Extended dynamic mode decomposition with dictionary learning using neural ordinary differential equations
This work addresses the challenge of data-driven analysis of nonlinear systems for researchers in dynamical systems and machine learning, but it is incremental as it builds on existing EDMD and NODE methods.
The paper tackles the problem of approximating the Koopman operator for nonlinear phenomena by proposing an algorithm that uses neural ordinary differential equations (NODEs) to find a parameter-efficient dictionary for extended dynamic mode decomposition (EDMD), showing superiority in parameter efficiency through numerical experiments.
Nonlinear phenomena can be analyzed via linear techniques using operator-theoretic approaches. Data-driven method called the extended dynamic mode decomposition (EDMD) and its variants, which approximate the Koopman operator associated with the nonlinear phenomena, have been rapidly developing by incorporating machine learning methods. Neural ordinary differential equations (NODEs), which are a neural network equipped with a continuum of layers, and have high parameter and memory efficiencies, have been proposed. In this paper, we propose an algorithm to perform EDMD using NODEs. NODEs are used to find a parameter-efficient dictionary which provides a good finite-dimensional approximation of the Koopman operator. We show the superiority of the parameter efficiency of the proposed method through numerical experiments.