[Turkmath:6708] RKHS Seminars will continue on November 12th, 2024! Online on Zoom!

[Omur Ugur]

Dear Friends,

We continue with the RKHS Seminars, now with an **expert** in the field, 
on **Tuesday, 12th of November** at 14:00 (Ankara, Turkey)!

The RKHS-Seminar Website is on https://iam.metu.edu.tr/rkhs-seminars. 
Please, do not miss the scheduled talk below (with a relatively 
extensive abstract)!

Again, for a quick reminder, the event's page (Machine Learning and 
Dynamical Systems meet in Reproducing Kernel Hilbert Spaces) is on:

- 
https://ougur.iam.metu.edu.tr/rkhs-seminars/2024/10/21/machine-learning-and-dynamical-systems-meet-in-rkhs.html

Best wishes,
Omur

### RKHS-Seminars

Title: Machine Learning and Dynamical Systems meet in Reproducing Kernel 
Hilbert Spaces

Speaker: Boumediene Hamzi (Dr., Department of Computing and Mathematical 
Sciences, Caltech)

Date / Time: Tuesday, November 12, 2024 / 14:00 (Ankara, Turkey)

Online on Zoom: https://turing-uk.zoom.us/j/5141752978

Abstract: Since its inception in the 19th century through the efforts of 
Poincare and Lyapunov, the theory of dynamical systems addresses the 
qualitative behavior of dynamical systems as understood from models. 
 From this perspective, the modeling of dynamical processes in 
applications requires a detailed understanding of the processes to be 
analyzed. This deep understanding leads to a model, which approximates 
the observed reality and is often expressed by a system of 
Ordinary/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, recommender systems and sentiment analysis in social 
media. The machine learning approach is invaluable in settings where no 
explicit model is formulated, but measurement data is available. This is 
frequently 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 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.

In the first part of the talk, we introduce simple methods to learn 
surrogate models for complex systems. We present variants of the method 
of Kernel Flows as simple approaches for learning the kernel that appear 
in the emulators we use in our work. First, we will talk about the 
method of parametric and nonparametric kernel flows for learning chaotic 
dynamical systems. We’ll also talk about learning dynamical systems from 
irregularly sampled time series as well as from partial observations. We 
will also introduce the methods of Sparse Kernel Flows and 
Hausdorff-metric based Kernel Flows (HMKFs) and apply them to learn 132 
chaotic dynamical systems. Finally, we extend the method of Kernel Mode 
Decomposition to design kernels in view of detecting critical 
transitions in some fast-slow random dynamical systems.

Then, we 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. 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. Finally, 
we 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.

#### Biography
Boumediene Hamzi is currently a Senior Scientist at the Department of 
Computing and Mathematical Sciences, Caltech. He is also co-leading the 
Research Interest Group on Machine Learning and Dynamical Systems at the 
Alan Turing Institute. Broadly speaking, his research is at the 
interface of Machine Learning and Dynamical Systems.


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Dr. Omur Ugur                  | Middle East Technical University
http://users.metu.edu.tr/ougur | Institute of Applied Mathematics
Tel.: +90(312) 210 29 87       |             06800 Ankara, Turkey
Fax : +90(312) 210 29 85       |
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