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Kuramoto Oscillators and Swarms on Manifolds for Geometry Informed Machine Learning

Vladimir Jacimovic

TL;DR

This work proposes a geometry-informed ML framework built on swarms of Kuramoto oscillators generalized to manifolds, including spheres, Lie groups, and hyperbolic spaces, to encode maps and learn coupled actions of transformation groups. It unifies geometric Riccati ODEs, gradient-flow structures, and directional statistics to enable probabilistic modeling, learning on non-Euclidean spaces, and physics-informed perspectives. Core contributions include (i) a systematic treatment of Kuramoto models on higher-dimensional manifolds and their gradient-flow properties, (ii) a survey of directional-statistics-based distributions compatible with swarming dynamics for latent-space modeling, and (iii) practical paradigms for training swarms in supervised, unsupervised, and RL settings, plus illustrative examples like Wahba’s problem and robotic arm rotations. The framework offers a principled, low-parameter approach to learning on curved spaces with potential impact on hyperbolic data, group actions, and geometry-aware ML applications.

Abstract

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.

Kuramoto Oscillators and Swarms on Manifolds for Geometry Informed Machine Learning

TL;DR

This work proposes a geometry-informed ML framework built on swarms of Kuramoto oscillators generalized to manifolds, including spheres, Lie groups, and hyperbolic spaces, to encode maps and learn coupled actions of transformation groups. It unifies geometric Riccati ODEs, gradient-flow structures, and directional statistics to enable probabilistic modeling, learning on non-Euclidean spaces, and physics-informed perspectives. Core contributions include (i) a systematic treatment of Kuramoto models on higher-dimensional manifolds and their gradient-flow properties, (ii) a survey of directional-statistics-based distributions compatible with swarming dynamics for latent-space modeling, and (iii) practical paradigms for training swarms in supervised, unsupervised, and RL settings, plus illustrative examples like Wahba’s problem and robotic arm rotations. The framework offers a principled, low-parameter approach to learning on curved spaces with potential impact on hyperbolic data, group actions, and geometry-aware ML applications.

Abstract

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.
Paper Structure (57 sections, 8 theorems, 55 equations)

This paper contains 57 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Some recent trends in theoretical ML
  3. Deep Learning via continuous-time controlled dynamical system
  4. Probabilistic modeling and inference in DL
  5. Deep Learning in non-Euclidean spaces
  6. Learning (coupled) actions of transformation groups
  7. Physics Informed ML
  8. Kuramoto model
  9. Kuramoto models from the geometric point of view
  10. Hyperbolic geometry of Kuramoto ensembles
  11. Kuramoto models with several globally coupled sub-ensembles
  12. Kuramoto models on higher-dimensional manifolds
  13. Non-Abelian Kuramoto models on Lie groups
  14. Kuramoto models on spheres
  15. Real Kuramoto models on spheres
  16. ...and 42 more sections

Key Result

Proposition 1

MMS Consider the system (Riccati_global) with an arbitrary complex-valued coupling function $f = f(z_1,\dots,z_N)$. There exists a one-parametric family $g_t \in {\mathbb G}$, such that

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Corollary 1
  • Remark 3
  • Remark 4
  • Proposition 2
  • Proposition 3
  • Corollary 2
  • Proposition 4
  • ...and 3 more