NeSyF: Neural-Symbolic Feedback for Joint Network Dynamics Prediction and Governing Equation Discovery

Predicting network dynamics while recovering their governing equations remains challenging because existing neural-symbolic approaches often use recovered equations as final outputs or post-hoc interpretations, leaving the interaction between prediction and equation discovery largely one-way. We propose a Neural-Symbolic Feedback(NeSyF) framework that couples network dynamics prediction and governing equation discovery. The framework decomposes network dynamics into self-dynamics and interaction dynamics, learns them with separate neural networks, and applies symbolic regression to recover explicit equations. These equations are then used as derivative-consistency supervision to refine the learned dynamics. Experiments across four network topologies and four nonlinear dynamical systems demonstrate accurate long-horizon prediction and explicit equation recovery, while experiments on two real-world epidemic datasets further demonstrate accurate state prediction and yield compact symbolic models of empirical dynamics. A rollout-error analysis provides a theoretical rationale for derivative-consistency feedback, while ablation studies demonstrate its contribution to long-horizon predictive performance.