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

Hai V. Nguyen, Jau-Uei Chen, Tan Bui-Thanh

TL;DR

DGNet integrates a model-constrained, graph-based learning framework with high-order discontinuous Galerkin discretizations to solve the compressible Euler equations. By representing the DG mesh as a dual graph and learning both a volume-correcting network and a Riemann-flux network, it achieves temporal-discretization invariance, conservation adherence, and mesh-discretization generalization. A key contribution is the data randomization strategy that regularizes the tangent-slope surrogate and implicitly aligns derivatives up to second order with the DG discretization, boosting long-term stability and out-of-distribution performance. Extensive 1D and 2D benchmarks (including shocks, vortices, and hypersonic flows) demonstrate competitive accuracy with classical DG while offering significant speedups and robust shock-capturing, even on unseen geometries. The work lays the groundwork for scalable 3D applications and potential acceleration through graph-encoded representations, enabling real-time or digital-twin style simulations for complex flows.

Abstract

Real-time accurate solutions of large-scale complex dynamical systems are critically needed for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications, particularly in digital twin contexts. In this work, we develop a model-constrained discontinuous Galerkin Network (DGNet) approach, a significant extension to our previous work [Model-constrained Tagent Slope Learning Approach for Dynamical Systems], for compressible Euler equations with out-of-distribution generalization. The core of DGNet is the synergy of several key strategies: (i) leveraging time integration schemes to capture temporal correlation and taking advantage of neural network speed for computation time reduction; (ii) employing a model-constrained approach to ensure the learned tangent slope satisfies governing equations; (iii) utilizing a GNN-inspired architecture where edges represent Riemann solver surrogate models and nodes represent volume integration correction surrogate models, enabling capturing discontinuity capability, aliasing error reduction, and mesh discretization generalizability; (iv) implementing the input normalization technique that allows surrogate models to generalize across different initial conditions, geometries, meshes, boundary conditions, and solution orders; and (v) incorporating a data randomization technique that not only implicitly promotes agreement between surrogate models and true numerical models up to second-order derivatives, ensuring long-term stability and prediction capacity, but also serves as a data generation engine during training, leading to enhanced generalization on unseen data. To validate the effectiveness, stability, and generalizability of our novel DGNet approach, we present comprehensive numerical results for 1D and 2D compressible Euler equation problems.

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

TL;DR

DGNet integrates a model-constrained, graph-based learning framework with high-order discontinuous Galerkin discretizations to solve the compressible Euler equations. By representing the DG mesh as a dual graph and learning both a volume-correcting network and a Riemann-flux network, it achieves temporal-discretization invariance, conservation adherence, and mesh-discretization generalization. A key contribution is the data randomization strategy that regularizes the tangent-slope surrogate and implicitly aligns derivatives up to second order with the DG discretization, boosting long-term stability and out-of-distribution performance. Extensive 1D and 2D benchmarks (including shocks, vortices, and hypersonic flows) demonstrate competitive accuracy with classical DG while offering significant speedups and robust shock-capturing, even on unseen geometries. The work lays the groundwork for scalable 3D applications and potential acceleration through graph-encoded representations, enabling real-time or digital-twin style simulations for complex flows.

Abstract

Real-time accurate solutions of large-scale complex dynamical systems are critically needed for control, optimization, uncertainty quantification, and decision-making in practical engineering and science applications, particularly in digital twin contexts. In this work, we develop a model-constrained discontinuous Galerkin Network (DGNet) approach, a significant extension to our previous work [Model-constrained Tagent Slope Learning Approach for Dynamical Systems], for compressible Euler equations with out-of-distribution generalization. The core of DGNet is the synergy of several key strategies: (i) leveraging time integration schemes to capture temporal correlation and taking advantage of neural network speed for computation time reduction; (ii) employing a model-constrained approach to ensure the learned tangent slope satisfies governing equations; (iii) utilizing a GNN-inspired architecture where edges represent Riemann solver surrogate models and nodes represent volume integration correction surrogate models, enabling capturing discontinuity capability, aliasing error reduction, and mesh discretization generalizability; (iv) implementing the input normalization technique that allows surrogate models to generalize across different initial conditions, geometries, meshes, boundary conditions, and solution orders; and (v) incorporating a data randomization technique that not only implicitly promotes agreement between surrogate models and true numerical models up to second-order derivatives, ensuring long-term stability and prediction capacity, but also serves as a data generation engine during training, leading to enhanced generalization on unseen data. To validate the effectiveness, stability, and generalizability of our novel DGNet approach, we present comprehensive numerical results for 1D and 2D compressible Euler equation problems.
Paper Structure (30 sections, 3 theorems, 61 equations, 38 figures, 5 tables)

This paper contains 30 sections, 3 theorems, 61 equations, 38 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. A brief review of the nodal discontinuous Galerkin Approach
  4. Discontinuous Galerkin graph neural network (DGNet) framework
  5. How to train $\texttt{DGNet}$$\ $ block $\Psi_\texttt{DGNet}\left( \boldsymbol{u} \right)$?
  6. The architecture design of the $\Psi_\texttt{DGNet}$ block
  7. Data randomization
  8. Error estimation
  9. Numerical results
  10. General settings and learning hyperparameters
  11. Data generation
  12. Training settings
  13. Simplified training for model-constrained approach, $\texttt{mcDGNet} \ $
  14. Selection of the "best" trained $\texttt{DGNet}$$\ $
  15. Noise level and slope limiters
  16. ...and 15 more sections

