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Turbulent flow control has numerous applications and building reduced-ordermodels (ROMs) of the flow and the associated feedback control laws is extremelychallenging. Despite the complexity of building data-driven ROMs forturbulence, the superior representational capacity of deep neural networks hasdemonstrated considerable success in learning ROMs. Nevertheless, thesestrategies are typically devoid of physical foundations and often lackinterpretability. Conversely, the Proper Orthogonal Decomposition (POD) basedGalerkin projection (GP) approach for ROM has been popular in many problemsowing to its theoretically consistent and explainable physical foundations.However, a key limitation is that the ordinary differential equations (ODEs)arising from GP ROMs are highly susceptible to instabilities due to truncationof POD modes and lead to deterioration in temporal predictions. In this work,we propose a \textit{differentiable programming} approach that blends thestrengths of both these strategies, by embedding neural networks explicitlyinto the GP ODE structure, termed Neural Galerkin projection. We demonstratethis approach on the isentropic Navier-Stokes equations for compressible flowover a cavity at a moderate Mach number. When provided the structure of theprojected equations, we show that the Neural Galerkin approach implicitlylearns stable ODE coefficients from POD coefficients and demonstratessignificantly longer and accurate time horizon predictions, when compared tothe classical POD-GP assisted by calibration. We observe that the key benefitsof this differentiable programming-based approach include increased flexibilityin physics-based learning, very low computational costs, and a significantincrease in interpretability, when compared to purely data-driven neuralnetworks.

Learning Stable Galerkin Models of Turbulence with Differentiable Programming