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

arXiv: https://arxiv.org/abs/2507.20072 · HTML: https://mlsanity.com/paper/2507.20072/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 66 references that are in the map.
- First posted on arXiv 26 Jul 2025.

## Builds on (most cited first, 14 of 14)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes](https://mlsanity.com/paper/2009.07444/) (2020)
- [Network Reconstruction From High Dimensional Ordinary Differential Equations](https://mlsanity.com/paper/1610.03177/) (2016)
- [Kernel Ordinary Differential Equations](https://mlsanity.com/paper/2008.02915/) (2020)
- [Adaptive Estimation for Nonlinear Systems using Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1707.01567/) (2017)
- [Naive Penalized Spline Estimators of Derivatives Achieve Optimal Rates of Convergence](https://mlsanity.com/paper/2208.10664/) (2022)
- [Green's matching: an efficient approach to parameter estimation in complex dynamic systems](https://mlsanity.com/paper/2403.14531/) (2024)
- [Functional-SVD for Heterogeneous Trajectories: Case Studies in Health](https://mlsanity.com/paper/2410.03619/) (2024)

## Built on by (most cited first, 1 of 1)

- [Smooth Flow Matching for Synthesizing Functional Data](https://mlsanity.com/paper/2508.13831/) (2025)

## Abstract (arXiv)

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.