Deterministic local reduced-order modelling for chaotic flows with cluster-based quantization

The long-term behavior of dissipative dynamical systems can be represented on manifolds that have fewer degrees of freedom than the phase space. In chaotic systems, the manifolds can have intricate shapes, which makes may make single reduced-order model (ROM) inaccurate. In this work, we construct a series of local ROMs, each of which models the dynamics of specific portions of the manifold. To do so, we quantize the manifold with cluster-based analysis, whose centroids identify local patches to create the manifold cartography. We develop both linear methods based on Galerkin projections and nonlinear methods based on Echo State Networks and Long Short-Term Memory networks. Our methodology is verified on the Kuramoto-Sivashinsky equations, and finally applied to the wake flow past the fluidic pinball in different flow regimes. Deterministic local ROMs open opportunities for efficient flow control and real-time prediction of turbulent systems.