A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws

Abstract: Hyperbolic conservation laws govern a wide range of transport-driven dynamics featuring shocks, contact discontinuities, and complex wave interactions, posing distinct challenges for deep-learning-based surrogate modeling. While classical numerical methods provide robust and physically admissible solutions, their computational cost restricts applicability in many-query tasks such as parametric studies and design optimization. Conversely, existing neural surrogates offer rapid inference but often fail to respect intrinsic PDE structures, leading to non-physical artifacts, rollout instability, and poor generalization. We present an interpretable, structure-preserving graph neural solver that bridges classical numerical principles with graph neural networks (GNNs). The network is designed as a learned reconstruction-and-flux operator rather than a black-box state updater, thereby inherently preserving key properties such as local conservation and upwinding. Inspired by Arbitrary high-order DERivatives schemes, we further recast message-passing GNNs as high-order space-time predictors, enabling conservative and stable neural updates with large time steps. Evaluation is performed on challenging supersonic flow benchmarks spanning broad parametric variations in geometry, initial/boundary conditions, and flow regimes. The neural solver achieves superior long-horizon rollout stability and accuracy compared with strong surrogate baselines, outperforms low-order discretizations, and delivers orders-of-magnitude runtime speedups over high-resolution simulations.

• The paper introduces CPGNet, a graph neural solver that embeds Godunov principles to enforce conservation and upwinding in shock-capturing hyperbolic PDEs.
• It utilizes directional message-passing and ADER-inspired space-time predictions to achieve high-order accuracy and reduce RMSE by up to 80% on challenging supersonic flow benchmarks.
• The solver achieves computational efficiency with inference speeds more than two orders of magnitude faster than traditional DGSEM methods while maintaining physical fidelity.

Authoritative Summary: Structure-Preserving Graph Neural Solvers for Parametric Hyperbolic Conservation Laws

Motivation and Context

Parametric hyperbolic conservation laws, such as the multidimensional Euler equations, underpin modeling of transport-driven phenomena exhibiting shocks, discontinuities, and intricate wave interactions. Traditional solvers relying on high-order finite volume and finite element discretizations provide robust and physically admissible solutions, but are computationally prohibitive for many-query applications, including design optimization and real-time decision support. Recent neural surrogates enable rapid inference but often fail to embed fundamental PDE structures—conservation, upwinding, entropy consistency—resulting in instability, smeared discontinuities, and poor generalization.

Structure-Preserving Neural Solver Architecture

This work introduces a structure-preserving, interpretable graph neural solver (CPGNet) grounded in the algorithmic principles of Godunov-type finite volume schemes and realized via expressive GNNs. The core innovations are:

• Learned Reconstruction-and-Flux Operator: The GNN is explicitly designed as a reconstruction-and-flux operator. Edge-wise decoding produces left/right interface states, which are supplied to a differentiable Riemann solver enforcing conservation and upwinding at the architectural level.
• Directional Message-Passing and Spatial Stencils: Successive message-passing layers emulate the wide, spatially-biased stencils used in high-order reconstructions (e.g., WENO, MUSCL). Edge-centric message propagation naturally incorporates one-sided biasing, crucial for non-oscillatory shock capturing.
• ADER-inspired Space-Time Prediction: The GNN is recast as a high-order space-time predictor, inspired by Arbitrary high-order DERivatives (ADER) schemes. This enables accurate, stable updates with time steps far exceeding explicit CFL limitations, allowing for substantial computational speedup during training and inference.

The architecture requires only point-cloud connectivity and nodal coordinates, eliminating reliance on explicit meshes. Geometric weights are learned to accommodate irregular spatial domains, maintaining compatibility with conservation-form PDEs on unstructured graphs.

Conservation and Upwinding Enforcement

Consistent with classical Godunov methods, numerical fluxes are antisymmetric and entropy-satisfying. The Rusanov flux is used in the reference implementation, guaranteeing robust upwind behavior.

Spatial reconstructions are encoded via deep message passing. Edge-wise decoders predict high-order interface states, which are coupled with learned geometric weights representing area/length ratios, facilitating mesh-free conservative updates. These design choices hard-constrain conservation and upwinding, preventing the propagation of non-physical artifacts and ensuring numerical stability, especially across discontinuities.

Training Regimens and Strategies

The solver is trained on trajectories generated by high-fidelity DGSEM simulations (up to fourth-order) across diverse parametric geometries, initial conditions, and Mach numbers. A two-stage curriculum is employed:

• One-Step Training: Immediate temporal mappings are optimized via MSE loss. Additive Gaussian noise is injected to improve robustness to distributional shifts during rollout.
• Multistep Fine-Tuning: Rollouts spanning multiple steps are supervised to minimize cumulative error and prevent temporal drift typical in autoregressive inference.

Training costs are comparable to baseline architectures, with modest additional expense from multistep fine-tuning justified by enhanced stability and accuracy.

Numerical Results and Empirical Evaluation

Four challenging supersonic flow benchmarks are employed: Supersonic Bump, Forward Step, Shock Diffraction, and Supersonic Cylinder. Extensive quantitative and qualitative analyses are conducted.

• Rollout Error and Stability: Across all datasets, CPGNet achieves considerable reductions (40–80%) in RMSE compared to strong surrogates (MeshGraphNet, GINO, GNOT), with up to 80% error reduction on select benchmarks. Long-horizon error accumulation and instability are markedly diminished.
• Qualitative Solution Fidelity: Final-time comparisons demonstrate CPGNet's capacity to resolve sharp shock fronts, contact discontinuities, and intricate wave interactions, preserving solution topology and suppressing artifacts and oscillations. The solver generalizes robustly across unseen geometries and parametrizations.
• Computational Efficiency: CPGNet inference is more than two orders of magnitude faster than high-resolution DGSEM, attaining solution fidelity exceeding low-order discretizations and rivaling high-order references.

Theoretical and Practical Implications

By embedding conservation and upwinding directly in the model architecture as hard constraints, this approach circumvents the limitations of black-box neural surrogates and soft physics-informed regularizations. The method achieves interpretable modeling, rollout stability, and high-fidelity predictions in regimes previously intractable for neural surrogates, particularly shock-dominated hyperbolic systems.

From a practical perspective, CPGNet enables scalable, many-query workflows in engineering, aerodynamics, and real-time control, with runtime speedups practically viable for large-scale deployment. Theoretically, the unification of Godunov principles and GNN expressivity suggests pathways for generalizable, structure-aware neural PDE solvers for arbitrary conservation laws on irregular domains.

Future developments may extend ADER-inspired neural operators to broader classes of PDEs, explore tighter integration with adaptive mesh refinement, and advance strategies for uncertainty quantification and robustness under distributional shift.

Conclusion

The structure-preserving graph neural solver framework detailed in this paper constitutes a rigorous integration of classical numerical principles and modern neural architecture design for parametric hyperbolic conservation laws (2604.15617). By hard-constraining conservation and upwinding in the architecture, leveraging ADER-inspired space-time prediction, and accommodating arbitrary geometries, the solver achieves superior accuracy, stability, and efficiency relative to established surrogates and low-order numerical schemes. These results underscore a viable paradigm for interpretable and reliable neural PDE solvers, with broad implications for scientific computing and engineering applications.