  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.
