Learning Rarefied Gas Dynamics with Physics-Enforced Neural Networks

Abstract

This study develops and validates neural network frameworks with physics-based constraints for surrogate modeling of rarefied gas dynamics across different levels of complexity. As a baseline, we first examine the BGK kinetic relaxation problem and show that reformulating the learning task in terms of the perturbation from the Maxwell Boltzmann equilibrium ensures stability and accuracy. Building upon this foundation, we employ Deep Operator Networks, DeepONets, with physical constraints to address two more challenging problems. The first is the prediction of the one-dimensional structure of a standing shock wave in a rarefied polyatomic gas at Mach 5, where the incorporation of physical constraints avoids overshoot and yields accurate predictions even for unseen viscosity ratios. The second is the modeling of two-dimensional rarefied hypersonic flow over a cylinder, where an ensemble of DeepONets trained on a sparse dataset obtained from the direct simulation Monte Carlo, DSMC, approach, generalizes successfully to both interpolation and extrapolation cases up to M equal to 10. A custom weighted loss function improves the prediction of pressure, while ensemble-based uncertainty quantification correctly identifies regions of high gradients such as shock waves. The results demonstrate that embedding physical constraints into neural operator architectures enables accurate, physically consistent, and computationally efficient surrogates, paving the way for their application to multi-dimensional high-speed rarefied flow problems.