Physics-informed neural networks for rarefied-gas dynamics: Thermal creep flow in the Bhatnagar–Gross–Krook approximation

Abstract

This work aims at accurately solve a thermal creep flow in a plane channel problem, as a class of rarefied-gas dynamics problems, using Physics-Informed Neural Networks (PINNs). We develop a particular PINN framework where the solution of the problem is represented by the Constrained Expressions (CE) prescribed by the recently introduced Theory of Functional Connections (TFC). CEs are represented by a sum of a free-function and a functional (e.g., function of functions) that analytically satisfies the problem constraints regardless to the choice of the free-function. The latter is represented by a shallow Neural Network (NN). Here, the resulting PINN-TFC approach is employed to solve the Boltzmann equation in the Bhatnagar–Gross–Krook approximation modeling the Thermal Creep Flow in a plane channel. We test three different types of shallow NNs, i.e., standard shallow NN, Chebyshev NN (ChNN), and Legendre NN (LeNN). For all the three cases the unknown solutions are computed via the extreme learning machine algorithm. We show that with all these networks we can achieve accurate solutions with a fast training time. In particular, with ChNN and LeNN we are able to match all the available benchmarks.