Graph Neural Networks (GNNs), especially message-passing neural networks (MPNNs), have emerged as powerful architectures for learning on graphs in diverse applications. However, MPNNs face challenges when modeling non-local interactions in graphs such as large conjugated molecules, and social networks due to oversmoothing and oversquashing. Although Spectral GNNs and traditional neural networks such as recurrent neural networks and transformers mitigate these challenges, they often lack generalizability, or fail to capture detailed structural relationships or symmetries in the data. To address these concerns, we introduce Matrix Function Neural Networks (MFNs), a novel architecture that parameterizes non-local interactions through analytic matrix equivariant functions. Employing resolvent expansions offers a straightforward implementation and the potential for linear scaling with system size. The MFN architecture achieves stateof-the-art performance in standard graph benchmarks, such as the ZINC and TU datasets, and is able to capture intricate non-local interactions in quantum systems, paving the way to new state-of-the-art force fields.

翻译：图神经网络（GNNs），尤其是消息传递神经网络（MPNNs），已成为在多样化应用中处理图数据的强大架构。然而，MPNNs在建模大共轭分子、社交网络等图中的非局部交互时，由于过度平滑和过度挤压等问题面临挑战。尽管谱图神经网络以及循环神经网络、Transformer等传统神经网络缓解了这些挑战，但它们往往缺乏泛化能力，或无法捕捉数据中的详细结构关系与对称性。为解决这些问题，我们引入了矩阵函数神经网络（MFNs）——一种通过解析矩阵等变函数参数化非局部交互的新型架构。采用预解展开方法可实现简便的算法实现，并具备随系统规模线性扩展的潜力。MFN架构在ZINC和TU数据集等标准图基准测试中达到最先进性能，并能捕捉量子系统中复杂的非局部交互，为开发新一代最先进的力场铺平了道路。