A Seminar by Yuanchao Xu (Kyoto University, Japan)
Abstract: Complex dynamical systems are challenging to model due to their high dimensionality, strong nonlinearity, and inherent uncertainty. This work develops data-driven methods to extract meaningful dynamics directly from observations using the Koopman operator framework, which provides a linear representation of nonlinear dynamics. We present two contributions addressing key limitations in existing approaches. First, ResKoopNet uses neural networks to automatically learn optimal dictionary functions by minimizing spectral residuals, which overcomes the spectral pollution and inclusion problems that cause traditional methods to miss important dynamical features or produce spurious results. Second, Stochastic Dynamic Mode Decomposition (SDMD) provides a numerically stable and theoretically rigorous framework for systems with noise by approximating the Koopman semigroup rather than its unbounded generator. We establish convergence guarantees in three limits: large data, vanishing sampling time, and large dictionary size, and demonstrate the effectiveness of both methods across diverse application, from deterministic settings to stochastic processes.
Biography: Yuanchao Xu is a Postdoctoral Researcher at Kyoto University. He received his Ph.D. in Applied Mathematics from the University of Alberta, M.Sc. in Mathematics from the University of Manitoba, and B.Sc. in Mathematics from the University of Oregon. His research focuses on the spectral analysis of Koopman operator and data-driven modeling of stochastic dynamical systems.