Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations (Communications in Computational Physics, 2024)
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 in allranges of Knudsen number. Our primary objective is to devise accurate APNN approaches for resolving multiscale kinetic equations, which is also efficient in the smallKnudsen number regime. The first APNN for linear transport equation is based oneven-odd decomposition, which relaxes the stringent conservation prerequisites whileconcurrently introducing an auxiliary deep neural network. We conclude that enforcing the initial condition for the linear transport equation with inflow boundaryconditions is crucial for this network. For the Boltzmann-BGK equation, the APNNincorporates the conservation of mass, momentum, and total energy into the APNNframework as well as exact boundary conditions. A notable finding of this study is thatapproximating the zeroth, first, and second moments—which govern the conservationof density, momentum, and energy for the Boltzmann-BGK equation, is simpler thanthe distribution itself. Another interesting phenomenon observed 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 ourproposed APNN methods.