# Machine learning identifies nullclines in oscillatory dynamical systems

arXiv: https://arxiv.org/abs/2503.16240 · HTML: https://mlsanity.com/paper/2503.16240/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 39 references that are in the map.
- First posted on arXiv 20 Mar 2025.

## Builds on (most cited first, 6 of 6)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Learning Symbolic Physics with Graph Networks](https://mlsanity.com/paper/1909.05862/) (2019)
- [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024)

## Abstract (arXiv)

We introduce CLINE (Computational Learning and Identification of Nullclines), a neural network-based method that uncovers the hidden structure of nullclines from oscillatory time series data. Unlike traditional approaches aiming at direct prediction of system dynamics, CLINE identifies static geometric features of the phase space that encode the (non)linear relationships between state variables. It overcomes challenges such as multiple time scales and strong nonlinearities while producing interpretable results convertible into symbolic differential equations. We validate CLINE on various oscillatory systems, showcasing its effectiveness.