The Boltzmann equation with the Bhatnagar-Gross-Krook collision model (Boltzmann-BGK equation) has been widely employed to describe multiscale flows, i.e., from the hydrodynamic limit to free molecular flow. In this study, we employ physics-informed neural networks (PINNs) to solve forward and inverse problems via the Boltzmann-BGK formulation (PINN-BGK), enabling PINNs to model flows in both the continuum and rarefied regimes. In particular, the PINN-BGK is composed of three sub-networks, i.e., the first for approximating the equilibrium distribution function, the second for approximating the non-equilibrium distribution function, and the third one for encoding the Boltzmann-BGK equation as well as the corresponding boundary/initial conditions. By minimizing the residuals of the governing equations and the mismatch between the predicted and provided boundary/initial conditions, we can approximate the Boltzmann-BGK equation for both continuous and rarefied flows. For forward problems, the PINN-BGK is utilized to solve various benchmark flows given boundary/initial conditions, e.g., Kovasznay flow, Taylor-Green flow, cavity flow, and micro Couette flow for Knudsen number up to 5. For inverse problems, we focus on rarefied flows in which accurate boundary conditions are difficult to obtain. We employ the PINN-BGK to infer the flow field in the entire computational domain given a limited number of interior scattered measurements on the velocity without using the (unknown) boundary conditions. Results for the two-dimensional micro Couette and micro cavity flows with Knudsen numbers ranging from 0.1 to 10 indicate that the PINN-BGK can infer the velocity field in the entire domain with good accuracy. Finally, we also present some results on using transfer learning to accelerate the training process. Specifically, we can obtain a three-fold speedup compared to the standard training process (e.g., Adam plus L-BFGS-B) for the two-dimensional flow problems considered in our work.

带有 Bhatnagar-Gross-Krook 碰撞模型（Boltzmann-BGK 方程）的 Boltzmann 方程已被广泛用于描述多尺度流动，即从流体动力学极限到自由分子流动。在这项研究中，我们采用物理信息神经网络 (PINN) 通过 Boltzmann-BGK 公式 (PINN-BGK) 解决正向和逆向问题，使 PINN 能够对连续介质和稀薄状态中的流动进行建模。特别地，PINN-BGK由三个子网络组成，即第一个用于逼近平衡分布函数，第二个用于逼近非平衡分布函数，第三个用于编码Boltzmann-BGK方程作为相应的边界/初始条件。通过最小化控制方程的残差以及预测和提供的边界/初始条件之间的不匹配，我们可以近似用于连续和稀薄流动的 Boltzmann-BGK 方程。对于前向问题，PINN-BGK 用于解决给定边界/初始条件的各种基准流，例如，科瓦兹奈流、泰勒-格林流、腔流和微库埃特流，克努森数高达 5。对于逆问题，我们专注于难以获得精确边界条件的稀薄流动。我们使用 PINN-BGK 来推断整个计算域中的流场，在不使用（未知）边界条件的情况下，给定数量有限的内部散射速度测量值。二维微库埃特和克努森数为 0.1 到 10 的微腔流的结果表明，PINN-BGK 可以很好地推断整个域中的速度场。最后，我们还介绍了使用迁移学习来加速训练过程的一些结果。具体来说，对于我们工作中考虑的二维流问题，与标准训练过程（例如，Adam 加 L-BFGS-B）相比，我们可以获得三倍的加速。