%0 Journal Article
%T 神经网络求解Boltzmann-BGK方程及其在微流中的应用<br>A Neural Network Method for the Boltzmann-BGK Equation with Applications in Microflows
%A 严玲
%A 张佩
%J Advances in Applied Mathematics
%P 995-1006
%@ 2324-8009
%D 2025
%I Hans Publishing
%R 10.12677/aam.2025.144222
%X 本文提出一种基于神经网络的BGK方程求解方法&#65292;特别关注在微流问题中的应用。首先&#65292;通过引入灵活辅助分布函数构造BGK方程的降维模型&#65292;从而有效降低方程维度。其次&#65292;设计全连接神经网络架构高效逼近降维分布函数&#65292;以避免时空离散化。接着针对微流问题中复杂的Maxwell边界条件&#65292;提出特殊设计的损失函数进行处理。此外&#65292;利用多尺度输入策略和Maxwellian分裂技术以提升逼近效率。最后&#65292;通过对一维Couette流和二维矩形风管流两个经典问题进行数值实验&#65292;验证了该方法的有效性。<br>We consider the neural representation to solve the BGK equation, especially focusing on the application in microscopic flow problems. Firstly, a new dimension reduction model of the BGK equation with the flexible auxiliary distribution functions is deduced to reduce the problem dimension. Then, a fully connected neural network is utilized to approximate the dimension-reduced distribution with extremely high efficiency and to avoid discretization in space and time. A specially designed loss function is employed to deal with the Maxwell boundary conditions in microflow problems. Moreover, strategies such as multi-scale input and Maxwellian splitting are applied to further enhance the approximation efficiency. Finally, two classical numerical experiments, including one-dimensional Couette flow and two-dimensional duct flow, are studied to demonstrate the effectiveness of this neural representation method.
%K BGK方程&#65292
%K 降维&#65292
%K Maxwell边界条件&#65292
%K 神经网络<br>BGK Equation
%K Dimension Reduction
%K Maxwell Boundary Condition
%K Neural Network
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=113396