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Approximation of the Koopman operator via Bernstein polynomials

Rishikesh Yadav, Alexandre Mauroy

TL;DR

This work introduces a Bernstein-polynomial-based finite-dimensional approximation of the Koopman operator, providing explicit uniform-norm error bounds expressed via the modulus of continuity and Lipschitz constants. By forming a matrix representation in the Bernstein basis, the approach yields a robust, noise-tolerant method for trajectory prediction and enables a data-driven extension through a coordinate change that maps irregular data to a regular lattice. The paper establishes convergence rates in one and multiple dimensions, with improved rates under $C^1$ regularity, and compares the Bernstein method to EDMD, highlighting the avoidance of the pseudoinverse and favorable noise robustness. The data-driven variant further broadens applicability to real-world datasets, while the conclusions point to future improvements via higher-order derivatives, iterated Bernstein operators, and alternative approximation operators. Overall, this framework offers a rigorous, scalable alternative for finite-dimensional Koopman approximations with explicit error control and practical trajectory-prediction capabilities.

Abstract

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.

Approximation of the Koopman operator via Bernstein polynomials

TL;DR

This work introduces a Bernstein-polynomial-based finite-dimensional approximation of the Koopman operator, providing explicit uniform-norm error bounds expressed via the modulus of continuity and Lipschitz constants. By forming a matrix representation in the Bernstein basis, the approach yields a robust, noise-tolerant method for trajectory prediction and enables a data-driven extension through a coordinate change that maps irregular data to a regular lattice. The paper establishes convergence rates in one and multiple dimensions, with improved rates under regularity, and compares the Bernstein method to EDMD, highlighting the avoidance of the pseudoinverse and favorable noise robustness. The data-driven variant further broadens applicability to real-world datasets, while the conclusions point to future improvements via higher-order derivatives, iterated Bernstein operators, and alternative approximation operators. Overall, this framework offers a rigorous, scalable alternative for finite-dimensional Koopman approximations with explicit error control and practical trajectory-prediction capabilities.

Abstract

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.
Paper Structure (20 sections, 13 theorems, 122 equations, 4 figures, 2 tables)

This paper contains 20 sections, 13 theorems, 122 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bernstein polynomials and Bernstein operator
  4. Properties.
  5. Koopman operator framework
  6. Modulus of continuity and Lipschitz continuity
  7. Modulus of continuity
  8. Lipschitz continuity
  9. Matrix representation of the Koopman operator
  10. Bounds on the approximation error
  11. One-dimensional case
  12. Approximation of the Koopman operator in the multivariate case
  13. Error propagation under the iteration of the Koopman operator approximation
  14. A first step towards the data-driven setting
  15. Bernstein approximation of the Koopman operator over a set of random points
  16. ...and 5 more sections

Key Result

Lemma 2.1

For all $f\in C([0,1]^m)$, with $u_l,x_l\in[0,1]$, $l=1,2,\dots,m.$

Figures (4)

  • Figure 1: Predicted trajectories of the Van der Pol system obtained with the Bernstein approximation of the Koopman operator, for different degrees of the polynomials.
  • Figure 2: Error bounds obtained with Theorem \ref{['Th1']} and Theorem \ref{['Th2']}, for the flow map considered in Example \ref{['ex:uni']}.
  • Figure 3: Error bounds computed with Theorem \ref{['theo_multi']} (full modulus of continuity), Theorem \ref{['theo_multi_2']} (partial modulus of continuity), and Theorem \ref{['theo_multi_3']} for the flow map considered in Example \ref{['ex:multi']}.
  • Figure 4: For the competitive Lotka-Volterra system, Bernstein and EDMD approximations of the Koopman operator are computed from the values of the flow at the data points (panel (a)). Predicted trajectories are obtained by iterating the approximation matrices (panel (b)). The Bernstein approximation yields an accurate prediction, even in the case of measurement noise (dashed curves).

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.2
  • proof
  • ...and 28 more