Minimal realization time-delay Koopman analysis for stochastic dynamical system identification

Numerical simulations and real-world monitoring data are increasingly prevalent in fields such as solid mechanics, fluid dynamics, and structural engineering. However, developing accurate models that capture the underlying system dynamics from sparse and noisy measurements remains a significant challenge. To address this, we propose a novel methodology called Minimal Realization Time-Delay Koopman (MRTK) analysis. This method combines time-delay embedding with Koopman operator theory to transform nonlinear dynamics into a linearized form. Additionally, it employs Singular Value Decomposition (SVD) to reduce model order, enhancing computational efficiency and accurately identifying system dynamics from sparse and noisy measurements. By explicitly modeling noise in the data, we demonstrate that the MRTK method serves as a generalized extension of Dynamic Mode Decomposition (DMD), encompassing variants such as Extended DMD and Total Least Squares DMD, while also establishing theoretical connections with both Hankel Alternative View of Koopman (HAVOK) analysis and Subspace DMD. We validate the proposed approach using simulated data from transitional channel flow and the Lorenz system, as well as real-world wind speed and main girder acceleration measurements from the Hangzhou Bay Bridge. The results show that integrating the identified reduced-order model with a Kalman filter enables real-time, accurate estimation and prediction from sparse data. The method achieves high predictive accuracy across all scenarios, with the maximum Normalized Mean Squared Error (NMSE) prediction error for the wind speed field being 1.911%, underscoring its potential to advance the prediction and control of complex systems.