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Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation

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Abstract

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high dimensionality of the Boltzmann equation, high-resolution simulations are often difficult to achieve numerically. The 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 simulations. We 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 problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other 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 equation. In 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 equation. Numerical 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.

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Acknowledgements

We thank Prof. Ruo Li from Peking University, and Dr. Jiequn Han from Flatiron Institute for their valuable suggestions. Zhengyi Li and Bin Dong are supported in part by Natural Science Foundation of Beijing Municipality (No. 180001) and National Natural Science Foundation of China (Grant No. 12090022). Yanli Wang is supported by the National Natural Science Foundation of China (Grant No. 12171026, U1930402 and 12031013).

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, People’s Republic of China

    Zhengyi Li

  2. Beijing International Center for Mathematical Research & Center for Machine Learning Research, Peking University, Beijing, People’s Republic of China

    Bin Dong

  3. Beijing Computational Science Research Center, Beijing, People’s Republic of China

    Yanli Wang

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  1. Zhengyi Li
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  2. Bin Dong
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Correspondence to Yanli Wang.

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Appendix

Appendix

In the appendix, we will provide the detailed initial condition of the first sample in the wave and mix problem.

The detailed initial condition for the first sample in Sect. 4.2 is in Table 10, the numerical results of which are plotted in Fig. 3. The detailed initial condition for the first sample in Sect. 4.3 is listed in Table 11, the numerical results of which are plotted in Fig. 7.

Table 10 The initial condition of the initial condition sample for the wave problem
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Table 11 The initial condition of the initial condition sample for the mix problem
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Li, Z., Dong, B. & Wang, Y. Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation. Commun. Math. Stat. 11, 59–101 (2023). https://doi.org/10.1007/s40304-022-00331-5

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