Neural network representation of microflows with BGK model

TL;DR

This work considers the neural representation to solve the Boltzmann-BGK equation, especially focusing on the application in microscopic flow problems, and proposes a network-based ansatz that can approximate the dimension-reduced distribution with extremely high efficiency.

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.

Figures (16)

• Figure 1: Schematic of DRNR for solving the Boltzmann-BGK equation. (a) Sketch of DRNR. In the neural network, the inputs are the spatial space $\boldsymbol{x}$ and time $t$. The shaded regions indicate the architecture of the neural networks $g^{\text{NN}}_{\theta},h^{\text{NN}}_{\theta},s^{\text{NN}}_{j,\theta}$, which are used to approximate the dimension-reduced distribution functions ${\bm g}, {\bm h}$, and ${\bm s}_j$. The detailed structure of the neural networks in the shaded region is shown in Fig. \ref{['fig:net_arch']}. The loss function contains three parts the initial condition (IC), boundary condition (BC), and residual of PDE (Eq). (b) Network architecture. For the discrete dimension-reduced distribution ${\bm g}, {\bm h}$, and ${\bm s}_j$, each of them corresponds to two separate neural networks, and has its pseudo macro variables. The multi-scale inputs and the Maxwellian splitting are utilized to improve the approximation efficiency.
• Figure 2: (1D Couette flow in Sec \ref{['sec:couette']}) Numerical solution of the density $\rho$, macroscopic velocity in $y-$axis $u_2$, the temperature $T$ and the heat flux $q_1$ of the Couette flow at steady state for ${\rm Kn} = 0.1, 1$ and $2.5$. Here, the left wall velocity $\boldsymbol{u}^W_L = (0,-0.5,0)$ and the right wall velocity $\boldsymbol{u}^W_R = (0,0.5,0)$. The black lines are the numerical solution obtained by DRNR, and the dashed red lines represent the reference solution obtained by DVM.
• Figure 3: (1D Couette flow in Sec \ref{['sec:couette']}) Numerical solution of the density $\rho$, macroscopic velocity in $y-$axis $u_2$, the temperature $T$ and the heat flux $q_1$ of the Couette flow at steady state for ${\rm Kn} = 0.1, 1$ and $2.5$. Here, the left wall velocity $\boldsymbol{u}^W_L = (0,-1,0)$ and the right wall velocity $\boldsymbol{u}^W_R = (0,1,0)$. The black lines are the numerical solution obtained by DRNR, and the dashed red lines represent the reference solution obtained by DVM.
• Figure 4: (1D Couette flow in Sec \ref{['sec:couette']}) Numerical solution of the density $\rho$, macroscopic velocity in $y-$axis $u_2$, the temperature $T$ and the heat flux $q_1$ of the Couette flow at steady state for ${\rm Kn} = 0.1, 1$ and $2.5$. Here, the left wall velocity $\boldsymbol{u}^W_L = (0,-2,0)$ and the right wall velocity $\boldsymbol{u}^W_R = (0,2,0)$. The black lines are the numerical solution obtained by DRNR, and the dashed red lines represent the reference solution obtained by DVM.
• Figure 5: (The variant of Couette flow in Sec \ref{['sec:couette']}) Numerical solution of the density $\rho$, macroscopic velocity $u_2, u_3$, the temperature $T$, the shear stress $\sigma_{13}$ and the heat flux $q_1$ of the variant of Couette flow at steady state for ${\rm Kn} = 0.1, 1$ and $2.5$. Here, the left wall velocity $\boldsymbol{u}^W_L = (0,1,0)$ and the right wall velocity $\boldsymbol{u}^W_R = (0,0,1)$. The black lines are the numerical solution obtained by DRNR, and the dashed red lines represent the reference solution obtained by DVM.

Theorems & Definitions (2)

• Remark 1
• Remark 2