Application of Deep Galerkin Method to Solve Compressible Navier-Stokes Equations

Abstract

Recently, the application of a deep-learning technique to fl uid analysis has been suggested. Additionally, a deep-learning-based method called the Deep Galerkin Method (DGM) has been suggested for solving a partial di ff erential equation. In DGM, a loss function for training a deep neural network is formulated so that di ff erential operators, boundary conditions, and initial conditions of the targeted partial di ff erential equation are satis fi ed. This study aims to extend and apply DGM to solving compressible Navier-Stokes equations and examine the feasibility of using DGM for fl uid analysis. In this paper, DGM is applied to two-dimensional Burgers equations with periodic boundary conditions, one-dimen-sional Navier-Stokes equations for a shock tube problem, and two-dimensional Navier-Stokes equations for the supersonic fl ow around a blunt body. The approximate solutions obtained using DGM show generally good agreement with that obtained using a fi nite di ff erence method.