Bayesian-based merging of data assimilation and machine learning to learn unsteady turbulence models from sparse data

In this presentation, we describe a Bayesian-based approach to learn unsteady turbulence-model corrections from sparse data. Relying on the Expectation-Maximization (EM) formalism, we rigorously justify performing such a learning task in two steps. In a first step, Data Assimilation (DA) techniques, more specifically ensemble Kalman filtering (EnKF), are employed to infer full flow descriptions from the considered sparse data (Expectation step). In a second step, the thus obtained full flow descriptions are gathered to form a training dataset that may be exploited by machine-learning (ML) tools to derive the sought model corrections (Maximization step). As such, and as justified by the EM approach, the present methodology enables a seemingly optimal combination of the respective strengths of DA and ML techniques, namely the ability of the former in state estimation and the ability of the latter in optimizing highly nonlinear model representations. Moreover, thanks to the use of the EnKF and its sequential treatment of data in time, the present approach is essentially non-intrusive and may deal with potentially chaotic flows and over arbitrary long time horizons. The potentialities of this methodology are illustrated in particular through learning corrective terms in an Unsteady Reynolds-Averaged Navier-Stokes (URANS) model from synthetic sparse velocity data of the turbulent flow around a circular bluff body.