Physics-Informed Neural Networks for Weakly Compressible Flows Using Galerkin-Boltzmann Formulation

TL;DR

This paper introduces a novel physics-informed neural network approach using Galerkin-Boltzmann equations to effectively solve weakly compressible flow problems, demonstrating promising accuracy and efficiency on benchmark tests.

Contribution

It develops a new PINN framework based on Galerkin-Boltzmann equations with reduced output dimension, improving applicability to weakly compressible flows.

Findings

Accurate solutions for benchmark flow problems

Effective handling of scale disparity in collision terms

Potential for inverse problem applications

Abstract

In this work, we study the Galerkin-Boltzmann formulation within a physics-informed neural network (PINN) framework to solve flow problems in weakly compressible regimes. The Galerkin-Boltzmann equations are discretized with second-order Hermite polynomials in microscopic velocity space, which leads to a first-order conservation law with six equations. Reducing the output dimension makes this equation system particularly well suited for PINNs compared with the widely used D2Q9 lattice Boltzmann velocity space discretizations. We created two distinct neural networks to overcome the scale disparity between the equilibrium and non-equilibrium states in collision terms of the equations. We test the accuracy and performance of the formulation with benchmark problems and solutions for forward and inverse problems with limited data. Our findings show the potential of utilizing the Galerkin-Boltzmann formulation in PINN for weakly compressible flow problems.