# Constructing interpretable principal curve using Neural ODEs

arXiv: https://arxiv.org/abs/2311.09274 · HTML: https://mlsanity.com/paper/2311.09274/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 29 references that are in the map.
- First posted on arXiv 15 Nov 2023.

## Builds on (most cited first, 7 of 7)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Autonomous Discovery of Unknown Reaction Pathways from Data by Chemical Reaction Neural Network](https://mlsanity.com/paper/2002.09062/) (2020)
- [Uncovering differential equations from data with hidden variables](https://mlsanity.com/paper/2002.02250/) (2020)
- [Finite Time Lyapunov Exponent Analysis of Model Predictive Control and Reinforcement Learning](https://mlsanity.com/paper/2304.03326/) (2023)
- [Automated, predictive, and interpretable inference of C. elegans escape dynamics](https://mlsanity.com/paper/1809.09321/) (2018)

## Abstract (arXiv)

The study of high dimensional data sets often rely on their low dimensional projections that preserve the local geometry of the original space. While numerous methods have been developed to summarize this space as variations of tree-like structures, they are usually non-parametric and "static" in nature. As data may come from systems that are dynamical such as a differentiating cell, a static, non-parametric characterization of the space may not be the most appropriate. Here, we developed a framework, the principal flow, that is capable of characterizing the space in a dynamical manner. The principal flow, defined using neural ODEs, directs motion of a particle through the space, where the trajectory of the particle resembles the principal curve of the dataset. We illustrate that our framework can be used to characterize shapes of various complexities, and is flexible to incorporate summaries of relaxation dynamics.