融合时空与频域图网络，高效求解不规则域上的力学方程
Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid
- 结合局部卷积与全局频谱特征，兼顾计算效率与长程依赖建模
- 在规则与不规则域上均实现高精度求解时间相关/无关偏微分方程
- 适用于复杂几何的科学计算问题，尤其适合工程仿真场景
科学机器学习因算子学习的兴起取得显著进展。然而，现有方法在处理非结构化网格和不规则域问题时仍面临挑战。空间图神经网络通过邻域局部卷积有望应对这些问题，但深层架构常出现过平滑和过挤压现象。相反，频谱图神经网络利用全局卷积捕捉域图中的广泛特征与长程依赖，却因特征值分解带来高昂计算成本。本文提出一种新方法——时空图神经算子（Sp²GNO），有效融合空间与频谱图神经网络。该框架克服了单一方法的局限性，实现任意几何上解算子的学习，可广泛应用于实际问题。Sp²GNO在规则与不规则域上对时变与时不变偏微分方程均表现优异。通过全面基准测试及来自计算力学与科学计算文献的实际应用验证了其有效性。
原文摘要 · Abstract (English)
Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural networks utilize local convolution in a neighborhood to potentially address these challenges, yet they often suffer from issues such as over-smoothing and over-squashing in deep architectures. Conversely, spectral graph neural networks leverage global convolution to capture extensive features and long-range dependencies in domain graphs, albeit at a high computational cost due to Eigenvalue decomposition. In this paper, we introduce a novel approach, referred to as Spatio-Spectral Graph Neural Operator (Sp$^2$GNO) that integrates spatial and spectral GNNs effectively. This framework mitigates the limitations of individual methods and enables the learning of solution operators across arbitrary geometries, thus catering to a wide range of real-world problems. Sp$^2$GNO demonstrates exceptional performance in solving both time-dependent and time-independent partial differential equations on regular and irregular domains. Our approach is validated through comprehensive benchmarks and practical applications drawn from computational mechanics and scientific computing literature.
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