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.

矩阵函数神经网络是一种新颖的架构，通过分析的矩阵等变函数参数化非局部交互。在标准图形基准测试中，MFN架构实现了最先进的性能，并能够捕捉量子系统中复杂的非局部交互，为新的最先进力场铺平了道路。

等变矩阵函数神经网络