Adjoint Optimization of the BGK Equation with an Embedded Neural Network for Reduced-Order Modeling of Hypersonic Flows

Numerical simulation of the Boltzmann equation, most commonly by the DSMC method, can provide highly accurate descriptions of supersonic, rarefied flows. However, it is computationally expensive in the transitional flow regime, with Knudsen numbers 0.01 ≲ Kn ≲ 0.1, motivating the use of simpler, but less accurate models such as the BGK equation. We develop and implement optimization methods for calibrating machine learning-based BGK models to high fidelity DSMC data. We derive and validate an adjoint equation for optimization over the BGK equation with a general form of the collision frequency. We train a neural network model for the collision frequency using automatic differentiation to implement the ad joint equations, and apply this model to the formation of a 1D shock.