Learning Galerkin Reduced Order Model Closures for Feedback Flow Control with Differentiable Programming

Turbulent flow control has numerous applications and building POD-based Galerkin projection reduced order models (GP-ROMs) of the flow and the associated feedback control laws are extremely challenging. However, a key limitation is that the ODEs arising from GP ROMs are highly susceptible to instabilities due to truncation of POD modes and lead to deterioration in accuracy. In this work, we propose a differentiable programming approach that learns stable GP-ROMs with a closure term for truncated modes, by embedding neural networks in the ODEs and integrating it with a feedback controller. We test this approach on the isentropic Navier-Stokes equations for compressible flow over a cavity at a moderate Mach number. We show that differentiable programming as a paradigm can learn arbitrary terms corresponding to different formulations of truncated mode closures in ODEs, while obeying its constraints. The results show significantly longer and accurate time horizon predictions and effective control when compared to the classical GP-ROM. Key benefits of the differentiable programming-based approach include superior physics-based learning, low computational costs, and a significant increase in interpretability when compared to purely data-driven vanilla neural networks.