Boumediene Hamzi (Johns Hopkins University)
Title: Machine Learning and dynamical systems meet in reproducing kernel Hilbert spaces
Abstract:The intersection of the fields of dynamical systems and machine learning is largely unexplored and the objective of this talk is to show that working in reproducing kernel Hilbert spaces offers tools for a data-based theory of nonlinear dynamical systems. We use the method of parametric and nonparametric kernel flows to predict some prototypical chaotic dynamical systems as well as geophysical observational data.
We also consider microlocal kernel design for detecting critical transitions in some fast-slow random dynamical systems. We then show how kernel methods can be used to approximate center manifolds, propose a data-based version of the center manifold theorem and construct Lyapunov functions for nonlinear ODEs.
We also introduce a data-based approach to estimating key quantities which arise in the study of nonlinear autonomous, control and random 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 Reproducing Kernel Hilbert Space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energies for nonlinear systems. We apply this approach to the problem of model reduction of nonlinear control systems. It is also shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.