arXiv cs.AI· Jinyoung Jeong, Joseph B. Choi, Xinlun Cheng, H. S. Udaykumar, Sanghun Choi, Stephen S. Baek·· 14 小时前AI 评分20
面向双曲输运方程图神经求解器的 CFL 感知边设计
CFL-aware edge design for graph neural solvers of hyperbolic transport
AI 导读
研究将边局部 Courant-Friedrichs-Lewy（CFL）信号显式引入图神经网络（GNN）的边特征，以提升其对双曲 PDE 的建模能力。在架构固定、仅改变边信息形式的条件下，当边表示已区分交互方向与空间尺度时，CFL 条件化可将最终步误差降低 10.4-14.4%，且步长越大改善越明显。改进主要来自抑制激波附近的振荡而非抹平解。
正文 · 原文
Abstract:Hyperbolic partial differential equations (PDEs) transport information at finite speed along characteristics, often producing shocks and discontinuities that are challenging to capture numerically. Graph neural networks (GNNs) offer learned update rules, but neighbor aggregation alone does not tell a GNN how far information must travel and along which edge. As a result, when a GNN model takes a time step larger than what was used in training, shock fronts smear and oscillations grow. In this work, we ask whether making the scale of transport explicit as an edge-local Courant-Friedrichs-Lewy (CFL) signal can enhance GNNs' ability to model hyperbolic PDEs. Holding the architecture fixed, we vary only the information given to each edge, from a raw edge form to forms that progressively expose the edge-local transport interaction through direction, length, and wave speed. Because all of these are computed from the same ingredients, what changes is not the amount of information but its form. The results show a consistent pattern, in which feature count is not the limiting factor, and appending CFL directly to a raw edge form produces almost no benefit. The CFL conditioning proves most effective when the edge information already exposes the transport interaction, reducing final-step error by 10.4-14.4%, and the improvement widens as the step size grows larger. Notably, this improvement arises not from smearing the solution but from suppressing oscillations near shocks. These results point to a specific design principle that CFL conditioning is most effective when the edge representation already separates the direction of interaction from spatial scale, so that the CFL value has a well-defined quantity to modulate.
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Fluid Dynamics (physics.flu-dyn) |
| Cite as: | arXiv:2610.33769 [cs.LG] |
| (or arXiv:2610.33769v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.33769 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Jinyoung Jeong [view email]
[v1]
Sat, 24 Oct 2026 16:20:20 UTC (14,569 KB)
来源：arXiv cs.AI · arxiv.org