A Model-Constrained Discontinuous Galerkin Network (DGNet) for Compressible Euler Equations with Out-Of-Distribution Generalization

Paper publication

Publication Link: https://www.sciencedirect.com/science/article/pii/S0045782525001847

Citation:

@article{van2025model,
  title={A model-constrained discontinuous Galerkin Network (DGNet) for compressible Euler equations with out-of-distribution generalization},
  author={Van Nguyen, Hai and Chen, Jau-Uei and Bui-Thanh, Tan},
  journal={Computer Methods in Applied Mechanics and Engineering},
  volume={440},
  pages={117912},
  year={2025},
  publisher={Elsevier}
}

Methodology

In this work, we developed a machine learning framework to solve shock-type PDEs, in particular, Compressible Euler equations. The core idea is motivated by the dual mesh between Discontinuous Galerkin (DG) method and Graph Neural Network (GNN).

Figure 1: The dual mesh Discontinuous Galerkin (DG) method and Graph Neural Network (GNN).

The training data flow is presented as shown in Figure 2. We integrate the data randomization and differentiable solvers to enhance the generalization of neural surrogate models.

Figure 2: The schematic of DGNet network architecture.

Numerical results

We work on 2D Euler equations

\[\begin{align*} \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} &= 0 \\ \frac{\partial (\rho u)}{\partial t} + \frac{\partial (\rho u^2 + p)}{\partial x} + \frac{\partial (\rho u v)}{\partial y} &= 0 \\ \frac{\partial (\rho v)}{\partial t} + \frac{\partial (\rho u v)}{\partial x} + \frac{\partial (\rho v^2 + p)}{\partial y} &= 0 \\ \frac{\partial E}{\partial t} + \frac{\partial (u(E + p))}{\partial x} + \frac{\partial (v(E + p))}{\partial y} &= 0, \end{align*}\]

where \(E\) is the total energy per unit volume:

\[E = \frac{p}{\gamma - 1} + \frac{\rho}{2}(u^2 + v^2)\]

Problem 1. Airfoil NACA0012

Training data is generated from Airfoil AoA = 3 and Mach = 0.8, in time interval [0,1.2]s
Test data is generated from Airfoil AoA = 3 and Mach = 0.8 and Airfoil AoA = 5 and Mach = 1.2 for time interval [0, 7.5]s

Figure 3: (Airfoil) Airfoil configuration AoA-3.

Figure 4: (Airfoil) predictions by DGNet for Airfoil NACA0012 of AoA = 5 and Mach = 1.2.

Problem 2. Euler configurations 6 & 12

Training data is generated from Euler configuration 6 with time interval [0,0.16]s
Test data is generated from Euler configuration 6 for time interval [0, 0.8]s and Euler configuration 12 for time interval [0, 0.25]s