Learning a generalized Navier-Stokes model beyond the continuum regime
Abstract
Navier-Stokes (NS) equations lose accuracy in the transition regime when the Knudsen number is large, while the Boltzmann equation provides an adequate kinetic description with a substantially higher computational cost. In this work, a data-driven generalized Navier-Stokes (GNS) model is proposed to extend the applicability of the NS equations to the transition regime. A new relationship between the non-equilibrium variables, such as the stress tensor and the heat flux, and the equilibrium variables, including the density, macroscopic velocity, and the temperature, together with the Knudsen number, is learned from the data formed by the numerical solution to the Boltzmann equation. The standard end-to-end learning is utilized to construct the loss function, based on which, a long-term end-to-end learning method is proposed for the loss function to reduce accumulated error and improve long-term predictive accuracy. Several numerical examples, including one-dimensional wave and Riemann problems as well as two-dimensional isentropic vortex and Taylor-Green vortex problems, are studied to validate the efficiency of this new GNS model.
Related
- Weighted Lq-Lr estimates of a generalized Oseen evolution operator in exterior domains, with applications to the Navier-Stokes flow past a rigid body
- Generalized Logarithmic Sobolev inequalities on Bounded Domains and exponential decay to Boltzmann states for Nernst--Planck--Navier--Stokes systems
- Beyond the L9/5 Vorticity Criterion in the Stationary Navier--Stokes Liouville Problem
- Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method
- Autoencoders vs. Numerical Analysis--Informed Manifold Learning for Navier--Stokes Flows
BibTeX
@misc{shao2026learning,
title = {{Learning a generalized Navier-Stokes model beyond the continuum regime}},
author = {Sihong Shao and Yanli Wang and Zhongwei Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07952},
eprint = {2610.07952},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07952},
note = {CC0},
}
RIS
TY - UNPB
TI - Learning a generalized Navier-Stokes model beyond the continuum regime
AU - Sihong Shao
AU - Yanli Wang
AU - Zhongwei Xu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07952
UR - https://arxiv.org/abs/2610.07952
AB - Navier-Stokes (NS) equations lose accuracy in the transition regime when the Knudsen number is large, while the Boltzmann equation provides an adequate kinetic description with a substantially higher computational cost. In this work, a data-driven generalized Navier-Stokes (GNS) model is proposed to extend the applicability of the NS equations to the transition regime. A new relationship between the non-equilibrium variables, such as the stress tensor and the heat flux, and the equilibrium variables, including the density, macroscopic velocity, and the temperature, together with the Knudsen number, is learned from the data formed by the numerical solution to the Boltzmann equation. The standard end-to-end learning is utilized to construct the loss function, based on which, a long-term end-to-end learning method is proposed for the loss function to reduce accumulated error and improve long-term predictive accuracy. Several numerical examples, including one-dimensional wave and Riemann problems as well as two-dimensional isentropic vortex and Taylor-Green vortex problems, are studied to validate the efficiency of this new GNS model.
ER -