# Zero-Shot Size Transfer for Neural ODEs on Sparse Random Graphs: Graphon Limits and Adjoint Convergence Source: [Zero-Shot Size Transfer for Neural ODEs on Sparse Random Graphs: Graphon Limits and Adjoint Convergence]() Collection index: [llms.txt]() Data updated: 2026-06-27T05:00:04.727Z Authors: Mingsong Yan, Zhida Wang, Sui Tang Published: 2026-06-25T06:53:20.000Z ## Abstract 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 $\\alpha\_n$, we prove trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions at rate $O((\\alpha\_n n)^{-1/2})$, 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Linear Heat Dynamics Prediction on Tent Graphon](): Linear Heat Dynamics Prediction; Tent Graphon; Transfer Error: 0.003. - [Linear Heat Dynamics Prediction on HSBM Graphon](): Linear Heat Dynamics Prediction; HSBM Graphon; Transfer Error: 0.4. - [SIS Dynamics Prediction on Tent Graphon](): SIS Dynamics Prediction; Tent Graphon; Transfer Error: 0.016. - [SIS Dynamics Prediction on HSBM Graphon](): SIS Dynamics Prediction; HSBM Graphon; Transfer Error: 0.012. - [Consensus Dynamics Prediction on Circular Graphon](): Consensus Dynamics Prediction; Circular Graphon; Transfer Error: 0.199. - [Consensus Dynamics Prediction on Power-law Graphon](): Consensus Dynamics Prediction; Power-law Graphon; Transfer Error: 0.145. - [Fisher–KPP Dynamics Prediction on Circular Graphon](): Fisher–KPP Dynamics Prediction; Circular Graphon; Transfer Error: 0.024. - [Fisher–KPP Dynamics Prediction on Power-law Graphon](): Fisher–KPP Dynamics Prediction; Power-law Graphon; Transfer Error: 2.7. - [Linear Heat Dynamics Prediction on Circular Graphon](): Linear Heat Dynamics Prediction; Circular Graphon; Transfer Error: 0.003. - [Linear Heat Dynamics Prediction on Power-law Graphon](): Linear Heat Dynamics Prediction; Power-law Graphon; Transfer Error: 0.004. - [SIS Dynamics Prediction on Circular Graphon](): SIS Dynamics Prediction; Circular Graphon; Transfer Error: 0.008. - [SIS Dynamics Prediction on Power-law Graphon](): SIS Dynamics Prediction; Power-law Graphon; Transfer Error: 0.011. ## Original sources and implementation [Original paper]() [PDF]()