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

Generally, reduced order models for fluid flows are intrusively built by projecting the governing equations onto a subspace often generated by the Proper Orthogonal Decomposition (POD). In this talk, we introduce a non-intrusive paradigm based on Machine Learning to build Closed Parametric Reduced Order Models (ML-CPROM) relevant to fluid dynamics. This method is purely data-driven as it operates on data regardless of their origin (DNS, RANS, or experiment). The key idea to building such models is to assimilate the derivatives of the temporal POD modes to a quadratic polynomial with a closure term. The closure term predicted by a Long-Short-Term-Memory neural network is added to account for errors that may stem from data noise, POD truncation, and time integration schemes. To address parameter variations, the model is updated by interpolation onto the quotient manifold of the set of maximal-rank matrices by the orthogonal group. The potential of the ML-CPROM method is assessed on examples of flow past a cylinder with variable Reynolds number, and the flow past an Ahmed-body with a variable rear slant angle. We show that the ML-CPORM succeeds in recovering the dynamics with good accuracy, even for parameter values on which it was not previously trained.