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Relaxed Equivariant Graph Neural Networks

Elyssa Hofgard, Rui Wang, Robin Walters, Tess Smidt

TL;DR

The paper addresses the limitation of strictly equivariant $E(3)$ neural networks in modeling systems with symmetry breaking. It introduces relaxed $E(3)$NNs, where learnable weights span a direct-sum of $O(3)$ irreps up to a chosen $l_{ ext{relaxed}}$, enabling data-driven symmetry breaking while preserving equivariance when restricted to scalars. Empirically, the approach interprets the learned weights as spherical signals and demonstrates correct symmetry-breaking factors in shape deformation tasks and the ability to recover electric and magnetic field structure in a charged-particle trajectory. This framework provides a principled, interpretable means to discover and quantify symmetry-breaking in 3D physical systems, with practical implications for materials science and physics-informed modeling, and includes code for reproducibility.

Abstract

3D Euclidean symmetry equivariant neural networks have demonstrated notable success in modeling complex physical systems. We introduce a framework for relaxed $E(3)$ graph equivariant neural networks that can learn and represent symmetry breaking within continuous groups. Building on the existing e3nn framework, we propose the use of relaxed weights to allow for controlled symmetry breaking. We show empirically that these relaxed weights learn the correct amount of symmetry breaking.

Relaxed Equivariant Graph Neural Networks

TL;DR

The paper addresses the limitation of strictly equivariant neural networks in modeling systems with symmetry breaking. It introduces relaxed NNs, where learnable weights span a direct-sum of irreps up to a chosen , enabling data-driven symmetry breaking while preserving equivariance when restricted to scalars. Empirically, the approach interprets the learned weights as spherical signals and demonstrates correct symmetry-breaking factors in shape deformation tasks and the ability to recover electric and magnetic field structure in a charged-particle trajectory. This framework provides a principled, interpretable means to discover and quantify symmetry-breaking in 3D physical systems, with practical implications for materials science and physics-informed modeling, and includes code for reproducibility.

Abstract

3D Euclidean symmetry equivariant neural networks have demonstrated notable success in modeling complex physical systems. We introduce a framework for relaxed graph equivariant neural networks that can learn and represent symmetry breaking within continuous groups. Building on the existing e3nn framework, we propose the use of relaxed weights to allow for controlled symmetry breaking. We show empirically that these relaxed weights learn the correct amount of symmetry breaking.
Paper Structure (17 sections, 1 theorem, 9 equations, 5 figures, 4 tables)

This paper contains 17 sections, 1 theorem, 9 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Related Work
  4. Methodology
  5. Definition of relaxed $E(3)$NN
  6. Interpreting Relaxed Weights as Spherical Signals
  7. Experiments
  8. Shape Deformations
  9. Charged Particle in an Electromagnetic Field
  10. Discussion
  11. Acknowledgements
  12. Symmetry Analysis for Experiment \ref{['exp:shape_def']}
  13. Character Tables
  14. Converting Relaxed Weights to Spherical Signal
  15. Experimental Details
  16. ...and 2 more sections

Key Result

Proposition 4.1

The relaxed $E(3)$NN is equivariant if and only if $\tilde{\theta}$ transforms as a scalar, i.e. only the term corresponding to the scalar irrep is non-zero.

Figures (5)

  • Figure 1: Visualization of tasks and corresponding spherical harmonic projections of the relaxed weights for the first (first row) and second (second row) layers. The spherical harmonic projections are plotted setting the scalar ($0_e$) term to zero for ease of viewing. A 2-layer relaxed $E(3)$NN network is trained to 1) map a cube to a cube, 2) map a cube to a rectangular prism, and 3) map a cube to a less symmetric object.
  • Figure 2: The number of nonzero values for each $\theta^{l,p}$ for the last layer in each network for the task of mapping the cube to a given output shape. Note that each irrep $l$ has dimension $2l+1$.
  • Figure 3: Sample trajectories of particles with random initial velocities and starting positions under an electric field in the $\hat{x}$ direction and a magnetic field in the $\hat{y}$ direction.
  • Figure 4: True and predicted fields for the relaxed network.
  • Figure 5: Loss for fully equivariant model and model with relaxed weights trained on the same task.

Theorems & Definitions (2)

  • Proposition 4.1
  • proof