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

TL;DR

This paper introduces Sparse Equation Matching (SEM), a derivative-free framework for discovering equations governing complex, general-order dynamical systems, demonstrated on simulations and EEG data to reveal neural connectivity.

Contribution

SEM unifies existing methods under a common integral-based sparse regression framework, enabling derivative-free equation discovery for complex dynamical systems.

Findings

SEM outperforms derivative-based methods in simulations.

Applied to EEG data, SEM identifies active brain regions.

Reveals task-specific neural connectivity patterns.

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 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.