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（APNN），用于处理多尺度含时动力学问题，涵盖具有扩散标度的线性输运方程和Bhatnagar-Gross-Krook（BGK）方程。我们的核心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，并得出结论：对具有流入边界条件的线性输运方程强制施加初始条件至关重要。这种基于奇偶性的APNN方法放宽了严格的守恒要求，同时引入了一个辅助深度神经网络。此外，我们通过强制执行精确边界条件，将Boltzmann-BGK方程的质量、动量和能量守恒定律纳入APNN框架——这是本文的第二项贡献。本研究最显著的发现是，对粒子密度分布的零阶、一阶和二阶矩进行逼近比逼近分布本身更为简便。训练过程中另一个值得关注的现象是：密度的收敛速度快于动量与能量的收敛速度。最后，我们通过若干基准算例验证了所提出的APNN方法的有效性。