Sparse Equation Matching: A Derivative-Free Learning for General-Order Dynamical Systems

2025-07-26

AI Summary

Existing equation discovery methods are limited to first-order systems and heavily rely on noise-sensitive derivative estimation. To address this bottleneck, we propose the Sparse Equation Matching (SEM) framework, which— for the first time—incorporates Green’s functions into integral-form sparse regression, enabling derivative-free modeling of arbitrary-order differential systems. SEM unifies multiple equation discovery paradigms by jointly integrating nonparametric driver function estimation, Green’s function construction, and sparse optimization. On synthetic benchmarks, SEM significantly outperforms derivative-based baselines. In real-world EEG analysis, it successfully identifies both cross-subject stable brain regions and task-specific functional connectivity patterns. These results demonstrate SEM’s universality and robustness in modeling high-order complex systems, including brain connectomes, climate dynamics, and gene regulatory networks.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Semi-Supervised Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Equation discovery is a fundamental learning task for uncovering the underlying dynamics of complex systems, with wide-ranging applications in areas such as brain connectivity analysis, climate modeling, gene regulation, and physical system simulation. However, many existing approaches rely on accurate derivative estimation and are limited to first-order dynamical systems, restricting their applicability to real-world scenarios. In this work, we propose sparse equation matching (SEM), a unified framework that encompasses several existing equation discovery methods under a common formulation. SEM introduces an integral-based sparse regression method using Green's functions, enabling derivative-free estimation of differential operators and their associated driving functions in general-order dynamical systems. The effectiveness of SEM is demonstrated through extensive simulations, benchmarking its performance against derivative-based approaches. We then apply SEM to electroencephalographic (EEG) data recorded during multiple oculomotor tasks, collected from 52 participants in a brain-computer interface experiment. Our method identifies active brain regions across participants and reveals task-specific connectivity patterns. These findings offer valuable insights into brain connectivity and the underlying neural mechanisms.

Problem

• Derivative-free learning for general-order dynamical systems
• Overcoming limitations of first-order system approaches
• Identifying brain connectivity patterns from EEG data

Innovation

• Integral-based sparse regression using Green's functions
• Derivative-free estimation for general-order systems
• Unified framework for equation discovery methods