# Kuramoto Oscillators and Swarms on Manifolds for Geometry Informed Machine Learning

arXiv: https://arxiv.org/abs/2405.09453 · HTML: https://mlsanity.com/paper/2405.09453/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 148 references that are in the map.
- First posted on arXiv 15 May 2024.

## Builds on (most cited first, 28 of 28)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Differentiable plasticity: training plastic neural networks with backpropagation](https://mlsanity.com/paper/1804.02464/) (2018)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [von Mises-Fisher Mixture Model-based Deep learning: Application to Face Verification](https://mlsanity.com/paper/1706.04264/) (2017)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [ODE2VAE: Deep generative second order ODEs with Bayesian neural networks](https://mlsanity.com/paper/1905.10994/) (2019)
- [Deep Learning Theory Review: An Optimal Control and Dynamical Systems Perspective](https://mlsanity.com/paper/1908.10920/) (2019)
- [Hyperbolic Deep Reinforcement Learning](https://mlsanity.com/paper/2210.01542/) (2022)
- [Skip-gram word embeddings in hyperbolic space](https://mlsanity.com/paper/1809.01498/) (2018)
- [Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning](https://mlsanity.com/paper/1808.02229/) (2018)
- [Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance](https://mlsanity.com/paper/2302.11024/) (2023)
- [Directional Statistics in Machine Learning: a Brief Review](https://mlsanity.com/paper/1605.00316/) (2016)
- [Consensus-Based Optimization on the Sphere: Convergence to Global Minimizers and Machine Learning](https://mlsanity.com/paper/2001.11988/) (2020)
- [Bingham Policy Parameterization for 3D Rotations in Reinforcement Learning](https://mlsanity.com/paper/2202.03957/) (2022)
- [A Wrapped Normal Distribution on Hyperbolic Space for Gradient-Based Learning](https://mlsanity.com/paper/1902.02992/) (2019)
- [Mean field repulsive Kuramoto models: Phase locking and spatial signs](https://mlsanity.com/paper/1803.02647/) (2018)
- [Reparameterizing Distributions on Lie Groups](https://mlsanity.com/paper/1903.02958/) (2019)

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

- [The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics](https://mlsanity.com/paper/2608.10234/) (2026)

## Abstract (arXiv)

We propose the idea of using Kuramoto models (including their higher-dimensional generalizations) for machine learning over non-Euclidean data sets. These models are systems of matrix ODE's describing collective motions (swarming dynamics) of abstract particles (generalized oscillators) on spheres, homogeneous spaces and Lie groups. Such models have been extensively studied from the beginning of XXI century both in statistical physics and control theory. They provide a suitable framework for encoding maps between various manifolds and are capable of learning over spherical and hyperbolic geometries. In addition, they can learn coupled actions of transformation groups (such as special orthogonal, unitary and Lorentz groups). Furthermore, we overview families of probability distributions that provide appropriate statistical models for probabilistic modeling and inference in Geometric Deep Learning. We argue in favor of using statistical models which arise in different Kuramoto models in the continuum limit of particles. The most convenient families of probability distributions are those which are invariant with respect to actions of certain symmetry groups.