The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.

翻译：本文旨在对图神经网络（GNNs）模型所基于的扩散问题提供系统的数学解释。我们方法的出发点是一个耗散泛函，该泛函导出的动力学方程使我们能够研究模型的对称性。我们讨论了守恒荷，并给出了一种保持电荷的数值方法来求解动力学方程。在任何动力系统以及图神经扩散（GRAND）中，了解荷值及其在演化流中的守恒性质，可提供一种理解GNNs及其他网络如何利用其学习能力运行的方法。