Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition

TL;DR

This paper introduces a data-driven neural network approach to learn Koopman invariant subspaces for spectral analysis of nonlinear dynamical systems, removing the need for manual feature selection.

Contribution

It proposes a novel neural network-based method to automatically learn Koopman invariant subspaces directly from data, enhancing spectral analysis capabilities.

Findings

Effective in nonlinear dynamical systems

Outperforms traditional methods in accuracy

Applicable to real-world data

Abstract

Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynamics, which is not always possible since we may not have any a priori knowledge about them. In this paper, we propose a fully data-driven method for Koopman spectral analysis based on the principle of learning Koopman invariant subspaces from observed data. To this end, we propose minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. We introduce an implementation with neural networks and evaluate performance empirically using nonlinear dynamical systems and applications.