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CATEGORIES:Academics,Lectures & conferences
DESCRIPTION:Speaker. Boumediene Hamzi\, Caltech.\n\nTitle. Machine Learnin
g and Dynamical Systems meet in Reproducing Kernel Hilbert Spaces\n\nAbstra
ct. Since its inception in the 19th century through the efforts of Poincare
and Lyapunov\, the theory of dynamical systems addresses the qualitative b
ehavior of dynamical systems as understood from models. From this perspecti
ve\, the modeling of dynamical processes in applications requires a detaile
d understanding of the processes to be analyzed. This deep understanding le
ads to a model\, which approximates the observed reality and is often expre
ssed by a system of Ordinary/Partial\, Underdetermined (Control)\, Determin
istic/Stochastic differential or difference equations. While models are ver
y precise for many processes\, for some of the most challenging application
s of dynamical systems (such as climate dynamics\, brain dynamics\, biologi
cal systems\, or the financial markets)\, the development of such models is
notably difficult. On the other hand\, the field of machine learning is co
ncerned with algorithms designed to accomplish a certain task\, whose perfo
rmance improves with the input of more data. Applications for machine learn
ing methods include computer vision\, stock market analysis\, speech recogn
ition\, recommender systems and sentiment analysis in social media. The mac
hine learning approach is invaluable in settings where no explicit model is
formulated\, but measurement data is available. This is frequently the cas
e in many systems of interest\, and the development of data-driven technolo
gies is becoming increasingly important in many applications. The intersect
ion of the fields of dynamical systems and machine learning is largely unex
plored\, and the objective of this talk is to show that working in reproduc
ing kernel Hilbert spaces offers tools for a data-based theory of nonlinear
dynamical systems.\n\n \n\nIn the first part of the talk\, we introduce si
mple methods to learn surrogate models for complex systems. We present vari
ants of the method of Kernel Flows as simple approaches for learning the ke
rnel 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-me
tric based Kernel Flows (HMKFs) and apply them to learn 132 chaotic dynamic
al 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.\n\n \n\nThen\, we introduce a data-based approach
to estimating key quantities which arise in the study of nonlinear autonom
ous\, control and random dynamical systems. Our approach hinges on the obse
rvation that much of the existing linear theory may be readily extended to
nonlinear systems - with a reasonable expectation of success- once the nonl
inear system has been mapped into a high or infinite dimensional Reproducin
g Kernel Hilbert Space. We develop computable\, non-parametric estimators a
pproximating controllability and observability energies for nonlinear syste
ms. We apply this approach to the problem of model reduction of nonlinear c
ontrol 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 meth
ods can be used to approximate center manifolds\, propose a data-based vers
ion of the center manifold theorem and construct Lyapunov functions for non
linear ODEs.
DTEND:20230929T190000Z
DTSTAMP:20240915T122042Z
DTSTART:20230929T180000Z
LOCATION:
SEQUENCE:0
SUMMARY:Machine Learning and Dynamical Systems meet in Reproducing Kernel H
ilbert Spaces with Boumediene Hamzi
UID:tag:localist.com\,2008:EventInstance_44370975227271
URL:https://calendar.fiu.edu/event/machine_learning_and_dynamical_systems_m
eet_in_reproducing_kernel_hilbert_spaces_with_boumediene_hamzi
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