Adaptive Cluster-Dynamic Mode Decomposition with Application to the Burgers' Equation

This paper proposes a new model reduction method that improves the prediction accuracy of dynamic modes decomposition with control (DMDc). DMD is a data-driven technique that extracts low-order models from high-dimensional complex dynamic systems with actuation. With DMDc, an input-output reduced-order model can be obtained for system identification and prediction. In this work, in order to better capture the nonlinear behavior, an adaptive clustering method is introduced to group the snapshots obtained by experimental data or numerical simulation into several sub-regions that display similar behavior to construct a reduced-order model. Cluster methods with DMDc are combined to construct the local reduced-order model. Furthermore, with the prediction process, new incoming data is fed into the clusters to update the cluster-DMDc reduced order model to obtain more accurate predictions. Time clustering is applied to the snapshots generated by the Burgers' equations with boundary actuation, and the adaptive cluster DMDc reduced-order model outperforms the standard DMDc.