2023/04/24 by Shaowu Pan, Karthik Duraisamy, Pan, Shaowu +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #37M10 #37M25 #47B33 #62F15 #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Fusion and Plasma Physics Studies #Machine Learning (cs.LG) #Mathematical Biology Tumor Growth #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2304.11860
openalex publication_date 2023/04/24 · openalex created_date 2023/04/27 · openalex updated_date 2026/07/28
The Koopman operator provides a linear perspective on non-linear dynamics by focusing on the evolution of observables in an invariant subspace. Observables of interest are typically linearly reconstructed from the Koopman eigenfunctions. Despite the broad use of Koopman operators over the past few years, there exist some misconceptions about the applicability of Koopman operators to dynamical systems with more than one disjoint invariant sets (e.g., basins of attractions from isolated fixed points). In this work, we first provide a simple explanation for the mechanism of linear reconstruction-based Koopman operators of nonlinear systems with multiple disjoint invariant sets. Next, we discuss the use of discrete symmetry among such invariant sets to construct Koopman eigenfunctions in a data efficient manner. Finally, several numerical examples are provided to illustrate the benefits of exploiting symmetry for learning the Koopman operator.