Galerkin Reduced Order Models for Compressible Flows with Differentiable Programming*

Incorporating active predictive control strategies in dynamical systems such as turbulent flows is an active research topic. Reduced Order Models (ROM) facilitate this goal since they can provide accurate and computationally cheap forecasts. Widely employed Galerkin projection-based ROMs (GP-ROM) can suffer from instabilities and inaccuracies over long time horizons, despite the use of calibration techniques. Here, we utilize the Neural Galerkin Projection (NeuralGP) procedure; a differentiable programming-based approach that blends the strengths of a purely data-driven neural network-based technique to the physics-driven GP-ROM formulation. In NeuralGP, the structure of the ROM ODE can be specified and the ROM coefficients are learned directly from the data. We demonstrate this for a transonic flow over a buffeting NACA0012 airfoil where the NeuralGP implicitly learns stable ROM coefficients and accurately predicts over significantly longer time horizons, compared to the GP-ROM. We then compare the GP-ROM and Neural GP coefficients to study their stability properties. Further, to facilitate full state estimation we also demonstrate the use of a deep learning-based compressed sensing tool that reproduces the flow state from a few randomly placed sensors.