A Closed Machine Learning Parametric Reduced Order Model Approach -- Application to Turbulent Flows

Generally, reduced order models of fluid flows are obtained by projecting the Navier-Stokes equations onto a reduced subspace spanned by vector functions that carry the meaningful information of the dynamics. A common method to generate such subspace is the Proper Orthogonal Decomposition. This projection strategy is intrusive since it assumes that the approximation model used to solve the high fidelity flow problem is known and accessible. In the present article, we propose a non-intrusive paradigm based on Machine Learning to build Closed Parametric Reduced Order Models (ML-PROM) relevant to fluid dynamics. With no prior knowledge requirement of the approximation model, this method is purely data-driven as it operates directly on data regardless their origin, DNS, RANS simulations or experiment. The key idea to build the ML-CPROM is to use the generally known form of Galerkin ROMs and assimilate the time variations of the temporal POD modes to a nonlinear quadratic form taking a closure right hand side member. This last term is introduced to account for uncaptured dynamics due to data noise, POD truncation errors and time integration schemes. For parameter variation, the ML-CPROM is updated by interpolation on the quotient manifold of the set of fixed rank matrices by the orthogonal group. The predicted closure term is afterwards calculated by a Long-Short-Term-Memory neural network. We assess the potential of the proposed approach on the parametric examples of the lid-driven cavity flow and the flow past a cylinder by varying the Reynolds number through the viscosity, and the Ahmed-Body flow by varying the geometry through the rear slant angle. We show on these examples that our method enables to recover the dynamics with a good accuracy, not only for the training parameter points, but also for unseen parameter values for which the model was not priory trained.
Comment: The paper is in process of review and many changes need to be done. I will be resubmitted to Arxiv once the review completed