Koopman theory and integrates it with Wasserstein gradient descent. Our key insight is…
Nonlinear dynamical systems are ubiquitous in science and engineering, yet analysis and prediction of these systems remains a challenge. Koopman operator theory circumvents some of these issues by considering the dynamics in the space of…
Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning…
This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural-network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a…
Koopman and Perron-Frobenius operators for dynamical systems have been getting popular in a number of fields in science these days. Properties of the Koopman operator essentially depend on the choice of function spaces where it acts.…
We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space…
This paper explores a simple question: can we model the internal transformations of a neural network using dynamical systems theory? We introduce Koopman autoencoders to capture how neural representations evolve through network layers,…
In this paper we prove new connections between two frameworks for analysis and control of nonlinear systems: the Koopman operator framework and contraction analysis. Each method, in different ways, provides exact and global analyses of…
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…
In this paper, we propose a Koopman operator based approach to describe the nonlinear dynamics of a quadrotor on SE(3) in terms of an infinite-dimensional linear system which evolves in the space of observable functions (lifted space) and…
We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In…
The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are…
This work explores the relationship between state space methods and Koopman operator-based methods for predicting the time-evolution of nonlinear dynamical systems. We demonstrate that extended dynamic mode decomposition with dictionary…
Koopman operator theory, a powerful framework for discovering the underlying dynamics of nonlinear dynamical systems, was recently shown to be intimately connected with neural network training. In this work, we take the first steps in…