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：本文提出了两种新型渐近保持神经网络(APNNs)，用于处理含扩散缩放的多尺度时变动力学问题，涵盖线性输运方程和Bhatnagar-Gross-Krook (BGK)方程。我们的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组。基于奇偶分解理论构建了神经网络架构，并得出关键结论：对于含入流边界条件的线性输运方程，强制执行初始条件至关重要。这种基于奇偶性的APNN方法放宽了严格的守恒性要求，同时引入辅助深度神经网络。此外，我们通过强制执行精确边界条件，将玻尔兹曼-BGK方程的质量、动量和能量守恒律整合到APNN框架中——这是本文的第二项贡献。本研究最显著的发现是：粒子密度分布函数的零阶、一阶和二阶矩的近似难度低于分布函数本身。训练过程中还观察到有趣现象：密度的收敛速度快于动量和能量。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。