Global Linearization of Nonlinear Dynamics via Koopman Operators: A Gentle Introduction, Applications, and Open Challenges – Hyukpyo Hong

Abstract:

A central challenge of modern dynamical systems theory is to make nonlinear systems tractable without sacrificing fidelity. Koopman operator theory pursues this goal by lifting nonlinear dynamics into a linear, but infinite dimensional, operator acting on a function space. This operator-theoretic perspective underlies a broad class of modern data-driven methods, from dynamic mode decomposition to equation discovery in scientific machine learning for fluid dynamics and neuroscience. Yet this power comes at a price: the operator’s infinite dimensionality poses a fundamental obstacle to computation and practical use, and finding tractable finite-dimensional approximations remains an open and active challenge. In this talk, I will first introduce the basic principles of Koopman operator theory and survey some of the results that have made it a cornerstone of modern dynamical systems analysis. I will then briefly describe two of my works on finite-dimensional Koopman representations. Finally, I will turn to my recent work on non-autonomous dynamical system learning, in collaboration with Prof. Dae Wook Kim.