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。

一种求解图神经扩散网络的电荷保持方法

A charge-preserving method for solving graph neural diffusion networks

Recently, SyncMap pioneered an approach to learn complex structures from
sequences as well as adapt to any changes in underlying structures. This is
achieved by using only nonlinear dynamical equations inspired by neuron group
behaviors, i.e., without loss functions. Here we propose Symmetrical SyncMap
that goes beyond the original work to show how to create dynamical equations
and attractor-repeller points which are stable over the long run, even dealing
with imbalanced continual general chunking problems (CGCPs). The main idea is
to apply equal updates from negative and positive feedback loops by symmetrical
activation. We then introduce the concept of memory window to allow for more
positive updates. Our algorithm surpasses or ties other unsupervised
state-of-the-art baselines in all 12 imbalanced CGCPs with various
difficulties, including dynamically changing ones. To verify its performance in
real-world scenarios, we conduct experiments on several well-studied structure
learning problems. The proposed method surpasses substantially other methods in
3 out of 4 scenarios, suggesting that symmetrical activation plays a critical
role in uncovering topological structures and even hierarchies encoded in
temporal data.

通过使用非线性动力学方程和对称激活，对于长期稳定性和解决不平衡连续通用分块问题（CGCPs）等困难，Symmetrical SyncMap 提出了一种学习复杂结构的方法，并引入记忆窗口的概念来允许更多正向更新。在实验证明了其在结构学习问题中的出色表现，揭示了存在于时间数据中的拓扑结构和层次编码。