Complex network data is prevalent in various real-world domains, including physical, technological, and biological systems. Despite this prevalence, predicting trends and understanding behavioral patterns in complex systems remain challenging due to poorly understood underlying mechanisms. While data-driven methods have advanced in uncovering governing equations from time series data, efforts to extract physical laws from network data are limited and often struggle with incomplete or noisy data. Additionally, they suffer from computational costs on network data, making it difficult to scale to real-world networks. To address these challenges, we introduce a novel approach called the Finite Expression Method (FEX) and its fast algorithm for learning dynamics on complex networks. FEX represents dynamics on complex networks using binary trees composed of finite mathematical operators. The nodes within these trees are trained through a combinatorial optimization process guided by reinforcement learning techniques. This unique configuration allows FEX to capture complex dynamics with minimal prior knowledge of the system and a small dictionary of mathematical operators. We also integrate a fast, stochastic algorithm into FEX, reducing the computational complexity from $O(N^2)$ to $O(N)$. Our extensive numerical experiments demonstrate that FEX excels in accurately identifying dynamics across diverse network topologies and dynamic behaviors.

翻译：复杂网络数据普遍存在于现实世界的各个领域，包括物理、技术和生物系统。尽管普遍存在，但由于底层机制理解不足，预测复杂系统的趋势和理解行为模式仍然具有挑战性。虽然数据驱动方法在从时间序列数据中揭示控制方程方面取得了进展，但从网络数据中提取物理定律的努力有限，并且常常难以处理不完整或有噪声的数据。此外，这些方法在处理网络数据时计算成本高昂，难以扩展到现实世界的网络。为了解决这些挑战，我们引入了一种称为有限表达式方法（FEX）的新方法及其用于学习复杂网络上动力学的快速算法。FEX使用由有限数学运算符组成的二叉树来表示复杂网络上的动力学。这些树中的节点通过强化学习技术指导的组合优化过程进行训练。这种独特的配置使FEX能够以最少的系统先验知识和一个小型数学运算符字典来捕捉复杂的动力学。我们还将一种快速的随机算法集成到FEX中，将计算复杂度从$O(N^2)$降低到$O(N)$。我们大量的数值实验表明，FEX在准确识别不同网络拓扑和动态行为中的动力学方面表现出色。