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 systems such as large conjugated molecules, metals, or amorphous materials. Although Spectral GNNs and traditional neural networks such as recurrent neural networks and transformers mitigate these challenges, they often lack extensivity, adaptability, generalizability, computational efficiency, 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 state-of-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（GNN），特别是消息传递神经网络（MPNN），已成为在多种应用中学习图数据的强大架构。然而，当对大型共轭分子、金属或非晶态材料等系统中的非局域相互作用进行建模时，MPNN面临挑战。尽管谱图神经网络与循环神经网络、Transformer等传统神经网络在一定程度上缓解了这些问题，但它们往往缺乏广延性、自适应性、泛化能力或计算效率，且无法捕捉数据中的精细结构关系或对称性。为解决上述不足，我们提出了矩阵函数神经网络（MFN）——一种通过解析矩阵等变函数参数化非局域相互作用的新型架构。采用预解展开方法不仅简化了实现过程，还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了当前最优性能，并能有效捕捉量子系统中复杂的非局域相互作用，为开发新一代高性能力场开辟了道路。