Learning a generalized Navier-Stokes model beyond the continuum regime

Sihong Shao; Yanli Wang; Zhongwei Xu

Home/Archive/arXiv

arXiv · 2610.07952

Learning a generalized Navier-Stokes model beyond the continuum regime

Sihong Shao·Yanli Wang·Zhongwei Xu

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.NA cs.NA physics.flu-dyn

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sihong Shao, Yanli Wang, Zhongwei Xu. 2026-10-06. Learning a generalized Navier-Stokes model beyond the continuum regime. https://arxiv.org/abs/2610.07952

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A mixed Petrov-Galerkin Cosserat rod finite element formulation

This paper presents a total Lagrangian mixed Petrov-Galerkin finite element formulation that provides a computationally efficient approach for analyzing Cosserat rods that is free of singularities and locking. To achieve a singularity-free orientation parametrization of the rod, the nodal kinematical unknowns are defined as the nodal centerline positions and unit quaternions. We apply Lagrange interpolation to all nodal kinematic coordinates, and in combination with a projection of non-unit quaternions, this leads to an interpolation with orthonormal cross-section-fixed bases. To eliminate locking effects such as shear locking, the variational Hellinger-Reissner principle is applied, resulting in a mixed approach with additional fields composed of resultant contact forces and moments. Since the mixed formulation contains the constitutive law in compliance form, it naturally incorporates constrained theories, such as the Kirchhoff-Love theory. This study specifically examines the influence of the additional internal force fields on the numerical performance, including locking mitigation and robustness. Using well-established benchmark examples, the method demonstrates enhanced computational robustness and efficiency, as evidenced by the reduction in required load steps and iterations when applying the standard Newton-Raphson method.

math.NA↗

arXiv2026

Beyond the summation-by-parts property: nullspace consistency, sparsity, and regularization for FSBP operators

We investigate the construction and performance of summation-by-parts (SBP) operators, which offer a powerful framework for the systematic development of structure-preserving numerical discretizations of partial differential equations. Previous approaches for the construction of SBP operators have usually relied on either local methods or sparse differentiation matrices, as commonly used in finite difference schemes. However, these methods often impose implicit requirements that are not part of the formal SBP definition. We demonstrate that adherence to the SBP definition alone does not guarantee the desired accuracy, and we make additional conditions explicit that SBP operators need to satisfy in order to achieve accuracy. While these conditions are known in the SBP literature, they are usually enforced only implicitly by the respective construction procedure. Specifically, we analyze the error minimization for an augmented basis, discuss the role of sparsity, and examine the importance of nullspace consistency in the construction of SBP operators. A dispersion and dissipation analysis shows that the loss of accuracy has two sources. First, a lack of nullspace consistency produces stationary modes, which are neither transported nor damped by the scheme and whose contribution to the error converges more slowly under mesh refinement. Second, the physical mode, i.e., the discrete approximation of an exact traveling wave, can be poorly resolved, and the resulting error dominates in long-time simulations. Furthermore, we show how these design criteria can be integrated into a recently proposed optimization-based construction procedure for function space SBP (FSBP) operators on arbitrary grids. Our findings are supported by numerical experiments that illustrate the improved accuracy of the numerical solutions obtained with the proposed SBP operators.

math.NA↗

arXiv2026

A discontinuous Galerkin method for elliptic-hyperbolic equations: the Tricomi problem

We present and analyze a discontinuous Galerkin method for the numerical solution of a class of second-order linear mixed-type partial differential equations, i.e. equations that change their nature from elliptic to hyperbolic through the computational domain. Well-posedness of the discrete problem is established via coercivity in an energy norm, achieved through the Morawetz multiplier technique. We derive $hp$-a priori error estimates in the energy norm, which we use to prove convergence rates for standard and quasi-Trefftz polynomial spaces. Numerical experiments validate the theoretical results.

math.NA↗