Localize, Edit, and Learn: Structure-aware Symbolic Regression for Network Dynamics

Describing network dynamics through the discovery of symbolic equations is critical for understanding complex systems. However, as a network's dynamics are jointly governed by its nodes' local processes and mutual interactions, where both encapsulate diverse symbolic components, existing symbolic regression methods struggle to precisely localize erroneous components and translate diagnostic evidence into controlled structural edits. Furthermore, despite the rise of large language models (LLMs) in evolutionary symbolic regression for network dynamics, a dedicated policy update paradigm is needed to let LLMs learn from historical edits and internalize such capability, instead of merely reusing experience through prompts. We introduce Network Symbolic Patch Evolution (NetSPE), a framework that connects localization, editing, and policy learning for equation generation. Guided by network structural priors, NetSPE uses residual contributions to localize a target component to be updated, and then instructs an LLM to generate constrained local replacements. We further propose Structural Patch Preference Optimization (SPPO), which adapts the editing policy using preferences among alternative edits generated under the same editing state and evaluated against their parent. Experiments on two real-world and one synthetic benchmarks show NetSPE's efficacy in its prediction accuracy, equation quality, and generalizability.