# CROM – Our approach to reduced-order modelling

Cluster-based reduced-order modelling (CROM) is a novel strategy to derive a low-dimensional model of a complex system in an unsupervised manner. We combine two well-known methods for the first time in fluid dynamics: cluster analysis (see e.g. Burkardt et al., 2006) and Markov models (see Schneider et al., 2007). Cluster analysis is a method from statistics and machine learning and its aim is finding hidden groups, called clusters, in data. Markov models are stochastic models and employed to model the cluster transitions. CROM does not require any prior knowledge of the system behavior. It only expects a time-resolved sequence of observations and a problem-dependent definition of a distance measure. By construction, CROM is a generally applicable approach and not restricted to fluid flows. The data can originate from any system.

CROM has numerous applications, e.g. the distillation of physical mechanisms of complex systems, identification of precursors to desirable and undesirable events, the comparison of dynamical models, etc. So far, CROM has been applied to analytical systems, like e.g. the Lorenz attractor, and numerical simulations of fluid dynamical problems with known behavior. Currently, we are exploring CROM's capabilities for experimental data and systems which are much more challenging.

* E. Kaiser, B. R. Noack, L. Cordier, A. Spohn, M. Segond, M. Abel, G. Daviller, J. Ă–sth, S. Krajnović and R. K. Niven (2014). Cluster-based reduced-order modelling of a mixing layer. *Journal of Fluid Mechanics*, **754**, pp 365-414.

© 2014 Cambridge University Press

## CROM applied to a mixing layer

CROM has been applied to a numerical simulation of a two-dimensional incompressible mixing layer flow undergoing vortex pairing. The mixing layer is a result from two streams of different velocities which meet in a horizontal plane and exhibits the typical roll-up of initial Kelvin-Helmholtz vortices and vortex pairing events further downstream.

See the results...