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.

Kinetic modeling of polyatomic heat and mass transfer in rectangular microchannels

“…In addition, the temperature and pressure driven flows in micro-channels have also been studied based on the regularized 13-moment equations flows in the transition regime and slip regime [23,24]. Very recently, in [25], the monatomic temperature driven flow in a plane channel has been simulated by applying physics-informed neural network (PINNs)-based approach showing a very good agreement with the available numerical data in the literature. Pressure and temperature driven gaseous flows through microchannels have also been studied experimentally [26][27][28][29][30][31][32].…”

Physics-Informed Neural Networks and Functional Interpolation for Data-Driven Parameters Discovery of Epidemiological Compartmental Models

“…Primarily, TFC is used for the solution of DEs because the CEs eliminate the "curse of the equation constraints" [39][40][41]. Moreover, TFC has already been used to solve different classes of optimal control space guidance problems such as energy optimal landing on large and small planetary bodies [42,43], fuel optimal landing on large planetary bodies [44], energy optimal relative motion problems subject to Clohessy-Wiltshire dynamics [45], and classes of transport theory problems, such as radiative transfer [46] and rarefied-gas dynamics [47]. For tackling DEs, the standard (or Vanilla as defined in [48,49]) TFC method employs a linear combination of orthogonal polynomials, such as Legendre or Chebyshev polynomials [39,40], as a free function.…”

Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley–Leverett problem

“…PINNs are trained to solve supervised learning tasks constrained by PDEs, such as the conservation laws in continuum theories of fluid and solid mechanics 16 , 22 – 24 . In addition to fluid and solid mechanism, PINNs have been used to solve a big amount of applications governed by differential equations such as radioactive transfer 25 , 26 , gas dynamics 27 , 28 , water dynamics 29 , Euler equation 30 , 31 , numerical integration 32 , chemical kinetics 33 , 34 and optimal control 35 , 36 .…”