In this paper, we present two novel Asymptotic-Preserving Neural Networks (APNNs) for tackling multiscale time-dependent kinetic problems, encompassing the linear transport equation and Bhatnagar-Gross-Krook (BGK) equation with diffusive scaling. Our primary objective is to devise efficient and accurate APNN approaches for resolving multiscale kinetic equations. We have established a neural network based on even-odd decomposition and concluded that enforcing the initial condition for the linear transport equation with inflow boundary conditions is crucial. This APNN method based on even-odd parity relaxes the stringent conservation prerequisites while concurrently introducing an auxiliary deep neural network. Additionally, we have incorporated the conservation laws of mass, momentum, and energy for the Boltzmann-BGK equation into the APNN framework by enforcing exact boundary conditions. This is our second contribution. The most notable finding of this study is that approximating the zeroth, first and second moments of the particle density distribution is simpler than the distribution itself. Furthermore, a compelling phenomenon in the training process is that the convergence of density is swifter than that of momentum and energy. Finally, we investigate several benchmark problems to demonstrate the efficacy of our proposed APNN methods.

翻译：本文提出了两种新颖的渐近保持神经网络（APNNs），用于处理具有扩散尺度特性的多尺度含时动力学问题，涵盖线性输运方程和Bhatnagar-Gross-Krook（BGK）方程。主要目标是设计高效且精确的APNN方法以求解多尺度动力学方程。我们基于奇偶分解构建了神经网络，并得出重要结论：对于具有流入边界条件的线性输运方程，必须强制执行初始条件。这种基于奇偶对称性的APNN方法放宽了严格的守恒条件，同时引入了一个辅助深度神经网络。此外，我们将玻尔兹曼-BGK方程的质量、动量和能量守恒定律纳入APNN框架，通过强制执行精确边界条件来实现，这是本文的第二项贡献。本研究最显著的发现是：粒子密度分布零阶、一阶和二阶矩的近似比分布本身更易处理。在训练过程中还有一个引人注目的现象：密度的收敛速度比动量和能量更快。最后，我们通过多个基准问题验证了所提出APNN方法的有效性。