# Characterizing Nonlinear Dynamics via Smooth Prototype Equivalences

arXiv: https://arxiv.org/abs/2503.10336 · HTML: https://mlsanity.com/paper/2503.10336/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 82 references that are in the map.
- First posted on arXiv 13 Mar 2025.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Improving Variational Auto-Encoders using Householder Flow](https://mlsanity.com/paper/1611.09630/) (2016)
- [Beyond Geometry: Comparing the Temporal Structure of Computation in Neural Circuits with Dynamical Similarity Analysis](https://mlsanity.com/paper/2306.10168/) (2023)
- [Evolving Semantic Prototype Improves Generative Zero-Shot Learning](https://mlsanity.com/paper/2306.06931/) (2023)
- [Algorithmic (Semi-)Conjugacy via Koopman Operator Theory](https://mlsanity.com/paper/2209.06374/) (2022)
- [Deep Learning of Conjugate Mappings](https://mlsanity.com/paper/2104.01874/) (2021)
- [Phase2vec: Dynamical systems embedding with a physics-informed convolutional network](https://mlsanity.com/paper/2212.03857/) (2022)
- [DFORM: Diffeomorphic vector field alignment for assessing dynamics across learned models](https://mlsanity.com/paper/2402.09735/) (2024)
- [Let's do the time-warp-attend: Learning topological invariants of dynamical systems](https://mlsanity.com/paper/2312.09234/) (2023)
- [Efficient pseudometrics for data-driven comparisons of nonlinear dynamical systems](https://mlsanity.com/paper/2409.18681/) (2024)

## Abstract (arXiv)

Characterizing the long term behavior of dynamical systems given limited measurements is a common challenge throughout the physical and biological sciences. This is a challenging task due to the sparsity and noise inherent to empirical observations, as well as the variability of possible long-term dynamics. We address this by introducing smooth prototype equivalences (SPE), a framework for matching sparse observations to prototypical behaviors using invertible neural networks which model smooth phase space deformations. SPE can localize the invariant sets describing long-term behavior of the observed dynamics through the learned mapping from prototype space to data space. Furthermore, SPE can classify dynamical regimes by comparing the data residual of the deformed measurements to prototype dynamics. Our method outperforms existing techniques in the classification of oscillatory systems and can efficiently identify invariant structures like limit cycles and fixed points in an equation-free manner, even when only a small, noisy subset of the phase space is observed. SPE further reveals driving genes in synthetic oscillators such as the repressilator regulatory circuit, and traces cyclic biological processes like the cell cycle trajectory directly from experimental high-dimensional single-cell gene expression data.