Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning Decentralized Swarms Using Rotation Equivariant Graph Neural Networks

Taos Transue, Bao Wang

TL;DR

This work addresses decentralized flocking by enforcing rotation and translation symmetries in learning-based controllers. By replacing the TDAGNN’s CNN with an $\mathrm{O}(2)$-equivariant CNN and applying symmetry-aware activations, the authors achieve comparable flocking performance with far fewer training data and parameters, and improved generalization across tasks. They develop a rigorous generalization analysis under fast-forward behavior cloning and demonstrate empirically that the equivariant model reduces generalization bounds and improves reliability in flocking, leader following, and obstacle avoidance. The approach offers a scalable, symmetry-consistent framework for learning decentralized swarm controllers with practical impact on autonomous fleets and sensor networks.

Abstract

The orchestration of agents to optimize a collective objective without centralized control is challenging yet crucial for applications such as controlling autonomous fleets, and surveillance and reconnaissance using sensor networks. Decentralized controller design has been inspired by self-organization found in nature, with a prominent source of inspiration being flocking; however, decentralized controllers struggle to maintain flock cohesion. The graph neural network (GNN) architecture has emerged as an indispensable machine learning tool for developing decentralized controllers capable of maintaining flock cohesion, but they fail to exploit the symmetries present in flocking dynamics, hindering their generalizability. We enforce rotation equivariance and translation invariance symmetries in decentralized flocking GNN controllers and achieve comparable flocking control with 70% less training data and 75% fewer trainable weights than existing GNN controllers without these symmetries enforced. We also show that our symmetry-aware controller generalizes better than existing GNN controllers. Code and animations are available at http://github.com/Utah-Math-Data-Science/Equivariant-Decentralized-Controllers.

Learning Decentralized Swarms Using Rotation Equivariant Graph Neural Networks

TL;DR

This work addresses decentralized flocking by enforcing rotation and translation symmetries in learning-based controllers. By replacing the TDAGNN’s CNN with an -equivariant CNN and applying symmetry-aware activations, the authors achieve comparable flocking performance with far fewer training data and parameters, and improved generalization across tasks. They develop a rigorous generalization analysis under fast-forward behavior cloning and demonstrate empirically that the equivariant model reduces generalization bounds and improves reliability in flocking, leader following, and obstacle avoidance. The approach offers a scalable, symmetry-consistent framework for learning decentralized swarm controllers with practical impact on autonomous fleets and sensor networks.

Abstract

The orchestration of agents to optimize a collective objective without centralized control is challenging yet crucial for applications such as controlling autonomous fleets, and surveillance and reconnaissance using sensor networks. Decentralized controller design has been inspired by self-organization found in nature, with a prominent source of inspiration being flocking; however, decentralized controllers struggle to maintain flock cohesion. The graph neural network (GNN) architecture has emerged as an indispensable machine learning tool for developing decentralized controllers capable of maintaining flock cohesion, but they fail to exploit the symmetries present in flocking dynamics, hindering their generalizability. We enforce rotation equivariance and translation invariance symmetries in decentralized flocking GNN controllers and achieve comparable flocking control with 70% less training data and 75% fewer trainable weights than existing GNN controllers without these symmetries enforced. We also show that our symmetry-aware controller generalizes better than existing GNN controllers. Code and animations are available at http://github.com/Utah-Math-Data-Science/Equivariant-Decentralized-Controllers.

Paper Structure

This paper contains 33 sections, 20 theorems, 81 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Organization
  4. Background
  5. Flock representation and definitions
  6. Centralized flocking
  7. Decentralized flocking
  8. ML-based decentralized flocking
  9. Time-delayed aggregation graph neural networks
  10. Imitaiton learning
  11. Rotation equivariance and translation invariance of flocking controllers
  12. Equivariant controllers for learning decentralized flocking
  13. Rotation equivariant convolution layers
  14. Rotation equivariant activations
  15. Further improving TDAGNN
  16. ...and 18 more sections

Key Result

theorem 3

Define the loss funciton ${\mathcal{L}}: {\mathcal{Y}} \times {\mathcal{Y}} \to [0, 1]$, let ${\mathcal{F}} = \{f: {\mathcal{X}} \to {\mathcal{Y}}\}$, and let ${\mathcal{D}}$ be a probability distribution over ${\mathcal{X}} \times {\mathcal{Y}}$. Let ${\mathcal{B}} = \{(x_{b}, y_{b})\}_{{b} = 1}^{B where ${{\mathcal{F}}_{\mathcal{L}}} = \{(x, y) \mapsto {\mathcal{L}}(f(x), y): f \in {\mathcal{F}}

Figures (13)

  • Figure 1: Snapshots at times $t_n$ of a flock of agents (orange dots) with velocities (blue arrows) and the flock's communication graph (light blue edges). When controlled by the time-dependent neighborhood decentralized flocking controller from Tanner et al. tannerStableFlockingMobile2003 (top row), the communication graph loses connectivity from time $t_{27}$ onward. In contrast, the ML-based decentralized flocking controller from Tolstaya et al. tolstayaLearningDecentralizedControllers2017b (bottom row), not trained on these agent initial conditions, successfully maintains communication graph connectivity and achieves asymptotic flocking.
  • Figure 2: Examples from the RandomDisk dataset of flock initial conditions. There are ${N} = 100$ agents (orange dots) with at least $\deg_{\min} = 2$ neighbors (indicated by light blue edges connecting them). The distance between an agent and its neighbors is between ${R_{\min}} = 0.1$ and ${R_{\mathrm{comm}}} = 1$. The agents' velocities (dark blue arrows) have magnitudes no larger than $2{v_{\mathrm{max}}} = 6$.
  • Figure 3: Performance of ML controllers in flocking as they train. They are evaluated on the RandomDisk validation set with $100$ agents. Each simulation is run for ${T} = 2/{\Delta t}$ time steps where ${\Delta t} = 10^{-2}$. The curves show the median values of the respective metrics' integrals and the colored areas show their corresponding interquartile ranges.
  • Figure 4: Velocity variance of the controllers in flocking over the simulation time with time step size ${\Delta t} = 10^{-3}$. They are evaluated on the RandomDisk test set with the number of agents ${N} \in \{50, 100, 200, 400\}$. The lines show the median metric values and the colored areas show the corresponding interquartile ranges.
  • Figure 5: Mean acceleration norm of the controllers in flocking over the simulation time with time step size ${\Delta t}$. They are evaluated on the RandomDisk test set with the number of agents ${N} \in \{50, 100, 200, 400\}$. The lines show the median metric values and the colored areas show the corresponding interquartile ranges.
  • ...and 8 more figures

Theorems & Definitions (44)

  • definition thmcounterdefinition: Asymptotic flocking choiEmergentDynamicsCuckerSmale2016
  • definition thmcounterdefinition: Agent ${R_{\min}}$-separation
  • definition thmcounterdefinition: Generalization gap
  • definition thmcounterdefinition: Empirical Rademacher complexity
  • theorem 3: ERC bounds generalization gap mohriFoundationsMachineLearning2012
  • definition thmcounterdefinition: Covering number
  • lemma thmcounterlemma: Bounding ERC bartlettSpectrallynormalizedMarginBounds2017
  • definition thmcounterdefinition: Input of TDAGNN as an MLP
  • definition thmcounterdefinition: Input of $\mathrm{O}(2)$ equivariant TDAGNN as an MLP
  • proposition thmcounterproposition: Generalization bound of TDAGNN
  • ...and 34 more