E(n)-equivariant Graph Neural Cellular Automata

TL;DR

This work introduces E(n)-GNCAs, isotropic-by-design Graph Neural Cellular Automata that enforce $E(n)$-equivariance to ensure frame-independence and locality. A single $EGC$ rule operates on state ${\mathbf S}=[{\mathbf X},{\mathbf H}]$ (optionally ${\mathbf V}$), yielding open-ended, globally coherent dynamics from purely local interactions. The approach is trained with a distance-based invariant loss $\mathcal{L}_{INV}$ for pattern formation and with a binary cross-entropy loss for graph autoencoding, demonstrating persistent pattern formation, effective graph autoencoding, and competitive dynamics-based simulation (e.g., Boids) across diverse graphs and dimensions. The results underscore the potential of isotropic, distributed neural automata for scalable self-organization, pattern synthesis, and geometry-aware learning in natural and engineered systems.

Abstract

Cellular automata (CAs) are notable 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. We note that existing GNCAs can violate the locality principle of CAs by leveraging global information and, furthermore, are anisotropic in the sense that their transition rules are not equivariant to isometries 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) isotropic pattern formation, (ii) graph auto-encoding, and (iii) simulation of E(n)-equivariant dynamical systems.

Figures (15)

• Figure 1: E(n)-GNCA commutative diagram: For any number of steps $t$ the transition rule $\tau_{\theta}$ is run, output coordinates ${\mathbf{X}}'$ and node features ${\mathbf{H}}'$ are respectively E(n)-equivariant and E(n)-invariant to isometries of input coordinates ${\mathbf{X}}$. Node features are represented with 3 colored dots.
• Figure 2: E(n)-GNCA convergence to a 2D grid (top), a 3D torus (middle) and the Stanford geometric bunny (bottom). The first 4 columns show E(n)-GNCA states at different time steps. The second to last column shows either a local or global damage of coordinates at $t=24$. Finally, the last column shows regeneration and persistency abilities by running the transition rule for 1000 extra steps after perturbation has occurred. Note that, if we were to apply any isometry at any point in time, convergence and persistency would still be guaranteed. We report ${\mathcal{L}}_{\mathsf{INV}}$ (cf. \ref{['eq:inv_loss']}) for the state in each figure. The nearest-neighbor edges of the Stanford bunny are not shown so as to avoid clutter. We report complete trajectories in \ref{['appendix:geograph']}. Best viewed digitally and zoomed in.
• Figure 3: E(n)-GNCA coordinates at different time steps for a test-set graph in comm-s. In each figure, we plot the ground-truth edges and report the binary cross-entropy (cf. \ref{['eq:bce']}).
• Figure 4: Boids simulation. First (Second) row shows a ground-truth (predicted) trajectory at different time steps. E(n)-GNCA learns a flocking behaviour similar to the target system, although with smoother and less precise trajectories.
• Figure A.1: Convergence to a Line. We report the loss value (\ref{['eq:inv_loss']}) in each figure. Global damage occurs at $t'=25$. Best viewed digitally and zoomed in.