Asymptotic-Preserving Neural Networks for Multiscale Kinetic Equations

TL;DR

This paper introduces two innovative Asymptotic-Preserving Neural Networks (APNNs) designed to efficiently solve multiscale kinetic equations, incorporating novel techniques like even-odd decomposition and conservation law enforcement.

Contribution

The paper develops two APNN methods for multiscale kinetic equations, integrating even-odd decomposition and conservation laws, and demonstrates their effectiveness on benchmark problems.

Findings

Approximating moments is easier than the full distribution.

Density converges faster than momentum and energy during training.

The methods perform well on benchmark problems.

Abstract

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.