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

TL;DR

This paper introduces a graph neural network-based solver for hyperbolic conservation laws that preserves physical properties, enabling fast, stable, and accurate simulations across various flow conditions.

Contribution

It develops a structure-preserving GNN framework that combines classical numerical principles with neural networks for efficient PDE solutions.

Findings

Achieves superior long-term stability and accuracy in supersonic flow benchmarks.

Outperforms traditional low-order discretizations and surrogate models.

Provides significant runtime speedups over high-resolution simulations.

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.