In this paper, we construct an asymptotic-preserving neural networks (APNNs) [21] for the linearized Boltzmann equation in the acoustic scaling and with uncertain parameters. Utilizing the micro-macro decomposition, we design the loss function based on the stochastic-Galerkin system conducted from the micro-macro equations. Rigorous analysis is provided to show the capability of neural networks in approximating solutions near the global Maxwellian. By employing hypocoercivity techniques, we demonstrate two key results: the existence of APNNs when the loss function approaches zero, and the convergence of the APNN approximated solution as the loss tends to zero, with the error exhibiting an exponential decay in time.
Spectral Gap Computations for Linearized Boltzmann Operators
The Institute for Computational Engineering and Sciences（计算工程与科学研究所）；Department of Mathematics（数学系）；ICES（计算工程与科学研究所）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Convergence Analysis of the Grad's Hermite Approximation to the Boltzmann Equation
在（Commun Pure Appl Math 2（4）：331-407，1949）中，Grad提出了一种Hermite级数展开，用于近似具有无界速度空间的动力学方程的解。然而，对于初边值问题，不恰当地施加边界条件会导致Grad的Hermite展开的不稳定性，这可能导致非收敛解。对于线性动力学方程，最近提出了一种（形式上）任意阶Hermite逼近的稳定边界条件的构造方法。在目前的工作中，我们研究了这些稳定的Hermite逼近的$L^2$ -收敛性，并证明了明确的收敛速度在适当的正则性假设的精确解。通过对稀薄气体动力学线性化BGK方程的数值实验，我们证实了所提出的收敛速度。
A vectorial lattice Boltzmann scheme for the incompressible Navier-Stokes equations
Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学）；Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C & Fédération de Mathématiques de CentraleSupélec（巴黎萨克雷大学）；Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche（意大利国家研究委员会计算应用研究所）；Laboratoire J. A. Dieudonné, Université Côte d’Azur, CNRS（蔚蓝海岸大学）；Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Università di Roma（罗马第一大学）
