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SUMMARY:Kernels Simplify Differential Equations - Jonghyeon Lee (CALTECH (
 California Institute of Technology))
DTSTART:20250806T130000Z
DTEND:20250806T140000Z
UID:TALK234691@talks.cam.ac.uk
DESCRIPTION:Many nonlinear ordinary and partial differential equations are
  difficult or time-consuming to solve and analyse. It is unsurprising that
  transforming them to equations with 'simpler' behaviour is an active fiel
 d of research\; this includes mapping them to linear differential equation
 s either locally or globally or approximating the solution with a relative
 ly small number of basis functions that capture the essential elements of 
 the behaviour. Kernel methods have considerable value in learning such tra
 nsformations because they are typically linearity in time complexity as a 
 function of the collocation points and have strong theoretical convergence
  results. In the first part of the talk\, we introduce the idea of general
 ized kernel regression to learn the Cole-Hopf transformation\, which maps 
 the nonlinear Burgers equation to the linear equation\, and a Poincare nor
 mal form of the Hopf bifurcation of the Brusselator. We then move on to di
 scussing the applications of kernels to recover the eigenfunctions of the 
 Koopman operator\, which maps a nonlinear ODE to a dynamical system in inf
 inite dimensions\, and applications including Lyapunov functions and quasi
 -potential functions of stochastic systems. Finally\, we conclude by propo
 sing a new kernelized reduced order model (KROM) which uses an empirical k
 ernel matrix to quickly solve nonlinear PDEs.\nThis message and any attach
 ment are intended solely for the addressee and may contain confidential in
 formation. If you have received this message in error\, please contact the
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 ed by the author of this email do not necessarily reflect the views of the
  University of Nottingham. Email communications with the University of Not
 tingham may be monitored where permitted by law.
LOCATION:Seminar Room 2\, Newton Institute
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