小图训练的神经微分图模型可直接部署到更大相似图上，理论证明其有效性。
Zero-Shot Size Transfer for Neural ODEs on Sparse Random Graphs: Graphon Limits and Adjoint Convergence
- 基于图随机极限理论，建立神经微分图模型在稀疏图上的零样本扩展框架
- 在稀疏随机图上，解轨迹收敛速度达O((α_n n)^{-1/2})，高概率成立
- 实验证明跨图类零样本迁移有效，适用于图规模变化场景
图神经微分方程（GNDE）通过图神经网络参数化神经微分方程的速度场，建模连续时间图动态。其局部、与规模无关的滤波器特性暗示了零样本规模迁移原则：在小图上训练后可直接部署于更大、相似的图上而无需重训练。本文针对从图随机过程（graphons）采样的稀疏随机图，建立了该原则的定量理论。考虑图随机神经微分方程（Graphon-NDE）及其伴随系统（adjoint Graphon-NDE），作为前向与伴随GNDE系统的无限节点极限，并证明其适定性。对于包含n个节点、稀疏参数为α_n的随机图，我们证明了GNDE解以概率接近1的方式，以速率O((α_n n)^{-1/2})收敛至Graphon-NDE解，忽略对数因子。同时建立了伴随系统（控制隐状态与参数梯度）的全局时间一致收敛界。进一步研究离散化-再优化（DTO）与优化-再离散化（OTD）训练方法。在显式欧拉离散化下，使用M步时，证明了两种方法渐近一致，隐状态与局部参数梯度差异分别达到O(1/M)与O(1/M^2)，忽略稀疏性与对数因子影响。在HSBM和帐篷图随机过程上的实验支持理论速率，零样本迁移实验在四个图随机过程类别上展示了学习后的GNDE在独立采样大图上的准确部署能力。
原文摘要 · Abstract (English)
Graph Neural Differential Equations (GNDEs) model continuous-time graph dynamics by parameterizing Neural ODE velocity fields with Graph Neural Networks. Their local, size-independent filters suggest a zero-shot size-transfer principle: train on a small graph and deploy on larger, similar graphs without retraining. We develop a quantitative theory for this principle on sparse random graphs sampled from graphons. We consider Graphon Neural Differential Equations (Graphon-NDEs) and adjoint Graphon-NDEs as the infinite-node limits of the forward and adjoint GNDE systems, and establish well-posedness. For an $n$-node random graph with sparsity parameter $α_n$, we prove trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions at rate $O((α_n n)^{-1/2})$, up to logarithmic factors, with high probability. We also establish uniform-in-time convergence bounds for adjoint systems governing hidden-state and parameter gradients. We further study discretize-then-optimize (DTO) and optimize-then-discretize (OTD) training. Under explicit Euler discretization with $M$ steps, we show that DTO and OTD are asymptotically consistent, with hidden-state and local parameter-gradient discrepancies of orders $O(1/M)$ and $O(1/M^2)$, respectively, up to sparsity and logarithmic factors. Experiments on HSBM and tent graphons support the theoretical rates, while zero-shot transfer experiments across four graphon classes demonstrate accurate deployment of learned GNDEs on larger independently sampled graphs.
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