Graph neural networks (GNNs) are widely used in domains like social networks
and biological systems. However, the locality assumption of GNNs, which limits
information exchange to neighboring nodes, hampers their ability to capture
long-range dependencies and global patterns in graphs. To address this, we
propose a new inductive bias based on variational analysis, drawing inspiration
from the Brachistochrone problem. Our framework establishes a mapping between
discrete GNN models and continuous diffusion functionals. This enables the
design of application-specific objective functions in the continuous domain and
the construction of discrete deep models with mathematical guarantees. To
tackle over-smoothing in GNNs, we analyze the existing layer-by-layer graph
embedding models and identify that they are equivalent to l2-norm integral
functionals of graph gradients, which cause over-smoothing. Similar to
edge-preserving filters in image denoising, we introduce total variation (TV)
to align the graph diffusion pattern with global community topologies.
Additionally, we devise a selective mechanism to address the trade-off between
model depth and over-smoothing, which can be easily integrated into existing
GNNs. Furthermore, we propose a novel generative adversarial network (GAN) that
predicts spreading flows in graphs through a neural transport equation. To
mitigate vanishing flows, we customize the objective function to minimize
transportation within each community while maximizing inter-community flows.
Our GNN models achieve state-of-the-art (SOTA) performance on popular graph
learning benchmarks such as Cora, Citeseer, and Pubmed.

本文提出了一种基于变分分析的归纳偏差方法以增强图神经网络的长距离依赖和全局模式捕捉能力，并采用总变差方法对图扩散模式与社区拓扑进行对齐，最后构建了一种能够预测图中传播流的生成对抗网络。最终，我们的方法在 Cora、Citeseer 和 Pubmed 等著名图学习数据集上实现了最新的性能水平。

在连续图扩散函数空间中重新思考和重新设计图神经网络

Re-Think and Re-Design Graph Neural Networks in Spaces of Continuous  Graph Diffusion Functionals

The idea of a finite collection of closed sets having "strongly regular
intersection" at a given point is crucial in variational analysis. We show that
this central theoretical tool also has striking algorithmic consequences.
Specifically, we consider the case of two sets, one of which we assume to be
suitably "regular" (special cases being convex sets, smooth manifolds, or
feasible regions satisfying the Mangasarian-Fromovitz constraint
qualification). We then prove that von Neumann's method of "alternating
projections" converges locally to a point in the intersection, at a linear rate
associated with a modulus of regularity. As a consequence, in the case of
several arbitrary closed sets having strongly regular intersection at some
point, the method of "averaged projections" converges locally at a linear rate
to a point in the intersection. Inexact versions of both algorithms also
converge linearly.

本文探讨了在可分析变差意义下，有限个闭集的强正则交集在构造效率算法时的应用和其中的正则性和收敛性质。