We introduce a hyperbolic closure for the Grad moment expansion of the
Bhatnagar-Gross-Krook's (BGK) kinetic model using a neural network (NN) trained
on BGK's moment data. This closure is motivated by the exact closure for the
free streaming limit that we derived in our paper on closures in transport
\cite{Huang2022-RTE1}. The exact closure relates the gradient of the highest
moment to the gradient of four lower moments. As with our past work, the model
presented here learns the gradient of the highest moment in terms of the
coefficients of gradients for all lower ones. By necessity, this means that the
resulting hyperbolic system is not conservative in the highest moment. For
stability, the output layers of the NN are designed to enforce hyperbolicity
and Galilean invariance. This ensures the model can be run outside of the
training window of the NN. Unlike our previous work on radiation transport that
dealt with linear models, the BGK model's nonlinearity demanded advanced
training tools. These comprised an optimal learning rate discovery, one cycle
training, batch normalization in each neural layer, and the use of the
\texttt{AdamW} optimizer. To address the non-conservative structure of the
hyperbolic model, we adopt the FORCE numerical method to achieve robust
solutions. This results in a comprehensive computing model combining learned
closures with methods for solving hyperbolic models. The proposed model can
capture accurate moment solutions across a broad spectrum of Knudsen numbers.
Our paper details the multi-scale model construction and is run on a range of
test problems.

我们引入一个神经网络在 Bhatnagar-Gross-Krook（BGK）动力学模型的 Grad 矩展开上训练得到的超协议封闭。该模型能够有效捕捉一系列 Knudsen 数下的矩解，并采用了多尺度模型构建和 FORCE 数值方法以解决超协议模型的非守恒性。