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.

翻译：方程发现是揭示复杂系统底层动力学的一项基础性学习任务，在脑连接分析、气候建模、基因调控和物理模拟等领域具有广泛应用。然而，现有许多方法依赖于精确的导数估计，且仅限于一阶动力系统，这限制了其在真实场景中的适用性。本文提出稀疏方程匹配（SEM），这是一个统一框架，将多种现有方程发现方法纳入一个共同的公式体系。SEM 引入了一种基于积分和格林函数的稀疏回归方法，能够对一般阶动力系统中的微分算子及其关联的驱动函数进行无导数估计。通过大量仿真实验，并与基于导数的方法进行性能基准测试，验证了 SEM 的有效性。随后，我们将 SEM 应用于一项脑机接口实验中从 52 名参与者记录的多项眼动任务期间的脑电图（EEG）数据。我们的方法识别了跨参与者的活跃脑区，并揭示了任务特异性的连接模式。这些发现为理解脑连接及其底层神经机制提供了有价值的见解。