  We propose a model-data asymptotic-preserving neural network(MD-APNN) method
to solve the nonlinear gray radiative transfer equations(GRTEs). The system is
challenging to be simulated with both the traditional numerical schemes and the
vanilla physics-informed neural networks(PINNs) due to the multiscale
characteristics. Under the framework of PINNs, we employ a micro-macro
decomposition technique to construct a new asymptotic-preserving(AP) loss
function, which includes the residual of the governing equations in the
micro-macro coupled form, the initial and boundary conditions with additional
diffusion limit information, the conservation laws, and a few labeled data. A
convergence analysis is performed for the proposed method, and a number of
numerical examples are presented to illustrate the efficiency of MD-APNNs, and
particularly, the importance of the AP property in the neural networks for the
diffusion dominating problems. The numerical results indicate that MD-APNNs
lead to a better performance than APNNs or pure data-driven networks in the
simulation of the nonlinear non-stationary GRTEs.
