Correcting noisy dynamic mode decomposition with Kalman filters

Abstract

Data-driven discovery has become a very important paradigm for science and engineering. As an effective finite-dimensional approximation for the Koopman operator, dynamic mode decomposition (DMD) is an efficient approach for data-driven modeling for nonlinear dynamical systems using observation data. In practical problems, the observation data is often polluted by noise. If we straightforwardly use the noisy data to construct a noisy DMD model, the resultant model may be inaccurate and lead to implausible prediction. To take account of the effect of noise, we derive the maximum-likelihood estimator (MLE) of the DMD matrix and explore the relation to Noise-corrected DMD. But it is difficult to efficiently compute the MLE of the DMD matrix in general. To avoid the difficulty, we consider a noise perturbation dynamical system for observation snapshots and use Kalman filters to update the noisy snapshots. The updated snapshots are then used to reconstruct DMD models. This Kalman filter DMD significantly improves the model accuracy and prediction ability compared with noisy DMD. When the noise covariance is unknown, we use Kalman filter adaptively estimate the noise covariance and develop ensemble Kalman filter DMD, which can be used to construct surrogate stochastic Koopman models for random dynamical systems. The Kalman filter DMD and ensemble Kalman filter DMD are used to reconstruct a few representative nonlinear dynamical systems. The numerical results show the significant correcting effect on noisy DMD.