用物理信息优化图神经网络连接，解决流体模拟中远距离依赖难题
PIORF: Physics-Informed Ollivier-Ricci Flow for Long-Range Interactions in Mesh Graph Neural Networks
- 基于物理相关性与图拓扑结合的新型重连方法
- 在3个流体基准数据集上最高提升26.2%性能
- 适合需要长程交互的复杂物理系统建模
基于图神经网络的数据驱动模拟器在非结构化网格物理系统建模中受到关注，但难以处理流体流动中的长程依赖，尤其在细化网格区域。这一问题被称为‘过挤压’（over-squashing），阻碍信息传播。现有图重连方法虽部分缓解此问题，但仅考虑图拓扑，忽略底层物理现象。本文提出物理信息引导的奥利维耶-里奇流（PIORF），融合物理相关性与图拓扑。PIORF利用奥利维耶-里奇曲率（ORC）识别瓶颈区域，并将这些区域与高速度梯度节点连接，实现长程交互并缓解过挤压。该方法计算高效，可扩展至更大规模模拟。在3个流体动力学基准数据集上的实验表明，PIORF始终优于基线模型和现有重连方法，性能最高提升26.2%。
原文摘要 · Abstract (English)
Recently, data-driven simulators based on graph neural networks have gained attention in modeling physical systems on unstructured meshes. However, they struggle with long-range dependencies in fluid flows, particularly in refined mesh regions. This challenge, known as the 'over-squashing' problem, hinders information propagation. While existing graph rewiring methods address this issue to some extent, they only consider graph topology, overlooking the underlying physical phenomena. We propose Physics-Informed Ollivier-Ricci Flow (PIORF), a novel rewiring method that combines physical correlations with graph topology. PIORF uses Ollivier-Ricci curvature (ORC) to identify bottleneck regions and connects these areas with nodes in high-velocity gradient nodes, enabling long-range interactions and mitigating over-squashing. Our approach is computationally efficient in rewiring edges and can scale to larger simulations. Experimental results on 3 fluid dynamics benchmark datasets show that PIORF consistently outperforms baseline models and existing rewiring methods, achieving up to 26.2 improvement.
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