# HypNO: A Graph-Based Neural Operator with Physics-Informed Message Passing for Hyperbolic Conservation Laws

arXiv: https://arxiv.org/abs/2607.20541 · HTML: https://mlsanity.com/paper/2607.20541/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 37 references that are in the map.
- First posted on arXiv 10 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws](https://mlsanity.com/paper/2401.01783/) (2024)
- [Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs](https://mlsanity.com/paper/2601.06388/) (2026)

## Abstract (arXiv)

We introduce HypNO, a graph-based neural operator for scalar hyperbolic conservation laws. HypNO operates directly on a space-time graph of finite-volume cells and uses adjacency-factored, physics-informed message passing to respect upwinding and entropy admissibility near shocks. We benchmark the architecture on the Lighthill-Whitham-Richards (LWR) and Aw-Rascle-Zhang (ARZ) traffic-flow models, a stress test for operator-learning methods because of their simultaneous global transport and shock formation. HypNO predicts solution snapshots accurately across a range of initial conditions while capturing the shocks and discontinuities of the solution.