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

Jiaqiang Li, Jianbin Tan, Xueqin Wang

TL;DR

The paper tackles learning governing equations for general-order dynamical systems from data without requiring accurate derivatives, introducing Sparse Equation Matching (SEM), a derivative-free, integral-based sparse regression framework built on Green's functions. SEM unifies gradient- and integral-based equation discovery by transforming ODEs into integral forms and estimating both the differential operator of order $K$ and the driving function via a sparsity-promoting objective. Through simulations of a nonlinear pendulum and a second-order dynamic directional model, as well as analysis of EEG data from 52 participants across three oculomotor tasks, SEM demonstrates superior predictive accuracy and reliable network discovery compared with derivative-based methods like SINDy. The approach yields interpretable task-specific brain connectivity patterns and highlights SEM's potential as a robust tool for high-order dynamics in neuroscience and beyond.

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.

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

TL;DR

The paper tackles learning governing equations for general-order dynamical systems from data without requiring accurate derivatives, introducing Sparse Equation Matching (SEM), a derivative-free, integral-based sparse regression framework built on Green's functions. SEM unifies gradient- and integral-based equation discovery by transforming ODEs into integral forms and estimating both the differential operator of order and the driving function via a sparsity-promoting objective. Through simulations of a nonlinear pendulum and a second-order dynamic directional model, as well as analysis of EEG data from 52 participants across three oculomotor tasks, SEM demonstrates superior predictive accuracy and reliable network discovery compared with derivative-based methods like SINDy. The approach yields interpretable task-specific brain connectivity patterns and highlights SEM's potential as a robust tool for high-order dynamics in neuroscience and beyond.

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.

Paper Structure

This paper contains 14 sections, 2 theorems, 71 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Equation Matching for General-Order Systems
  4. Equation Matching with Green's Functions
  5. Sparse Equation Matching
  6. Simulation
  7. Data Analysis
  8. Discussion
  9. Technical Proof
  10. Proof of Theorem \ref{['theo:inte']}
  11. Proof of Theorem \ref{['thm-optimizer']}
  12. Supporting Results
  13. Nonlinear Pendulum with Misspecified Library
  14. Directed Brain Networks and Centrality Measures

Key Result

Theorem 1

Assume that each function $f_i(\bm{x}, t)$ in ode is continuous in $t$ and locally Lipschitz continuous in $\bm{x}$: Then, for any integer $k$ such that $1 \leq k \leq K - 1$, the differential equation ode is equivalent to the following integral equation: where $g_{ik}$ is a function in $\mathrm{Ker}\!\left(\frac{\mathrm{d}^k}{\mathrm{d}t^k}\right)$. Here,

Figures (15)

  • 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.
  • Figure 2: Relative error rate (RER) of the estimated driving functions under various sample sizes $n$ and contamination levels $\gamma$ (main title of each subgraph).
  • Figure 3: True and average estimated vector fields for the nonlinear pendulum under the setting of $n = 150$ and $\gamma = 0.05$. 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 $\gamma$ (indicated in the title of each subfigure).
  • ...and 10 more figures

Theorems & Definitions (13)

  • Example 1: Brain Connectivity Discovery
  • Example 2: Climate Dynamics Discovery
  • Example 3: Interaction Law Discovery
  • Remark 1: Choice of the Reproducing Kernel
  • Remark 2: Asymptotic Properties of Derivative Estimators via Penalized Splines
  • Theorem 1: Integral Form of Differential Equations
  • Definition 1: Order-$k$ Equation Matching
  • Remark 3: Connections to Gradient- and Integral-Based Methods for Equation Discovery
  • Remark 4: Derivative-Free Equation with Minimal Complexity
  • Remark 5: Connections to Network Discovery of Dynamical Systems
  • ...and 3 more