Discovering Generalizable Governing Equations for Graph Dynamical Systems with Interpretable Neural Networks

The discovery of symbolic governing equations is a central goal in science; yet, it remains challenging particularly for graph dynamical systems, where the network topology further shapes the system behavior. While artificial intelligence offers powerful tools for modeling these dynamics, the field lacks a rigorous comparative benchmark to assess the true scientific utility of the discovered laws. To address this challenge, this work proposes a novel evaluation pipeline designed to rigorously assess state-of-the-art symbolic regression models for graph equation discovery. Moving beyond simple fitting metrics, this framework evaluates discovered laws based on their long-term trajectory stability and, critically, their out-of-distribution generalization to unseen graph topologies. We benchmark established methods, including sparse regression and MLP-based architectures, and introduce the Graph Kolmogorov-Arnold Network-ODE (GKAN-ODE) model, a novel adaptation of KANs explicitly tailored for this domain, augmented by hyperparameter-free multiplicative nodes and a new Spline-Wise symbolic regression algorithm. Across a suite of synthetic and real-world graph dynamical systems, we numerically demonstrate through extensive experiments that neural-based approaches, particularly the GKAN-ODE model, recover exact ground-truth equations and achieve trajectory errors up to two orders of magnitude lower than the baseline methods on out-of-distribution test graphs.