Sparse Equation Matching: A Derivative-Free Learning for General-Order Dynamical Systems
This addresses a bottleneck in equation discovery for complex systems like brain connectivity analysis, offering a derivative-free approach that extends applicability beyond first-order systems.
The paper tackles the problem of equation discovery for general-order dynamical systems without requiring derivative estimation, proposing Sparse Equation Matching (SEM) which uses Green's functions for integral-based sparse regression. The method was validated through simulations and applied to EEG data from 52 participants, identifying active brain regions and task-specific connectivity patterns.
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.