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

arXiv: https://arxiv.org/abs/2604.15617 · HTML: https://mlsanity.com/paper/2604.15617/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 86 references that are in the map.
- First posted on arXiv 17 Apr 2026.

## Builds on (most cited first, 30 of 33)

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- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [RiemannONets: Interpretable Neural Operators for Riemann Problems](https://mlsanity.com/paper/2401.08886/) (2024)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [Graph Neural PDE Solvers with Conservation and Similarity-Equivariance](https://mlsanity.com/paper/2405.16183/) (2024)

## Abstract (arXiv)

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.