Data-driven analysis for dynamical systems using Koopman operators in Besov spaces.

Reference No.	2024a034
Type/Category	Grant for General Research-Short-term Joint Research
Title of Research Project	Data-driven analysis for dynamical systems using Koopman operators in Besov spaces.
Principal Investigator	Isao Ishikawa（Center for Data Science, Ehime University・Associate Professor）
Research Period	June 6,2024. ～ June 7,2024.
Keyword(s) of Research Fields	Koopman operators, composition operators, Besov spaces, dynamical systems, topological data analysis
Abstract for Research Report	This research proposal aims to develop the theory of Koopman operators on Besov spaces and establish a theoretical foundation for the data-driven analysis of dynamical systems. Besov spaces are function spaces that play a central role in the theory of partial differential equations and function space theory. They allow for a precise description of the smoothness and decay of functions, which has recently led to applications in the theoretical analysis of machine learning. Koopman operators are linear operators defined on function spaces by the operation of composing functions with dynamical systems from the right. There are two main expected outcomes from this research proposal. The first is to uncover the mathematical properties of Koopman operators in Besov spaces and their relationship with the behavior of dynamical systems, thereby constructing a valid theoretical basis for data analysis of such systems. Although Koopman operators contain all the information of a dynamical system, this information can only be extracted through functional analysis methods. The first outcome addresses the problem of extracting information from dynamical systems through Koopman operators, laying an important foundation. The second outcome aims to develop new data-driven estimation algorithms for Koopman operators using the extensive knowledge accumulated in solving problems in partial differential equations theory within Besov spaces. Known decompositions of various functions in Besov spaces suggest the potential for discovering new methods for extracting features of dynamical systems. The research will involve intensive discussions with experts in partial differential equations and function space theory, Ikeda and Taniguchi. Additionally, the research leader believes that the information obtained from Koopman operators includes the topological structure of dynamical systems. Therefore, Professor. Ike, who is affiliated with the Kyushu University Institute for Mathematics for Industry and is an expert in topological data analysis, will also participate in the discussions. The goal is to discover new relationships between topological data analysis and the theory of Koopman operators.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Masahiro Ikeda（RIKEN・Research Scientist） Koichi Taniguchi（Tohoku University・Assistant Professor）