A Fast Algorithm for the Finite Expression Method in Learning Dynamics on Complex Networks

TL;DR

The paper tackles the challenge of discovering interpretable dynamical laws for large complex networks from time-series data, where interactions scale as $O(N^2)$ and data may be noisy or incomplete. It introduces the Finite Expression Method (FEX), which represents self- and interaction-dynamics as binary trees of simple operators and uses a reinforcement-learning controller to select the operator structure, with continuous optimization of the operator parameters. A stochastic variant (Stochastic-FEX) batches node interactions to reduce cost to $O(N)$ during learning, while preserving accuracy, and a convergence result is provided for the stochastic algorithm. Across Hindmarsh-Rose, FitzHugh-Nagumo, and coupled Rössler networks on scale-free graphs, FEX achieves accurate recovery of governing equations and robust performance under noise, low data resolution, and topology perturbations, outperforming Two-Phase SINDy and ARNI. The approach offers an interpretable, scalable framework for extracting network dynamics with potential real-world impact in physics, biology, and engineering.

Abstract

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.

Figures (7)

• Figure 1: (A) Based on the input data of the time series and adjacency matrix, FEX performs combinatorial optimization to infer the mathematical structures of the dynamics and then implements fine-tuning on the top-performing candidates in the pool to identify the optimal dynamics. (B) FEX implements combinatorial optimization by training a controller $\chi$ to output probability mass function for each node in the FEX binary tree.
• Figure 2: Representation of the components of our implementation with FHN dynamics. (A) In the offline stage, we generate time series data according to the governing law of the system, potentially with noise or low resolution. (B) In the online phase, based on the time series data, we use two FEX binary trees to infer the dynamics of the system.
• Figure 3: Computational rule of a FEX binary tree. Each unary operator is equipped with a weight $\alpha$ and bias $\beta$ to enhance expressiveness. For tree depth greater than one (i.e., $L > 1$), the computation is implemented by recursion.
• Figure 4: True HR dynamics and the dynamics identified by FEX.
• Figure 5: True FHN dynamics and the dynamics identified by FEX.

Theorems & Definitions (4)

• Definition 3.1: Mathematical expression liang2022finite
• Definition 3.2: $k$-finite expression liang2022finite
• Definition 3.3: Finite expression method
• Theorem 1