Complex-Valued GNNs for Distributed Basis-Invariant Control of Planar Systems

2026-04-02

AI Summary

This work addresses the limitation of existing distributed graph neural network (GNN) controllers, which rely on a global coordinate frame and thus struggle in GPS- or compass-denied environments. To overcome this, the authors propose a complex-domain GNN architecture that encodes 2D geometric relationships and local coordinate transformations using complex numbers. Each layer employs complex-valued linear mappings with phase-equivariant activation functions, ensuring global invariance to the choice of local coordinate systems. This approach is the first to integrate complex-domain modeling and phase-equivariant mechanisms into distributed GNN-based control. Evaluated on formation tracking tasks, it significantly outperforms real-valued baselines, achieving notable improvements in data efficiency, tracking accuracy, and cross-coordinate generalization—eliminating the need for reference-frame alignment.

Abstract

Graph neural networks (GNNs) are a well-regarded tool for learned control of networked dynamical systems due to their ability to be deployed in a distributed manner. However, current distributed GNN architectures assume that all nodes in the network collect geometric observations in compatible bases, which limits the usefulness of such controllers in GPS-denied and compass-denied environments. This paper presents a GNN parametrization that is globally invariant to choice of local basis. 2D geometric features and transformations between bases are expressed in the complex domain. Inside each GNN layer, complex-valued linear layers with phase-equivariant activation functions are used. When viewed from a fixed global frame, all policies learned by this architecture are strictly invariant to choice of local frames. This architecture is shown to increase the data efficiency, tracking performance, and generalization of learned control when compared to a real-valued baseline on an imitation learning flocking task.

Problem

• distributed control
• basis invariance
• graph neural networks
• geometric observations
• planar systems

Innovation

• Complex-Valued GNN
• Basis Invariance
• Phase-Equivariant Activation
• Distributed Control
• Geometric Learning