A Physics-Informed Neural Network Based on the Boltzmann Equation with Multiple-Relaxation-Time Collision Operators

Abstract

The Boltzmann equation with multiple-relaxation-time (MRT) collision operators has been widely employed in kinetic theory to describe the behavior of gases and liquids at the macro-level. Given the successful development of deep learning and the availability of data analytic tools, it is a feasible idea to try to solve the Boltzmann-MRT equation using a neural network-based method. Based on the canonical polyadic decomposition, a new physics-informed neural network describing the Boltzmann-MRT equation, named the network for MRT collision (NMRT), is proposed in this paper for solving the Boltzmann-MRT equation. The method of tensor decomposition in the Boltzmann-MRT equation is utilized to combine the collision matrix with discrete distribution functions within the moment space. Multiscale modeling is adopted to accelerate the convergence of high frequencies for the equations. The micro–macro decomposition method is applied to improve learning efficiency. The problem-dependent loss function is proposed to balance the weight of the function for different conditions at different velocities. These strategies will greatly improve the accuracy of the network. The numerical experiments are tested, including the advection–diffusion problem and the wave propagation problem. The results of the numerical simulation show that the network-based method can obtain a measure of accuracy at O10−3.

Keywords

• deep neural networks
• Boltzmann equation
• multiple-relaxation-time model
• canonical polyadic decomposition