arXiv subjects

Weiqi Guan

Publications and source records attributed to Weiqi Guan.

4 recordsLinked to original sources

arXiv2026

Constrained variational problems on perturbed lattice graphs

In this paper, we solve some constrained variational problems on perturbed lattice graphs $G$. The first problem addresses the existence of ground state normalized solutions to Schrödinger equations \begin{equation*} \left\{ \begin{aligned} &-Δ_{G} u+λu=\vert u\vert^{p-2}u,x\in G &\Vert u\Vert_{l^2(G)}^2=a. \end{aligned} \right. \end{equation*} We prove that if the graph is obtained by deleting finite edges in lattice graphs while maintaining connectivity, then there exists a threshold $α_G\in[0,\infty)$ such that there do not exist ground state normalized solution if $0 α_G.$ If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exist $E^{'}$ and $a_1$ such that for all $a>a_1,$ there do not exist ground state normalized solutions. The second problem concerns the existence of an extremal function for the Sobolev inequality. If the graph $G$ is obtained by deleting finite edges in lattice graphs while maintaining connectivity, for the Sobolev super-critical regime, we prove that there exists an extremal function. for the Sobolev critical regime, we prove that there exists $G$ such that extremal can be attained. If the graph is obtained by adding finite edges $E^{'}$ to lattice graphs, we prove that there exists $E^{'}$ such that there does not exist an extremal function.

math.AP↗

arXiv2025

Solving Oversmoothing in GNNs via Nonlocal Message Passing: Algebraic Smoothing and Depth Scalability

The relationship between Layer Normalization (LN) placement and the oversmoothing phenomenon remains underexplored. We identify a critical dilemma: Pre-LN architectures avoid oversmoothing but suffer from the curse of depth, while Post-LN architectures bypass the curse of depth but experience oversmoothing. To resolve this, we propose a new method based on Post-LN that induces algebraic smoothing, preventing oversmoothing without the curse of depth. Empirical results across five benchmarks demonstrate that our approach supports deeper networks (up to 256 layers) and improves performance, requiring no additional parameters. Key contributions: Theoretical Characterization: Analysis of LN dynamics and their impact on oversmoothing and the curse of depth. A Principled Solution: A parameter-efficient method that induces algebraic smoothing and avoids oversmoothing and the curse of depth. Empirical Validation: Extensive experiments showing the effectiveness of the method in deeper GNNs.

cs.LG↗

arXiv2025

Measuring Over-smoothing beyond Dirichlet energy

While Dirichlet energy serves as a prevalent metric for quantifying over-smoothing, it is inherently restricted to capturing first-order feature derivatives. To address this limitation, we propose a generalized family of node similarity measures based on the energy of higher-order feature derivatives. Through a rigorous theoretical analysis of the relationships among these measures, we establish the decay rates of Dirichlet energy under both continuous heat diffusion and discrete aggregation operators. Furthermore, our analysis reveals an intrinsic connection between the over-smoothing decay rate and the spectral gap of the graph Laplacian. Finally, empirical results demonstrate that attention-based Graph Neural Networks (GNNs) suffer from over-smoothing when evaluated under these proposed metrics.

cs.LG↗

arXiv2025

Existence of normalized solutions to nonlinear Schrödinger equations with potential on lattice graphs

We study the existence of ground state normalized solution of the following Schrödinger equation: \begin{equation*} \begin{cases} -Δu+V(x)u+λu=f(x,u), & x\in\mathbb{Z}^d \\ \Vert u\Vert_2^2=a \end{cases} \end{equation*} where $V(x)$ is trapping potential or well potential, $f(x,u)$ satisfies Berestycki-Lions type condition and other suitable conditions. We show that there always exists a threshold $α\in[0,\infty)$ such that there do not exist ground state normalized solutions for $a\in (0,α)$, and there exists a ground state normalized solution for $a\in(α,\infty)$. Furthermore, we prove sufficient conditions for the positivity of $α$ that $α=0$ if $f(x,u)$ is mass-subcritical near 0, and $α>0$ if $f(x,u)$ is mass-critical or mass-supercritical near 0.

math.AP↗