Due to the over-smoothing issue, most existing graph neural networks can only
capture limited dependencies with their inherently finite aggregation layers.
To overcome this limitation, we propose a new kind of graph convolution, called
Graph Implicit Nonlinear Diffusion (GIND), which implicitly has access to
infinite hops of neighbors while adaptively aggregating features with nonlinear
diffusion to prevent over-smoothing. Notably, we show that the learned
representation can be formalized as the minimizer of an explicit convex
optimization objective. With this property, we can theoretically characterize
the equilibrium of our GIND from an optimization perspective. More
interestingly, we can induce new structural variants by modifying the
corresponding optimization objective. To be specific, we can embed prior
properties to the equilibrium, as well as introducing skip connections to
promote training stability. Extensive experiments show that GIND is good at
capturing long-range dependencies, and performs well on both homophilic and
heterophilic graphs with nonlinear diffusion. Moreover, we show that the
optimization-induced variants of our models can boost the performance and
improve training stability and efficiency as well. As a result, our GIND
obtains significant improvements on both node-level and graph-level tasks.

本文提出了一种新的图卷积方法 Graph Implicit Nonlinear Diffusion (GIND)，通过自适应聚合具有非线性扩散的特征来防止过度平滑，并以明确的凸优化目标的最小化表示所学到的表示。实验表明，GIND 在捕获长程依赖方面表现出色，并在节点级和图级任务上取得了显着的改进。

优化引导的图隐式非线性扩散

Optimization-Induced Graph Implicit Nonlinear Diffusion

As a counterpoint to classical stochastic particle methods for diffusion, we
develop a deterministic particle method for linear and nonlinear diffusion. At
first glance, deterministic particle methods are incompatible with diffusive
partial differential equations since initial data given by sums of Dirac masses
would be smoothed instantaneously: particles do not remain particles. Inspired
by classical vortex blob methods, we introduce a nonlocal regularization of our
velocity field that ensures particles do remain particles, and we apply this to
develop a numerical blob method for a range of diffusive partial differential
equations of Wasserstein gradient flow type, including the heat equation, the
porous medium equation, the Fokker-Planck equation, the Keller-Segel equation,
and its variants. Our choice of regularization is guided by the Wasserstein
gradient flow structure, and the corresponding energy has a novel form,
combining aspects of the well-known interaction and potential energies. In the
presence of a confining drift or interaction potential, we prove that
minimizers of the regularized energy exist and, as the regularization is
removed, converge to the minimizers of the unregularized energy. We then
restrict our attention to nonlinear diffusion of porous medium type with at
least quadratic exponent. Under sufficient regularity assumptions, we prove
that gradient flows of the regularized energies converge to solutions of the
porous medium equation. As a corollary, we obtain convergence of our numerical
blob method, again under sufficient regularity assumptions. We conclude by
considering a range of numerical examples to demonstrate our method's rate of
convergence to exact solutions and to illustrate key qualitative properties
preserved by the method, including asymptotic behavior of the Fokker-Planck
equation and critical mass of the two-dimensional Keller-Segel equation.

本研究基于 Wasserstein 梯度流结构和非局部正则化的思想，提出了一种基于数值 blob 方法的确定性粒子方法，用于解决非线性扩散问题，通过数值实验验证了该方法的收敛性和关键定性特性

一种扩散的 blob 方法

A blob method for diffusion

We study the numerical behaviour of a particle method for gradient flows
involving linear and nonlinear diffusion. This method relies on the
discretisation of the energy via non-overlapping balls centred at the
particles. The resulting scheme preserves the gradient flow structure at the
particle level, and enables us to obtain a gradient descent formulation after
time discretisation. We give several simulations to illustrate the validity of
this method, as well as a detailed study of one-dimensional
aggregation-diffusion equations.

研究了一个基于粒子方法的渐变流数值行为，给出了非线性扩散下一维聚集扩散方程的详细研究和模拟验证。

梯度流的粒子方法数值研究

Numerical Study of a Particle Method for Gradient Flows

Gradient flows in the Wasserstein space have become a powerful tool in the
analysis of diffusion equations, following the seminal work of Jordan,
Kinderlehrer and Otto (JKO). The numerical applications of this formulation
have been limited by the difficulty to compute the Wasserstein distance in
dimension >= 2. One step of the JKO scheme is equivalent to a variational
problem on the space of convex functions, which involves the Monge-Amp\`ere
operator. Convexity constraints are notably difficult to handle numerically,
but in our setting the internal energy plays the role of a barrier for these
constraints. This enables us to introduce a consistent discretization, which
inherits convexity properties of the continuous variational problem. We show
the effectiveness of our approach on nonlinear diffusion and crowd-motion
models.

该研究实现了对于凸函数空间的 Jenkins-Sturges-Synder 方案的可靠离散化，为非线性扩散问题和人流运动建模提供了有效的数值模拟结果。