Cluster Globally, Model Locally: Clusterwise modeling of nonlinear dynamics

Abstract Data-driven reduced-order modeling opens new avenues of understanding, predicting, controlling, and optimizing system behavior. Simple systems may have state spaces in which sparse human-interpretable dynamical systems can be identified. This approach has been pioneered by S. Brunton et al. (2016, PNAS) with Sparse Identification of Nonlinear Dynamics (SINDy). Complex systems, however, cannot be expected to benefit from such simple analytical descriptions. Yet, smoothness may be exploited by analytical local descriptions. In this paper, we identify a clusterwise polynomial dynamics from time-resolved snapshot data. The full state space is partitioned into clusters with a reduced-order polynomial description for each cluster and a global patching strategy. The resulting clusterwise modeling is entirely data-driven and requires no prior knowledge of the system dynamics. We illustrate the approach on the well-known chaotic Lorenz and Rössler systems, on the more challenging chaotic fluid flow dynamics of higher state-space dimensions, on a noisy electrocardiogram (ECG) signal, and finally on the time evolution of the monthly sunspot number. Clusterwise modeling offers a powerful and interpretable paradigm for dynamical modeling. Nonlinear dynamics can be approximated by assembling many simple local models of different resolutions, opening new paths to understand and control intricate nonlinearities.