Kernel methods for the approximation of some key quantities of nonlinear systems

Jake Bouvrie; Boumediene Hamzi

doi:10.3934/jcd.2017001

Article Contents

2017, Volume 4: 1-19. Doi: 10.3934/jcd.2017001

This issue Previous Article Next Article Parameterization method for unstable manifolds of delay differential equations

Kernel methods for the approximation of some key quantities of nonlinear systems

Jake Bouvrie^1, and
Boumediene Hamzi^2, ,

1.
Laboratory for Computational and Statistical Learning, Massachusetts Institute of Technology, Cambridge, MA, USA
2.
Department of Mathematics, AlFaisal University, Riyadh, KSA

* Corresponding author:Boumediene Hamzi
* Corresponding author:Boumediene Hamzi

Published: March 2018

BH thanks the European Commission and the Scientific and the Technological Research Council of Turkey (Tubitak) for financial support received through a Marie Curie Fellowship, and JB gratefully acknowledges support under NSF contracts NSF-IIS-08-03293 and NSF-CCF-08-08847 to M. Maggioni.
Parts of this work were done while the authors were at the Department of Mathematics of Duke University and then while the second author was with the Departments of Mathematics of Imperial College London, Yildiz Technical University and Ko¸c University for Marie Curie Fellowships and at the Fields Institute.

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Abstract

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success -once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we embed a nonlinear system in a reproducing kernel Hilbert space where linear theory can be used to develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.

Keywords:

Nonlinear systems,
machine learning,
kernel methods,
gramians,
controllability energy,
observability energy,
stationary solution of the Fokker-Planck equation.

Mathematics Subject Classification: Primary: 37M99, 37H99, 65P99, 62-07, 60-08; Secondary: 93A99, 65C50, 65C60.

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Figure 1. Comparison between the exact (red), the kernel-based (green) and the empirical (black) steady-state distribution

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