Sparse Equation Matching: A Derivative-Free Learning for General-Order Dynamical Systems
Authors: Jiaqiang Li, Jianbin Tan, Xueqin Wang
Organizations: School of Management, University of Science and Technology of China, Anhui, Hefei, China · Department of Biostatistics and Bioinformatics, Duke University, Durham, NC, USA
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.
Figures & tables
Figure 1: True trajectories and their derivatives, along with corresponding estimates obtained from noisy observations using RKHS regression. In these examples, the signal-to-noise ratio is set to 0.15 , and the sample sizes are n=50,150,250 , with observation points evenly spaced over the time interval [0,10] . Blue triangles denote the observed data points.
Algorithm 1 Sparse Equation Matching for General-Order Dynamical Systems
Figure 2: Relative error rate (RER) of the estimated driving functions under various sample sizes n and contamination levels γ (main title of each subgraph).
Figure 3: True and average estimated vector fields for the nonlinear pendulum under the setting of n=150 and γ=0.05 . The color indicates the magnitude of the arrows.
Figure 4: (A) The directed interaction network; Each node represents a neural mass unit whose dynamics are governed by local excitation, inter-node directional coupling, and proportional damping that stabilizes oscillatory amplitudes. The direction between two nodes indicates that the trajectory of one node is affected by that of the other according to the dynamic directional model. (B) The trajectories generated from the dynamic directional model.
Figure 5: Mean accuracy of the estimated networks under different sample sizes n and noise levels γ (indicated in the title of each subfigure).
Figure 6: True and average estimated vector fields for X19 under three different values of b , based on 2000 simulation runs with n=150 and γ=0.05 . The color indicates the magnitude of the arrows.
Figure 7: Relative error rate (RER) of the estimated driving functions for different numbers of observed time points n=15,20,25,30 under contamination level γ=0.05 . SEM performs pre-estimation using an RKHS approach, whereas SEM-FPCA employs FPCA for pre-estimation.
Figure 8: Smoothed and predicted dynamics of the eye-blinking task for EEG channels C3, CP5, and FT7 from participants h=18,25,37 . The dashed vertical lines mark the transition from the training interval ( t<4 ) to the validation interval ( t≥4 ).
Figure 9: Boxplots of relative prediction errors (RPE) for 52 participants across three oculomotor tasks: eye blinking (blinking), horizontal eye movement (horizontal), and vertical eye movement (vertical). The RPE is evaluated on the held-out validation interval [4,5] .
Figure 10: Directed brain networks for the eye-blinking, horizontal eye movement, and vertical eye movement tasks. Edge color represents the empirical probability of occurrence of each edge across participants.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 11: True and average estimated vector fields for the pendulum system with nonlinear damping under the setting n=150 and γ=0.05 . The color indicates the magnitude of the velocity vectors.
Figure 12: Directed brain networks for different tasks: (A) resting state, (B) eye-blinking, (C) horizontal eye movement, (D) vertical eye movement. Edge color indicates the empirical probability of each edge occurring across participants.
Figure 13: A comparison of centrality measures. The diagrams display rank-ordered distributions of betweenness centrality, closeness centrality, in-degree, and out-degree for the network associated with the eye-blinking task, as shown in Figure 10 .
Figure 14: A comparison of centrality measures. Diagrams show rank-ordered distributions of betweenness, closeness, in degree and out degree for the network of horizontal eyeball movement task shown in Figure 10
Figure 15: A comparison of centrality measures. Diagrams show rank-ordered distributions of betweenness, closeness, in degree and out degree for the network of vertical eyeball movement task shown in Figure 10
Data Science and Analytics Institute, University of Oklahoma, Norman, OK, USA · School of Industrial and Systems Engineering, University of Oklahoma, Norman, OK, USA