Estimate of Koopman modes and eigenvalues with Kalman Filter

TL;DR

This paper introduces an adaptive Ensemble Kalman filter-based approach to accurately estimate Koopman modes and eigenvalues from noisy measurements, improving dynamic mode decomposition in complex systems.

Contribution

The paper develops a novel adaptive method using the Ensemble Kalman filter to enhance DMD accuracy under noisy conditions and extend it to non-autonomous systems.

Findings

Effective in extracting modes from noisy data

Accurately recovers short-time eigenvalues and eigenvectors

Proven to work on both autonomous and non-autonomous systems

Abstract

Dynamic mode decomposition (DMD) is a data-driven method of extracting spatial-temporal coherent modes from complex systems and providing an equation-free architecture to model and predict systems. However, in practical applications, the accuracy of DMD can be limited in extracting dynamical features due to sensor noise in measurements. We develop an adaptive method to constantly update dynamic modes and eigenvalues from noisy measurements arising from discrete systems. Our method is based on the Ensemble Kalman filter owing to its capability of handling time-varying systems and nonlinear observables. Our method can be extended to non-autonomous dynamical systems, accurately recovering short-time eigenvalue-eigenvector pairs and observables. Theoretical analysis shows that the estimation is accurate in long term data misfit. We demonstrate the method on both autonomous and non-autonomous dynamical systems to show its effectiveness.