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...

Collaboration is key

As collaborators from France, Germany and Australia we join our personal expertise in different fields like turbulence control, climate modelling, entropic methods and machine learning techniques. Together we focus on reduced-order modelling of complex systems and created CROM – benefitting from our interdisciplinary viewpoints. We always look for new horizons. Feel free to contact us.

Eurika Kaiser
Bernd R. Noack
Laurent Cordier
Andreas Spohn
Marc Segond
Markus W. Abel
Guillaume Daviller
Robert K. Niven

Curious what CROM will accomplish for your own data?

General requirements

The time series of a measurement consists of decimal numbers anm where n is the number of the measured variable and m is the number of the time step. Thus, a11 to aN1 form a vector of all variables measured at the first time step.

The following formats are designed as ASCII files which contain a matrix of numbers. The decimal numbers must be written with a dot as the decimal mark – e.g. 11235.813 – and without any other separator like spaces or commas. Consecutive values on the same line have to be separated by whitespaces.

The native format

The first column contains the number of the time step m. The remaining columns contain the measured values.

1 a11 a21 ... aN1
2 a12 a22 ... aN2
... ... ... ... ...
M a1M a2M ... aNM

The xAMC format (more info)

The first column contains the number of the time step m. The second column contains the number of the variable n. The third column contains the corresponding measured value.

1 1 a11
1 2 a21
... ... a...1
1 N aN1
2 1 a12
2 2 a22
... ... a...2
2 N aN2
... ... ...
M 1 a1M
M 2 a2M
... ... a...M

We welcome everyone to try out CROM. Actually, we are eager to see how far we can take this modelling approach! Drop us a note with your data attached and we will send you the results in a few days. And of course, you can trust us to use your data only to assemble the CROM results.

Before you proceed, however, check that:

you have time-resolved data with a constant sampling time Δt

which can be provided in one of the data formats described to the right.

If you can fulfill these prerequisites, go ahead!

Hi Eurika,
CROM got me interested and I am eager to apply it to my own data. To make it a little easier for you to give me a good interpretation of the results let me give you a rough picture about where the data is from:

Here you got my data in the format:

(The file size is limited to 500MB. For larger files please contact me directly.)
Please send me the results to the email address: