Correcting Navier-Stokes-Fourier System for Rarefied Flows with Non-Linear Super-Stencils

Conventional particle-based methods for simulating rarefied gas flows, such as Direct Simulation Monte Carlo (DSMC), become prohibitively expensive in the near-continuum regime. While considerable efforts have been made to replace kinetic descriptions with generalized hydrodynamic models, such as the Burnett and Grads 13-moment equations, these approaches have achieved limited practical success. We propose a Non-Linear Super-Stencil (NLSS) model correction to augment the Navier-Stokes-Fourier (NSF) system for rarefied flow simulations. NLSS employs a compact stencil that samples local macroscopic flow features and maps them to corrective source terms in the original NSF system via a neural network, effectively capturing non-equilibrium effects beyond the continuum regime. We demonstrate the NLSS framework on planar Couette flow, introducing a novel boundary condition to accommodate velocity and temperature slip near solid walls without requiring a priori knowledge. After training on a limited dataset spanning a range of only wall velocities and Knudsen numbers, the corrected NSF system accurately predicts macroscopic flow behavior and wall slips in unseen conditions, including cases with different wall temperatures. This highlights the potential of NLSS as a generalizable surrogate for rarefied flow simulation.