Neural network representation of microflows with BGK model

TL;DR

This paper introduces a neural network approach to efficiently solve the Boltzmann-BGK equation for microscopic flow problems by reducing problem dimension and employing specialized network strategies.

Contribution

A novel dimension reduction model combined with a neural network ansatz and tailored loss functions for microscopic flow simulations.

Findings

High accuracy in 1D and 2D flow problems

Effective handling of Maxwell boundary conditions

Significant computational efficiency improvements

Abstract

We consider the neural representation to solve the Boltzmann-BGK equation, especially focusing on the application in microscopic flow problems. A new dimension reduction model of the BGK equation with the flexible auxiliary distribution functions is first deduced to reduce the problem dimension. Then, a network-based ansatz that can approximate the dimension-reduced distribution with extremely high efficiency is proposed. Precisely, fully connected neural networks are utilized to avoid discretization in space and time. A specially designed loss function is employed to deal with the complex Maxwell boundary in microscopic flow problems. Moreover, strategies such as multi-scale input and Maxwellian splitting are applied to enhance the approximation efficiency further. Several classical numerical experiments, including 1D Couette flow and Fourier flow problems and 2D duct flow and in-out flow problems are studied to demonstrate the effectiveness of this neural representation method.