Discover network dynamics with neural symbolic regression

Yu et al., 2025

Discovering network dynamics—automatically

Complex systems are everywhere: from ecosystems and gene networks to epidemics and social behavior. Each consists of many interacting components whose collective dynamics are notoriously difficult to model. Traditionally, scientists have relied on intuition and simplified equations—but for most real systems, the governing laws remain unknown.

Zihan Yu, Jingtao Ding & Yong Li  introduce ND2, a neural symbolic regression framework that discovers network dynamics directly from data. The key insight is to reduce the overwhelming search space of possible equations—normally exploding with the number of nodes—to an equivalent one-dimensional symbolic problem. ND2 combines graph neural networks and transformers (via a pretrained model called NDformer) with Monte Carlo tree search to automatically uncover concise, interpretable formulas that explain how networks evolve.

Applied to ten benchmark systems, ND2 correctly recovered the governing equations of classics such as Kuramoto oscillators, FitzHugh–Nagumo neurons, and Lotka–Volterra populations. More impressively, when tested on real data, it corrected existing biological models—reducing prediction errors by 60% in gene regulation and 56% in microbial communities—by revealing hidden higher-order interactions. In epidemic networks, ND2 uncovered transmission laws that explain cross-country differences in intervention effects and even captured the same power-law patterns across scales.

This study demonstrates how machine-driven discovery can bridge observational data and theoretical understanding, advancing complexity science from describing what we observe to deriving why it happens—directly from the data itself.

Abstract:

Network dynamics are fundamental to analyzing the properties of high-dimensional complex systems and understanding their behavior. Despite the accumulation of observational data across many domains, mathematical models exist in only a few areas with clear underlying principles. Here we show that a neural symbolic regression approach can bridge this gap by automatically deriving formulas from data. Our method reduces searches on high-dimensional networks to equivalent one-dimensional systems and uses pretrained neural networks to guide accurate formula discovery. Applied to ten benchmark systems, it recovers the correct forms and parameters of underlying dynamics. In two empirical natural systems, it corrects existing models of gene regulation and microbial communities, reducing prediction error by 59.98% and 55.94%, respectively. In epidemic transmission across human mobility networks of various scales, it discovers dynamics that exhibit the same power-law distribution of node correlations across scales and reveal country-level differences in intervention effects. These results demonstrate that machine-driven discovery of network dynamics can enhance understandings of complex systems and advance the development of complexity science.

https://www.nature.com/articles/s43588-025-00893-8