Koopman operator learning using invertible neural networks

TL;DR

This paper introduces FlowDMD, a novel method using invertible neural networks to learn Koopman invariant subspaces, enabling more accurate state reconstruction without manual observable selection.

Contribution

The paper proposes FlowDMD, which leverages invertible neural networks to automatically learn Koopman invariant subspaces, improving accuracy over existing methods.

Findings

FlowDMD outperforms state-of-the-art methods in numerical experiments.

The invertibility of neural networks enhances the accuracy of Koopman operator learning.

FlowDMD effectively reconstructs state variables in nonlinear systems.

Abstract

In Koopman operator theory, a finite-dimensional nonlinear system is transformed into an infinite but linear system using a set of observable functions. However, manually selecting observable functions that span the invariant subspace of the Koopman operator based on prior knowledge is inefficient and challenging, particularly when little or no information is available about the underlying systems. Furthermore, current methodologies tend to disregard the importance of the invertibility of observable functions, which leads to inaccurate results. To address these challenges, we propose the so-called FlowDMD, aka Flow-based Dynamic Mode Decomposition, that utilizes the Coupling Flow Invertible Neural Network (CF-INN) framework. FlowDMD leverages the intrinsically invertible characteristics of the CF-INN to learn the invariant subspaces of the Koopman operator and accurately reconstruct state variables. Numerical experiments demonstrate the superior performance of our algorithm compared to state-of-the-art methodologies.