Zero-Shot Size Transfer for Graph Neural ODEs Proven

Graph Neural Differential Equations (GNDEs) are models that capture continuous-time dynamics on graphs by using Graph Neural Networks to parameterize velocity fields. A key, intuitive principle for GNDEs is "zero-shot size transfer," meaning a model trained on a smaller graph could be directly applied to larger, similar graphs without needing retraining. This research provides a rigorous quantitative theory to support this principle, specifically for sparse random graphs derived from graphons. The study introduces Graphon Neural Differential Equations (Graphon-NDEs) and their adjoint counterparts as the theoretical infinite-node limits of GNDE systems, proving their well-posedness. For an $n$-node random graph, the researchers demonstrate that GNDE solutions converge to Graphon-NDE solutions at a rate of $O((\alpha_n n)^{-1/2})$, with high probability, up to logarithmic factors. This convergence is also established for adjoint systems, which are crucial for calculating gradients during training. Furthermore, the paper investigates the consistency between "discretize-then-optimize" (DTO) and "optimize-then-discretize" (OTD) training approaches. It shows that these methods are asymptotically consistent, with discrepancies in hidden-state and parameter gradients decreasing with the number of discretization steps. Experimental results on various graphon classes confirm the theoretical rates and validate the effectiveness of zero-shot transfer, allowing learned GNDEs to be accurately deployed on larger, independently sampled graphs.