Extended dynamic mode decomposition with dictionary learning using neural ordinary differential equations

TL;DR

This paper introduces a novel approach combining neural ordinary differential equations with extended dynamic mode decomposition to efficiently approximate the Koopman operator for analyzing nonlinear phenomena.

Contribution

The paper proposes a new algorithm that uses NODEs to learn a parameter-efficient dictionary for EDMD, improving approximation of the Koopman operator.

Findings

Demonstrates the superiority of the method's parameter efficiency

Shows improved approximation of the Koopman operator in numerical experiments

Validates the approach on nonlinear dynamical systems

Abstract

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.