Complementary Deep - Reduced Order Model
Résumé
Reducing simulation time is critical for applications such as closed loop control or iterative
design op- timisation. In this context, model reduction techniques have become a
growing area of research in the last decades. While research efforts have mainly been
centered around feature based approaches like POD, BPOD or DMD, direct approaches
leveraging Deep Neural Networks have been proposed in recent years with great success.
Despite these promising results, neural network architectures provide little to no physical
guarantees, and have limited interpretability. On the other hand, feature based methods
often reconstruct the final solution through a linear combination of modes embedded with
physical constraints. However, this often comes at the cost of loss of information and
increased error rates.
POD-Galerkin models are a perfect example of this trade-off between physical guarantees
and performance loss. These models have been shown to be very efficient for the reduction
of linear systems, but they are extremely limited when applied to nonlinear systems such
as the Navier-Stokes equations. For example, Noack et al. have shown that a simple 3
equations model was able to capture the oscillatory dynamics of a flow over a cylinder,
but failed to correctly predict the transition time and trajectory from a steady point of
the system to its oscillatory regime.
To address these shortcomings, we propose to add a closure term to POD-Galerkin models
to correct their dynamics. Observing that the information lost during the projection on
the POD basis can be retrieved by considering the past states of the system, we use
simple neural networks in combination with delay differential equations to reconstruct
the required correction. We show that a satisfactory model can be trained through the
Neural ODE framework to learn a memory based correction from simulation data. The
final architecture can be compared to a time-continuous recurrent neural network.
With this approach, we preserve the simple structure and low computational cost of
Galerkin models while improving their performance. Using the 3 modes model example
from [4], we show that the corrected ROM reproduces perfectly the original transition
trajectory, and generalises well to unseen initial conditions. On-going work is concerned
with validating and improving the proposed approach by applying it on the more challenging
chaotic pinball case.
Origine | Fichiers produits par l'(les) auteur(s) |
---|