Machine-learned R13 moment closures for shock-dominated rarefied gas flows

This work introduces R13-ML, a machine-learning–enhanced regularized 13-moment framework for shock-dominated rarefied gas flows in the transition regime. The analytically derived higher-order closures and collisional production terms of classical R13 are replaced by fully connected neural networks trained on high-fidelity Direct Simulation Monte Carlo (DSMC) data. The networks learn mappings for the third- and fourth-order moments ( m i j k , R i j ) and the production terms ( Q i j , Q i ) from a physics-based, ten-dimensional feature set, combining normalization by local mean free path and sound speed with Gaussian filtering, polynomial interpolation, and a weighted Huber loss. The trained surrogates are embedded in a high-order discontinuous Galerkin spectral element solver for dynamic online evaluation of the closures. Within the one-dimensional training range Ma = 1.2 – 8.0 , R13-ML reproduces higher-order moments and collision integrals in close agreement with DSMC, while errors of the analytical R13 closures exceed an order of magnitude for Ma ≳ 5 . For one-dimensional shocks, the model remains accurate at Ma = 9 and stable at Ma = 12 , where classical R13 develops severe oscillations. Robust generalization is further demonstrated for unsteady shock-interaction problems and for two-dimensional rarefied flow over a cylinder at Ma = 7 , where R13-ML captures the spatial topology and peak values of higher-order moments with stress and heat-flux deviations below about 6% relative to DSMC.