The issue of the relaxation to equilibrium has been at the core of the kinetic theory of rarefied gas dynamics. In the paper, we introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. We impose the varied types of boundary conditions including the inflow-type and the reflection-type boundaries as well as the varied diffusion and friction coefficients and study the boundary effects on the asymptotic behaviors. These include the predictions on the large-time behaviors of the pointwise values of the particle distribution and the macroscopic physical quantities including the total kinetic energy, the entropy, and the free energy. We also provide the theoretical supports for the pointwise convergence of the neural network solutions to the a priori analytic solutions. We use the library PyTorch, the activation function tanh between layers, and the Adam optimizer for the Deep Learning algorithm.

弛豫平衡问题一直是稀薄气体动力学动力学理论的核心。在本文中，我们介绍了有界区间内动力学福克-普朗克方程的深度神经网络（DNN）近似解，并研究了解的大时间渐近行为以及其他物理相关的宏观量。我们施加不同类型的边界条件，包括流入型和反射型边界以及不同的扩散和摩擦系数，并研究边界对渐近行为的影响。其中包括对粒子分布的点值的大时间行为的预测以及宏观物理量（包括总动能、熵和自由能）。我们还为神经网络解与先验解析解的逐点收敛提供理论支持。我们使用PyTorch库、层间激活函数tanh以及用于深度学习算法的Adam优化器。