Numerical methods to evaluate Koopman matrix from system equations

Abstract

The Koopman operator is beneficial for analyzing nonlinear and stochastic dynamics; it is linear but infinite-dimensional, and it governs the evolution of observables. The extended dynamic mode decomposition (EDMD) is one of the famous methods in the Koopman operator approach. The EDMD employs a data set of snapshot pairs and a specific dictionary to evaluate an approximation for the Koopman operator, i.e., the Koopman matrix. In this study, we focus on stochastic differential equations, and a method to obtain the Koopman matrix is proposed. The proposed method does not need any data set, which employs the original system equations to evaluate some of the targeted elements of the Koopman matrix. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD are performed for a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.

22 pages, 4 figures

References in corpus (5)

• Applied Koopmanism
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• Duality and hidden symmetries in interacting particle systems
• Duality in stochastic processes from the viewpoint of basis expansions
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Cited by in corpus (3)

• Embedding stochastic differential equations into neural networks via dual processes
• Statistics for stochastic differential equations and approximations of resolvent
• Integrated utilization of equations and small dataset in the Koopman operator: applications to forward and inverse problems