Physics-Informed Neural Networks for rarefied-gas dynamics: Poiseuille flow in the BGK approximation

We present a new accurate approach to solving a class of problems in the theory of rarefied–gas dynamics using a Physics-Informed Neural Networks framework, where the solution of the problem is approximated by the constrained expressions introduced by the Theory of Functional Connections. The constrained expressions are made by a sum of a free function and a functional that always analytically satisfies the equation constraints. The free function used in this work is a Chebyshev neural network trained via the extreme learning machine algorithm. The method is designed to accurately and efficiently solve the linear one-point boundary value problem that arises from the Bhatnagar–Gross–Krook model of the Poiseuille flow between two parallel plates for a wide range of Knudsen numbers. The accuracy of our results is validated via the comparison with the published benchmarks.