Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems
Minchan Jeong, Jongha (Jon) Ryu, Se-Young Yun, Gregory Wornell
The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems through linear operator theory. Recent advances in dynamic mode decomposition (DMD) have shown that trajectory data can be used to identify dominant modes of a system in a data-driven manner. Building on this idea, deep learning methods such as VAMPnets and DPNets have been proposed to learn the leading singular subspaces of the Koopman operator. However, these methods often rely on numerically unstable operations during objective computation, including singular value decomposition and matrix inversion, which introduce biased gradient estimates and limit scalability to large systems.In this work, we propose a scalable and conceptually simple method for learning the top-$L$ singular functions of the Koopman operator for stochastic dynamical systems based on a low-rank approximation framework. Our approach eliminates the need for unstable linear algebraic operations and integrates easily into modern deep learning pipelines. Empirical results demonstrate that the learned singular subspaces are both reliable and effective for downstream tasks such as multi-step prediction.