Learning Invariance Preserving Moment Closure Model for Boltzmann-BGK Equation

作     者：Zhengyi Li Bin Dong Yanli Wang 

作者机构：School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China Beijing International Center for Mathematical Research&Center for Machine Learning ResearchPeking UniversityBeijingPeople’s Republic of China Beijing Computational Science Research CenterBeijingPeople’s Republic of China 

出 版 物：《Communications in Mathematics and Statistics》 (数学与统计通讯（英文）)

年 卷 期：2023年第11卷第1期

页      面：59-101页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：supported in part by Natural Science Foundation of BeijingMunicipality(No.180001) National Natural Science Foundation of China(Grant No.12090022) supported by the National Natural Science Foundation of China(Grant No.12171026,U1930402 and 12031013) 

主　　题：Boltzmann equation Moment closure Machine learning Neural networks Invariance preserving 

摘      要：As one of the main governing equations in kinetic theory,the Boltzmann equation is widely utilized in aerospace,microscopic flow,*** high-resolution simulation is crucial in these related ***,due to the high dimensionality of the Boltzmann equation,high-resolution simulations are often difficult to achieve *** moment method which was first proposed in Grad(Commun Pure Appl Math 2(4):331-407,1949)is among the popular numerical methods to achieve efficient high-resolution *** can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation,which effectively reduces the dimensionality of the ***,one of themain challenges is that it leads to an unclosed moment system,and closure is needed to obtain a closedmoment *** is truly an art in designing closures for moment systems and has been a significant research field in kinetic *** than the traditional human designs of closures,the machine learning-based approach has attracted much attention lately in Han et al.(Proc Natl Acad Sci USA 116(44):21983-21991,2019)and Huang et al.(J Non-Equilib Thermodyn 46(4):355-370,2021).In this work,we propose a machine learning-based method to derive a moment closure model for the Boltzmann-BGK *** particular,the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances,i.e.,the Galilean invariance,reflecting invariance,and scaling invariance,inherited from the original Boltzmann-BGK equation and playing an important role in the correct simulation of the Boltzmann *** simulations on the 1D-1D examples including the smooth and discontinuous initial condition problems,Sod shock tube problem,the shock structure problems,and the 1D-3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method