Statistical machine learning tools for probabilistic closures of turbulence models

Turbulent flow remains a challenging subject, despite extensive efforts to find analytical descriptions. Modeling small scales of motion is crucial for saving time and resources in numerical simulations, particularly in industrial applications. Here, we attempt to model small scales of motion by creating closures for a Shell model of turbulence, more specifically for Sabra. Shell models are infinite dimensional dynamical systems that retain most key properties of Navier-Stokes equation, such as energy cascade and intermittency, while being computationally treatable. To account for Sabra's intermittent fluctuations, we employ a set of scaling relations that recover a hidden symmetry and leave us with universal statistics across the inertial range. On data from these rescaled variables, we then adapt and apply two machine learning tools, a variational auto-encoder and sparse identification of non-linear dynamics. We estimate the data's densities and dynamics in order to then generate new instances of data. Given a mid-inertial range cutoff scale, we evolve reduced models in time, resolving only scales of motion larger than the cutoff scale, closing the reduced model with data generated by the machine learning tools. We compare statistics of our reduced models against a Sabra's fully resolved simulation to evaluate each closure's performances. Our results show improvement regarding previous work, and all our closures are probabilistic and cutoff-independent.