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

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.

A Novel Hybrid Approach for Retinal Vessel Segmentation with Dynamic Long-Range Dependency and Multi-Scale Retinal Edge Fusion Enhancement

Accurate retinal vessel segmentation provides essential structural information for ophthalmic image analysis. However, existing methods struggle with challenges such as multi-scale vessel variability, complex curvatures, and ambiguous boundaries. While Convolutional Neural Networks (CNNs), Transformer-based models and Mamba-based architectures have advanced the field, they often suffer from vascular discontinuities or edge feature ambiguity. To address these limitations, we propose a novel hybrid framework that synergistically integrates CNNs and Mamba for high-precision retinal vessel segmentation. Our approach introduces three key innovations: 1) The proposed High-Resolution Edge Fuse Network is a high-resolution preserving hybrid segmentation framework that combines a multi-scale backbone with the Multi-scale Retina Edge Fusion (MREF) module to enhance edge features, ensuring accurate and robust vessel segmentation. 2) The Dynamic Snake Visual State Space block combines Dynamic Snake Convolution with Mamba to adaptively capture vessel curvature details and long-range dependencies. An improved eight-directional 2D Snake-Selective Scan mechanism and a dynamic weighting strategy enhance the perception of complex vascular topologies. 3) The MREF module enhances boundary precision through multi-scale edge feature aggregation, suppressing noise while emphasizing critical vessel structures across scales. Experiments on three public datasets demonstrate that our method achieves state-of-the-art performance, particularly in maintaining vascular continuity and effectively segmenting vessels in low-contrast regions. This work provides a robust method for clinical applications requiring accurate retinal vessel analysis. The code is available at https://github.com/frank-oy/HREFNet.