[Turkmath:5945] (no subject)

Fri Jan 13 15:05:14 UTC 2023

Dear Colleagues!

You are cordially invited to the  Weekly Online  Seminar “Analysis and
Applied Mathematics” on

Date: Tuesday, January 17, 2023

  Time: 14.00-15.00 (Istanbul) = 12.00-13.00 (Ghent) = 17.00-18.00
(Almaty)

Zoom link:
https://us02web.zoom.us/j/6678270445?pwd=SFNmQUIvT0tRaH-lDaVYrN3l5bzJVQT09,
Conference ID: 667 827 0445, Access code: 1

Speaker:  Prof. Dr. Boumediene Hamzi
Johns Hopkins University (USA) and Alan Turing Institute (UK)
  Title: Machine Learning and Dynamical Systems Meet in Reproducing Kernel
Hilbert Spaces
Abstract: Since its inception in the 19th century through the efforts of
Poincaré 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 is an approximation of 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. - 2 - In this talk, we use the method of parametric and
nonparametric kernel flows to predict some chaotic dynamical systems. When
trained on geophysical observational data, for example, the weekly averaged
global sea-surface temperature, considerable gains are also observed by the
proposed technique in comparison to classical partial differential
equation-based models in terms of forecast computational cost and accuracy.
When trained on publicly available reanalysis data for the daily
temperature of the North-American continent, we see significant
improvements over classical baselines such as climatology and
persistence-based forecast techniques. Although our experiments concern
specific examples, the proposed approach is general, and our results
support the viability of kernel methods (with learned kernels) for
interpretable and computationally efficient geophysical forecasting for a
large diversity of processes. 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. We also show how kernel methods can be used to detect
critical transitions for some multi scale dynamical systems. This is joint
work with Jake Bouvrie (MIT, USA), Matthieu Darcy (Caltech), Edward
DeBrouwer (KU Leuven), Peter Giesl (University of Sussex, UK), Christian
Kuehn (TUM, Munich/Germany), Jonghyeon Lee (Caltech), Romit Malik (ANNL),
Sameh Mohamed (SUTD, Singapore), Houman Owhadi (Caltech), Martin Rasmussen
(Imperial College London), Kevin Webster (Imperial College London), Bernard
Hasasdonk and Dominik Wittwar (University of Stuttgart), Gabriele Santin
(Fondazione Bruno Kessler).
Abstracts and forthcoming talks can be found on our webpage
https://sites.google.com/view/aam-seminars
With my best wishes
*Prof. Dr. Allaberen Ashyralyev *
*Department of Mathematics, Bahcesehir University, 34353, Istanbul, Turkey*
*  and **Near East University, Lefkoşa(Nicosia), Mersin 10 Turkey*
*Peoples' Friendship University of Russia (RUDN University),** Ul Miklukho
Maklaya 6, Moscow 117198, Russian Federation *
*Institute of Mathematics and Mathematical Modelling, 050010, Almaty,
Kazakhstan*
*e-mail: allaberen.ashyralyev at eng.bau.edu.tr
<allaberen.ashyralyev at neu.edu.tr> and **aallaberen at gmail.com
<aallaberen at gmail.com> *
 http://akademik.bahcesehir.edu.tr/web/allaberenashyralyev

https://sites.google.com/view/aam-seminars
https://content.sciendo.com/view/journals/ejaam/ejaam-overview.xml
*icaam-online.org <http://icaam-online.org>*

*https://www.genealogy.math.ndsu.nodak.edu/id.php?id=95872&fChrono=1
<https://www.genealogy.math.ndsu.nodak.edu/id.php?id=95872&fChrono=1>*
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