Data-driven low-dimensional dynamic model of Kolmogorov flow
- URL: http://arxiv.org/abs/2210.16708v2
- Date: Tue, 1 Aug 2023 16:38:44 GMT
- Title: Data-driven low-dimensional dynamic model of Kolmogorov flow
- Authors: Carlos E. P\'erez De Jes\'us, Michael D. Graham
- Abstract summary: Reduced order models (ROMs) that capture flow dynamics are of interest for decreasing computational costs for simulation.
This work presents a data-driven framework for minimal-dimensional models that effectively capture the dynamics and properties of the flow.
We apply this to Kolmogorov flow in a regime consisting of chaotic and intermittent behavior.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Reduced order models (ROMs) that capture flow dynamics are of interest for
decreasing computational costs for simulation as well as for model-based
control approaches. This work presents a data-driven framework for
minimal-dimensional models that effectively capture the dynamics and properties
of the flow. We apply this to Kolmogorov flow in a regime consisting of chaotic
and intermittent behavior, which is common in many flows processes and is
challenging to model. The trajectory of the flow travels near relative periodic
orbits (RPOs), interspersed with sporadic bursting events corresponding to
excursions between the regions containing the RPOs. The first step in
development of the models is use of an undercomplete autoencoder to map from
the full state data down to a latent space of dramatically lower dimension.
Then models of the discrete-time evolution of the dynamics in the latent space
are developed. By analyzing the model performance as a function of latent space
dimension we can estimate the minimum number of dimensions required to capture
the system dynamics. To further reduce the dimension of the dynamical model, we
factor out a phase variable in the direction of translational invariance for
the flow, leading to separate evolution equations for the pattern and phase. At
a model dimension of five for the pattern dynamics, as opposed to the full
state dimension of 1024 (i.e. a 32x32 grid), accurate predictions are found for
individual trajectories out to about two Lyapunov times, as well as for
long-time statistics. Further small improvements in the results occur at a
dimension of nine. The nearly heteroclinic connections between the different
RPOs, including the quiescent and bursting time scales, are well captured. We
also capture key features of the phase dynamics. Finally, we use the
low-dimensional representation to predict future bursting events, finding good
success.
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