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Simulation of turbulent flows at high Reynolds number is a computationallychallenging task relevant to a large number of engineering and scientificapplications in diverse fields such as climate science, aerodynamics, andcombustion. Turbulent flows are typically modeled by the Navier-Stokesequations. Direct Numerical Simulation (DNS) of the Navier-Stokes equationswith sufficient numerical resolution to capture all the relevant scales of theturbulent motions can be prohibitively expensive. Simulation atlower-resolution on a coarse-grid introduces significant errors. We introduce amachine learning (ML) technique based on a deep neural network architecturethat corrects the numerical errors induced by a coarse-grid simulation ofturbulent flows at high-Reynolds numbers, while simultaneously recovering anestimate of the high-resolution fields. Our proposed simulation strategy is ahybrid ML-PDE solver that is capable of obtaining a meaningful high-resolutionsolution trajectory while solving the system PDE at a lower resolution. Theapproach has the potential to dramatically reduce the expense of turbulent flowsimulations. As a proof-of-concept, we demonstrate our ML-PDE strategy on atwo-dimensional turbulent (Rayleigh Number $Ra=10^9$) Rayleigh-B\'enardConvection (RBC) problem.

Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations