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Koopman Kernels for Learning Dynamical Systems

Koopman Operator Theory Workshop: Fundamentals, Approximations and Applications "Koopman Kernels for Learning Dynamical Systems" Speaker: Petar Bevanda, Technical University of Munich In recent years, the development of the Koopman operator theory has seen applications in various fields of research, from fluid dynamics to control theory to pure mathematics, and also to fundamentals of physics. There are also tantalizing connections to training of deep neural networks. The Koopman Operator Theory Workshop was held in an interdisciplinary setting to foster discussions on further developments and investigations into the role that this recently rediscovered object could play in theoretical and data-driven research. The Koopman operator plays a role both in fundamental discussions, such as the relation between quantum and classical mechanics , and in the development of different application that entail a mixture of physics-driven and data-driven modeling. This may lead to the possible extension of mathematical treatment to phenomena and areas that are not amenable to mathematical description right now, because we are missing the fundamental relation that we think are at the base of their behavior . Examples can be found in social and economic studies, but also in biology, and in other fields, that traditionally have not been investigated using explicit mathematical tools. The emerging relation between the Koopman operator theory, and artificial intelligence data driven methods, seems to be pointing at the possibility of using the theory as a basis to understand the training and performance of deep learning algorithms. This relation is revealing a profound connection between Koopman operator theory and deep learning algorithms. Progress is being made in applications of Koopman operator theory to atmospheric, climate, and ocean dynamics. With the recent advances of using machine learning methods in forecasting, interpretability of Koopman operator-based techniques is particularly attractive to reconcile physics-driven and data-driven methodologies. The Koopman Operator Theory Workshop was held in the Aragonese Castle (Piazza Castello, Otranto) on May 20-21, 2024.