# Automatically identifying ordinary differential equations from data

arXiv: https://arxiv.org/abs/2304.11182 · HTML: https://mlsanity.com/paper/2304.11182/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 65 references that are in the map.
- First posted on arXiv 21 Apr 2023.

## Builds on (most cited first, 8 of 8)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [Data-driven Discovery of Cyber-Physical Systems](https://mlsanity.com/paper/1810.00697/) (2018)

## Abstract (arXiv)

Discovering nonlinear differential equations that describe system dynamics from empirical data is a fundamental challenge in contemporary science. Here, we propose a methodology to identify dynamical laws by integrating denoising techniques to smooth the signal, sparse regression to identify the relevant parameters, and bootstrap confidence intervals to quantify the uncertainty of the estimates. We evaluate our method on well-known ordinary differential equations with an ensemble of random initial conditions, time series of increasing length, and varying signal-to-noise ratios. Our algorithm consistently identifies three-dimensional systems, given moderately-sized time series and high levels of signal quality relative to background noise. By accurately discovering dynamical systems automatically, our methodology has the potential to impact the understanding of complex systems, especially in fields where data are abundant, but developing mathematical models demands considerable effort.