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 (Sp2GNO) 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. Sp2GNO 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.

中文翻译：

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用于求解不规则域和非结构化网格计算力学问题的时空谱图神经算子


随着操作员学习的出现，科学机器学习取得了重大进展。然而，现有方法在应用于非结构化网格和不规则域的问题时遇到了困难。空间图神经网络利用邻域中的局部卷积来潜在地解决这些挑战，但它们在深度架构中经常遇到过度平滑和过度压缩等问题。相反，谱图神经网络利用全局卷积来捕获域图中的广泛特征和长距离依赖关系，尽管由于特征值分解，计算成本很高。在本文中，我们介绍了一种称为空间光谱图神经运算符 （Sp2GNO） 的新方法，它有效地整合了空间和光谱 GNN。该框架减轻了单个方法的局限性，并支持跨任意几何结构学习解运算符，从而满足各种实际问题。Sp2GNO 在求解规则和不规则域上的瞬态和瞬态偏微分方程方面表现出卓越的性能。我们的方法通过来自计算力学和科学计算文献的综合基准和实际应用进行了验证。