# Zero-Shot Size Transfer for Neural ODEs on Sparse Random Graphs: Graphon Limits and Adjoint Convergence

arXiv: https://arxiv.org/abs/2606.26662 · HTML: https://mlsanity.com/paper/2606.26662/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 51 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs](https://mlsanity.com/paper/1902.10298/) (2019)
- [Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows](https://mlsanity.com/paper/2005.13420/) (2020)
- [A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks](https://mlsanity.com/paper/2304.14994/) (2023)
- [Higher-Order Graphon Neural Networks: Approximation and Cut Distance](https://mlsanity.com/paper/2503.14338/) (2025)
- [Global Convergence in Neural ODEs: Impact of Activation Functions](https://mlsanity.com/paper/2509.22436/) (2025)
- [Correcting Auto-Differentiation in Neural-ODE Training](https://mlsanity.com/paper/2306.02192/) (2023)
- [On the Convergence and Size Transferability of Continuous-depth Graph Neural Networks](https://mlsanity.com/paper/2510.03923/) (2025)

## Abstract (arXiv)

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 αn, we prove trajectory-wise convergence of GNDE solutions to Graphon-NDE solutions at rate O((αn 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/M2), 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.