Asian Journal of Mathematics and Computer Research

Graph neural ordinary differential equation models provide a continuous-depth framework for learning representations from graph-structured data, but their practical application may be affected by unstable numerical behaviour and unreliable fixed-point convergence. This study presents a topology-aware implicit Jacobian regularisation framework for stable fixed-point graph neural ODEs. Node-feature evolution is represented as a continuous-time dynamical system, while equilibrium representations are obtained through fixed-point iterations. The proposed regularisation term penalises differences between the local Jacobians of adjacent nodes, thereby encouraging neighbouring nodes to exhibit consistent sensitivity to feature perturbations in accordance with the graph topology. Unlike global Lipschitz constraints, the method acts locally and is intended to preserve model flexibility while promoting stable equilibrium dynamics. The optimisation objective combines the task-specific loss with Jacobian consistency regularisation, and gradients are computed through implicit differentiation and automatic differentiation without fully unrolling the equilibrium iterations. The framework is implemented using a graph-coupled drift function and evaluated on representative homophilic, heterophilic, and molecular graph-learning benchmarks. The reported findings indicate faster fixed-point convergence, improved robustness to feature and structural perturbations, and competitive predictive performance relative to the evaluated baseline models. The results support the use of graph-informed local Jacobian alignment as a practical stabilisation mechanism for continuous-depth and equilibrium-based graph neural networks.