Machine Learning and Dynamical Systems Meet in Reproducing Kernel Hilbert Spaces
Boumediene Hamzi
Department of Mathematics, Imperial College London

Since its inception in the 19th century through the efforts of Poincaré and Lyapunov, the theory ofdynamical systems addresses the qualitative behaviour of dynamical systems as understood frommodels. From this perspective, the modeling of dynamical processes in applications requires adetailed understanding of the processes to be analyzed. This deep understanding leads to a model,
which is an approximation of the observed reality and is often expressed by a system ofOrdinary/Partial, Underdetermined (Control), Deterministic/Stochastic differential or difference
equations. While models are very precise for many processes, for some of the most challenging
applications of dynamical systems (such as climate dynamics, brain dynamics, biological systems
or the financial markets), the development of such models is notably difficult. On the other hand,the field of machine learning is concerned with algorithms designed to accomplish a certain task,
whose performance improves with the input of more data. Applications for machine learning
methods include computer vision, stock market analysis, speech recognition, recommendersystems and sentiment analysis in social media. The machine learning approach is invaluable insettings where no explicit model is formulated, but measurement data is available. This isfrequently the case in many systems of interest, and the development of data-driven technologies
is becoming increasingly important in many applications. The intersection of the fields ofdynamical systems and machine learning is largely unexplored and the objective of this talk is toshow that working in reproducing kernel Hilbert spaces offers tools for a data-based theory ofnonlinear dynamical systems. In this talk, we introduce a data-based approach to estimating key
quantities which arise in the study of nonlinear autonomous, control and random dynamicalsystems. Our approach hinges on the observation that much of the existing linear theory may bereadily extended to nonlinear systems - with a reasonable expectation of success- once thenonlinear system has been mapped into a high or infinite dimensional Reproducing Kernel Hilbert
Space. In particular, we develop computable, non-parametric estimators approximatingcontrollability and observability energies for nonlinear systems. We apply this approach to theproblem of model reduction of nonlinear control systems. It is also shown that the controllabilityenergy estimator provides a key means for approximating the invariant measure of an ergodic,
stochastically forced nonlinear system. We also show how kernel methods can be used to detect
critical transitions for some multi scale dynamical systems. We also use the method of kernel flows
to predict some chaotic dynamical systems. Finally, we show how kernel methods can be used toapproximate center manifolds, propose a data-based version of the centre manifold theorem andconstruct Lyapunov functions for nonlinear ODEs. This is joint work with Jake Bouvrie (MIT,USA), Peter Giesl (University of Sussex, UK), Christian Kuehn (TUM, Munich/Germany), Romit
Malik (ANNL), Sameh Mohamed (SUTD, Singapore), Houman Owhadi (Caltech), Martin
Rasmussen (Imperial College London), Kevin Webster (Imperial College London), Bernard
Hasasdonk, Gabriele Santin and Dominik Wittwar (University of Stuttgart).