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），用于求解具有扩散缩放的多尺度时间依赖动力学问题，包括线性输运方程和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，以解决多尺度动力学方程。我们基于奇偶分解建立了一个神经网络，并得出结论：对于具有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要求，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、动量和能量守恒律通过强制执行精确边界条件整合到APNN框架中。这是我们的第二个贡献。本研究最显著的发现是：近似粒子密度分布的零阶、一阶和二阶矩比近似分布本身更为简单。此外，训练过程中一个引人注目的现象是：密度收敛速度比动量和能量更快。最后，我们通过若干基准问题验证了所提出的APNN方法的有效性。