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 Grad first proposed in 1949 [12] is among 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 [13, 19]. 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 smooth and discontinuous initial condition problems, Sod shock tube problem, and the shock structure problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.
翻译:作为动能理论的主要主导方程式之一,博尔茨曼方程式在航空航天、微观流等方程式中被广泛使用。它的高分辨率模拟在这些相关领域至关重要。然而,由于博尔茨曼方程式的高度维度,高分辨率模拟往往难以在数字上实现。1949年首次提出的格列[12]方法是获得高效高分辨率模拟的流行数字方法之一。我们可以从当时的方法中得出治理方程式,方法是在博尔茨曼方程式的两侧抽取时间,从而有效地减少问题的维度。然而,其中一项主要挑战是,它导致一个不闭合的时点系统,而需要关闭来获得一个闭合的时点系统。高分辨率模拟在1949年首次提出的时间方法[12]是用来实现高效高分辨率模拟的流行数字方法。除了传统的人类设计外,基于机器学习的方法最近[13]、19] 。在这项工作中,我们提出了一个基于机器学习的方法,为博尔茨曼-GK方程式的初始方程式的闭合点模型,在正轨平方程式的初始平方程式中,在深度平方程式的平方程式的平方程式中,以精确关系关系关系关系关系上显示一个深度平的平的平方程式的平局关系。在深度平的平的平方程式的平方程式的平方程式的平方程式的平方程式的平方程式的平方程式的平方程式的平方程式的平差关系,在反映的平方程式的平方程式的平方程式的平方程式的平方程式关系,在在在在在在深度平方程式的平方程式的平方程式的平方程式的平方程式的平方程式的平方程式的平方程式的平面关系中,在深度关系上,在反映的平方程式的平方程式的平方关系上显示的平方程式的平面关系上,在精确的平方关系上显示了。