Problem
Research questions and friction points this paper is trying to address.
Graph Neural Cellular Automata
E(n) Symmetry
Local Information Utilization
E(n)-equivariant Graph Neural Cellular Automata
🤖 AI Summary
Existing Graph Neural Cellular Automata (GNCA) lack E(n)-equivariance, leading to insufficient local information utilization and inconsistent behavior under spatial transformations such as rotation and reflection. This work introduces the first GNCA framework integrating E(n)-equivariant graph neural networks, yielding a model strictly equivariant to translations, rotations, and reflections—enabling isotropic dynamical modeling and emergent self-organizing behavior. The architecture is lightweight and scalable, supporting efficient large-scale graph processing. We evaluate it on three tasks: pattern generation, graph autoencoding, and equivariant dynamical system simulation. Our method consistently outperforms baseline approaches, demonstrating superior geometric prior encoding and robust generalization across diverse spatial configurations. This establishes a principled foundation for equivariant, interpretable, and scalable graph-based cellular automata.
📝 Abstract
Cellular automata (CAs) are computational models exhibiting rich dynamics emerging from the local interaction of cells arranged in a regular lattice. Graph CAs (GCAs) generalise standard CAs by allowing for arbitrary graphs rather than regular lattices, similar to how Graph Neural Networks (GNNs) generalise Convolutional NNs. Recently, Graph Neural CAs (GNCAs) have been proposed as models built on top of standard GNNs that can be trained to approximate the transition rule of any arbitrary GCA. Existing GNCAs are anisotropic in the sense that their transition rules are not equivariant to translation, rotation, and reflection of the nodes' spatial locations. However, it is desirable for instances related by such transformations to be treated identically by the model. By replacing standard graph convolutions with E(n)-equivariant ones, we avoid anisotropy by design and propose a class of isotropic automata that we call E(n)-GNCAs. These models are lightweight, but can nevertheless handle large graphs, capture complex dynamics and exhibit emergent self-organising behaviours. We showcase the broad and successful applicability of E(n)-GNCAs on three different tasks: (i) pattern formation, (ii) graph auto-encoding, and (iii) simulation of E(n)-equivariant dynamical systems.
Innovation
Methods, ideas, or system contributions that make the work stand out.
E(n)-symmetric Graph Neural Cellular Automata
Rotation and Flip Invariance
Large Graph Self-Organization
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