Kernels Simplify Differential Equations

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• Jonghyeon Lee (CALTECH (California Institute of Technology))
• Tuesday 05 August 2025, 10:30-11:30
• Seminar Room 2, Newton Institute.

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RCL - Representing, calibrating & leveraging prediction uncertainty from statistics to machine learning

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 field of research; this includes mapping them to linear differential equations either locally or globally or approximating the solution with a relatively small number of basis functions that capture the essential elements of the behaviour. Kernel methods have considerable value in learning such transformations 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 generalized kernel regression to learn the Cole-Hopf transformation, which maps the nonlinear Burgers equation to the linear equation, and a Poincare normal form of the Hopf bifurcation of the Brusselator. We then move on to discussing the applications of kernels to recover the eigenfunctions of the Koopman operator, which maps a nonlinear ODE to a dynamical system in infinite dimensions, and applications including Lyapunov functions and quasi-potential functions of stochastic systems. Finally, we conclude by proposing a new kernelized reduced order model (KROM) which uses an empirical kernel matrix to quickly solve nonlinear PDEs. This message and any attachment are intended solely for the addressee and may contain confidential information. If you have received this message in error, please contact the sender and delete the email and attachment. Any views or opinions expressed by the author of this email do not necessarily reflect the views of the University of Nottingham. Email communications with the University of Nottingham may be monitored where permitted by law.

This talk is part of the Isaac Newton Institute Seminar Series series.

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