Approximation of the Koopman operator via Bernstein polynomials
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916634372866048 |
|---|---|
| author | Yadav, Rishikesh Mauroy, Alexandre |
| author_facet | Yadav, Rishikesh Mauroy, Alexandre |
| contents | The Koopman operator approach provides a powerful linear description of nonlinear dynamical systems in terms of the evolution of observables. While the operator is typically infinite-dimensional, it is crucial to develop finite-dimensional approximation methods and characterize the related approximation errors with upper bounds, preferably expressed in the uniform norm. In this paper, we depart from the traditional use of orthogonal projection or truncation, and propose a novel method based on Bernstein polynomial approximation. Considering a basis of Bernstein polynomials, we construct a matrix approximation of the Koopman operator in a computationally effective way. Building on results of approximation theory, we characterize the rates of convergence and the upper bounds of the error in various contexts including the cases of univariate and multivariate systems, and continuous and differentiable observables. The obtained bounds are expressed in the uniform norm in terms of the modulus of continuity of the observables. Finally, the method is extended to a data-driven setting through a proper change of coordinates. Numerical experiments show that the method is robust to noise and demonstrates good performance for trajectory prediction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02438 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of the Koopman operator via Bernstein polynomials Yadav, Rishikesh Mauroy, Alexandre Dynamical Systems 37L65, 37M15, 41A36, 41A25, 47B33 The Koopman operator approach provides a powerful linear description of nonlinear dynamical systems in terms of the evolution of observables. While the operator is typically infinite-dimensional, it is crucial to develop finite-dimensional approximation methods and characterize the related approximation errors with upper bounds, preferably expressed in the uniform norm. In this paper, we depart from the traditional use of orthogonal projection or truncation, and propose a novel method based on Bernstein polynomial approximation. Considering a basis of Bernstein polynomials, we construct a matrix approximation of the Koopman operator in a computationally effective way. Building on results of approximation theory, we characterize the rates of convergence and the upper bounds of the error in various contexts including the cases of univariate and multivariate systems, and continuous and differentiable observables. The obtained bounds are expressed in the uniform norm in terms of the modulus of continuity of the observables. Finally, the method is extended to a data-driven setting through a proper change of coordinates. Numerical experiments show that the method is robust to noise and demonstrates good performance for trajectory prediction. |
| title | Approximation of the Koopman operator via Bernstein polynomials |
| topic | Dynamical Systems 37L65, 37M15, 41A36, 41A25, 47B33 |
| url | https://arxiv.org/abs/2403.02438 |