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.

这篇论文旨在对基于图神经网络（GNNs）模型的扩散问题进行系统的数学解释。我们的方法的出发点是一个耗散性函数，导致了动力学方程，从而使我们能够研究模型的对称性。我们讨论了守恒荷和一种保持荷守恒的数值方法来求解动力学方程。在任何动力学系统以及图神经扩散（GRAND）中，了解荷值及其在演化过程中的守恒情况可以帮助我们理解 GNNs 和其他网络的学习能力。