无需导数估计，可发现任意阶动态系统方程
Sparse Equation Matching: A Derivative-Free Learning for General-Order Dynamical Systems
- 基于格林函数的积分稀疏回归，避免导数计算
- 在多类动态系统上优于传统导数依赖方法
- 适用于脑电数据建模，揭示任务相关脑区连接
方程发现是揭示复杂系统内在动态的核心学习任务，在脑连接分析、气候建模、基因调控和物理仿真等领域有广泛应用。然而，现有方法多依赖精确的导数估计，且仅限于一阶系统，限制了其在真实场景中的应用。本文提出稀疏方程匹配（SEM），一个统一框架，涵盖多种现有方程发现方法。SEM采用基于格林函数的积分稀疏回归，实现对任意阶动态系统中微分算子及其驱动函数的无导数估计。通过大量模拟实验验证了SEM的有效性，其性能优于基于导数的方法。进一步将SEM应用于52名参与者在脑机接口实验中记录的多个眼动任务脑电信号，成功识别出跨被试的活跃脑区，并揭示了任务特异性的连接模式。这些结果为脑连接机制提供了重要见解。
原文摘要 · Abstract (English)
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 simulation. However, many existing approaches rely on accurate derivative estimation and are limited to first-order dynamical systems, restricting their applicability in 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 approach 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.
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