  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 with 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.
