Fourier-Hermite Kinetic Network for Gas Transport Under High Knudsen Numbers

Abstract

This work presents the Fourier-Hermite Kinetic Network (FHKN), a novel sequence-to-state temporal-spectral architecture designed to accelerate the prediction of rarefied gas transport in porous media. Traditional lattice Boltzmann method (LBM) simulations in complex geometries require massive computational overhead due to slow transient relaxation. To bypass auto-regressive time-stepping, the FHKN extracts dominant relaxation frequencies from the initial transient iterations of discrete velocity distribution functions (fi) via a 1D temporal Fourier transform. The architecture employs a truncated Hermite expansion block that isolates the 0th, 1st, and 2nd order moments. This mathematically aligns with the LBM Gauss-Hermite quadrature, acting as a low-pass kinetic filter that discards high-frequency non-equilibrium noise and maps directly to the macroscopic Navier-Stokes limit. Trained on optimized C++ LBM data from simple random granular domains, FHKN significantly outperforms isolated Fourier and Hermite networks, achieving a minimum point-wise R2 of 0.90 for velocity fields while demonstrating zero-shot scalability to grids with four times the training resolution. To validate engineering utility, the model was subjected to out-of-distribution blind tests on highly tortuous geometries generated via random Gaussian noise. FHKN successfully mapped early transient data to final steady-state fields, predicting macroscopic permeability within a 9% error margin and effectively bypassing the up to 300,000 iterations required for LBM convergence. This framework provides a physically-aligned, highly efficient feature-extraction method for modeling rarefied transport without relying on iterative spatial solvers.