用图神经网络解决任意几何上的偏微分方程，精度和效率双提升。
Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries
- 构建多尺度图与物理图，动态捕捉复杂空间关系。
- 在6个基准上超越现有最优模型，对非规则几何更鲁棒。
- 适合需要高精度求解复杂物理系统的科研与工程人员。
偏微分方程（PDEs）描述了众多科学现象，但传统计算方法在处理复杂非线性系统和不规则几何时常受限。本文提出AMG方法——一种面向任意几何的多图神经算子框架，通过新型GraphFormer架构结合先进的图技术与动态注意力机制，实现对多样空间域及复杂数据依赖关系的精准建模。该方法利用多尺度图处理不同频率特征，并引入物理图编码内在物理规律。在六个基准测试中，AMG持续优于现有最先进模型，显著突破传统方法仅限均匀网格的局限。结果表明，定制化的图神经算子具有变革性潜力，可有效应对传统PDE求解器面临的挑战。代码与数据集已公开于https://github.com/lizhihao2022/AMG。
原文摘要 · Abstract (English)
Partial Differential Equations (PDEs) underpin many scientific phenomena, yet traditional computational approaches often struggle with complex, nonlinear systems and irregular geometries. This paper introduces the AMG method, a Multi-Graph neural operator approach designed for efficiently solving PDEs on Arbitrary geometries. AMG leverages advanced graph-based techniques and dynamic attention mechanisms within a novel GraphFormer architecture, enabling precise management of diverse spatial domains and complex data interdependencies. By constructing multi-scale graphs to handle variable feature frequencies and a physics graph to encapsulate inherent physical properties, AMG significantly outperforms previous methods, which are typically limited to uniform grids. We present a comprehensive evaluation of AMG across six benchmarks, demonstrating its consistent superiority over existing state-of-the-art models. Our findings highlight the transformative potential of tailored graph neural operators in surmounting the challenges faced by conventional PDE solvers. Our code and datasets are available on https://github.com/lizhihao2022/AMG.
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