Key Result

Theorem 1

Let the data $\left\{ \boldsymbol{u}^i \right\}_{i=1}^{n_t}$ be randomized as in eq:add_noise for every epoch. There holds:

Figures (38)

  • Figure 1: Left figure shows the duality of a Discontinuous Galerkin (DG) mesh (black) and graph neural network (GNN) architecture (red nodes and orange edges). Right figure illustrates the $k$-th element $\mathcal{D}^k$ (triangle with purple boundary) and its set of faces $\Gamma^e \subset \partial \mathcal{D}^k$; $\mathcal{D}^{\mathcal{N}\left( k \right)}$ (bottom right) represents an immediate neighboring element of the $k$-th element; $\Gamma^{\mathcal{N}\left( e \right)} \subset \mathcal{D}^{\mathcal{N}\left( k \right)}$ denotes a neighboring face of $\Gamma^e$. The node attribute of the GNN $k$-th vertex is a collection of all nodal values of the $k$-th DG element. The edge feature for the GNN edge between the $k$-th vertex and its neighboring vertex $\mathcal{N}(k)$ represents the numerical flux between the DG elements $\mathcal{D}^k$ and $\mathcal{D}^{\mathcal{N}\left( k \right)}$; For illustration, circles represent the nodal value of $2^{nd}-$order DG elements.
  • Figure 2: The schematic of constructing the $\texttt{mcDGNet} \ $ loss \ref{['eq:mcDGGNNTagent_loss']} using $2^\text{nd}$ order strong stability-preserving Runge-Kutta (2nd-SSP-RK) time integration scheme. To embed the data randomization technique, the random noise vector $\boldsymbol{\varepsilon} \sim \mathcal{N}\left( \boldsymbol{0}, \delta^2 \Sigma_{ {\boldsymbol{u}}^{{i,0}}} \right)$ is added to the input of the neural network, where $\Sigma_{ {\boldsymbol{u}}^{{i,0}}}=\mathop{\mathrm{diag}}\nolimits\left( \left( {\boldsymbol{u}}^{{i,0}} \right)^2 \right)$. $\mathcal{F}$ is the DG spatial discretization operator. $S$ is the slope limiter operator tu2005slope applied at the end of each stage of 2nd-SSP-RK scheme.
  • Figure 3: Description of $\Psi_\texttt{DGNet}$ architecture block. For the $k$-th element $\mathcal{D}^k$, at the node level, the element nodal values $\boldsymbol{u}^k$ serve as node attributes in the $\texttt{DGNet}$$\ $ . The volume flux vector ${\boldsymbol{f}}^k$ is evaluated from $\boldsymbol{u}^k$ and then normalized to $\overline{\boldsymbol{f}}^k$ as in \ref{['eq:normalized_vol']}. The volume neural network $\Psi_\text{vol}:\mathbb{R}^{N_p} \mapsto \mathbb{R}^{N_p}$ modifies the normalized flux to learn the corrections. The output is denormalized by multiplying factor ${\eta}_{i,q}^k$ to get the corrected volume flux, $\tilde{\boldsymbol{f}}_{i,q}^k$. Finally, we apply the collocation-type integration rule to compute the volume term. At the edge level, the edge network $\Psi_\text{flux}$ represents the graph neural network edges and surrogates the Riemann solver. It computes the numerical fluxes at the shared face points of the element $k$ and its corresponding element $\mathcal{N}\left( k \right)$. The input for $\Psi_\text{flux} : \mathbb{R}^{d+1} \mapsto \mathbb{R}^{1}$ comprises the normalized average flux terms and normalized state jump, as described in \ref{['eq:normalized_flux']}. The output of the numerical flux network is denormalized by multiplying the factor ${\psi}_{q,j}^e$ to evaluate the numerical flux. Subsequently, the approximated numerical fluxes undergo a collocation-type integration rule for evaluating the flux term. The flux and volume terms are added up, returning the tangent slope for the $k$-th element.
  • Figure 4: Visualization of data enrichment effect from the data randomization strategy. Data randomization expands a single training data point to the area of the ball with a radius of $\delta$. As a result, test samples are more likely to be discovered during the training process.
  • Figure 5: 1D Sod and Lax shock tube problems: predicted density solutions, $\rho$, for four different cases using the $\texttt{DGNet}$$\ $ network trained with Model 1 Sod shock tube training data. Exact stands for the exact solution, DG for the DG solution, and WE-PINNs for the result from liu2024discontinuity. Top Left: Sod shock tube in Model 1 at $T = 0.2s$. Top Right: Sod shock tube in Model 2 at $T = 0.2s$. Bottom Left: Lax shock tube in Model 1 at $T = 0.13s$. Bottom Right: Lax shock tube in Model 2 at $T = 0.13s$.
  • ...and 33 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof
  • Remark 5
  • Theorem 3