RelaxNet: A structure-preserving neural network to approximate the Boltzmann collision operator

Abstract

The extremely high dimensionality and nonlinearity in the Boltzmann equation bring tremendous difficulties to the study of rarefied gas dynamics. This paper addresses a neural network-based surrogate model that provides a structure-preserving approximation for the fivefold collision integral. The notion originates from the similarity in structure between the BGK-type relaxation model and residual neural network (ResNet) when a particle distribution function is treated as the input to the neural network function. Therefore, we extend the ResNet architecture and construct what we call the relaxation neural network (RelaxNet). Specifically, two feed-forward neural networks with physics-informed connections and activations are introduced as building blocks in RelaxNet, which provide bounded and physically realizable approximations of the equilibrium distribution and velocity-dependent relaxation time respectively. The evaluation of the collision term is thus significantly accelerated due to the fact that the convolution in the fivefold integral is replaced by tensor multiplication in the neural network. We fuse the mechanical advection operator and the RelaxNet-based collision operator into a unified model named the universal Boltzmann equation (UBE). We prove that UBE preserves the key structural properties in a many-particle system, i.e., positivity, conservation, invariance, H-theorem, and correct fluid dynamic limit. These properties promise that RelaxNet is superior to strategies that naively approximate the right-hand side of the Boltzmann equation using a machine learning model. A novel sampling strategy based on closure hierarchies of the moment system of the Boltzmann equation is developed to generate reliable and unbiased sampling for the supervised learning. The construction of the RelaxNet-based UBE and its solution algorithm are demonstrated in detail. Several numerical experiments, where the ground-truth datasets are produced by the Shakhov model, velocity-dependent ν-BGK model, and the full Boltzmann equation, are investigated. The capability of the current approach for simulating non-equilibrium flow physics is validated through satisfactory in- and out-of-distribution performance. The open-source codes to reproduce the numerical results are available under the MIT license [1].