Research map / Probabilistic and scientific modeling / Dynamical systems
Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes relaxed group convolution with trainable equivariant weights to detect and quantify symmetry breaking in physical systems (crystal phase transitions, turbulence isotropy/homogeneity breaking, pendulum time-reversal breaking), with theory on gradient-based emergence of broken symmetry.
Problem
Standard equivariant networks enforce full symmetry and cannot describe symmetry breaking in physical data, so the asymmetries present in data need to be discovered and quantified.
Approach
Use relaxed group convolutions whose group-element-dependent weights start equal (preserving equivariance) and are trained by loss gradients; the learned departures from equality reveal which symmetries are preserved or broken, with theoretical proof that they stay equal exactly on the stabilizer of both input and output.
Key results
- On 3D super-resolution, R-Equiv achieves MAE 2.44e-2 on channel flow vs 2.54e-2 for Equiv and 2.60e-2 for Conv; on isotropic flow 1.00e-1 vs 1.12e-1 (Equiv) and 1.22e-1 (Conv).
- Relaxed octahedral weights for BaTiO3 cubic-to-tetragonal and cubic-to-orthorhombic transitions recover C4v and C2v preserved symmetry groups.
- Equivariance error for isotropy decreases toward zero at high wavenumber, consistent with the Kolmogorov isotropy hypothesis; relaxed time-reflection weights stay equal for frictionless pendulum but diverge with friction.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use relaxed group equivariant layers instead of trilinear upsampling | 3D super-resolution model · 3D channel flow | MAE: 5.24 -> 2.44±0.01 | Helped |
| Use relaxed group equivariant layers instead of Conv | 3D super-resolution model · 3D channel flow | MAE: 2.60±0.05 -> 2.44±0.01 | Helped |
| Use relaxed group equivariant layers instead of Equiv | 3D super-resolution model · 3D channel flow | MAE: 2.54±0.03 -> 2.44±0.01 | Helped |
| Use relaxed group equivariant layers instead of trilinear upsampling | 3D super-resolution model · 3D isotropic flow | MAE: 5.25 -> 1.00±0.01 | Helped |
| Use relaxed group equivariant layers instead of Conv | 3D super-resolution model · 3D isotropic flow | MAE: 1.22±0.04 -> 1.00±0.01 | Helped |
| Use relaxed group equivariant layers instead of Equiv | 3D super-resolution model · 3D isotropic flow | MAE: 1.12±0.02 -> 1.00±0.01 | Helped |
Surprising findings
- Relaxed group convolution gives the best super-resolution performance even on isotropic flow, which has distributional rotational symmetry.
- LieGAN failed to identify meaningful generators from random initialization on the fluid data, and MSR failed to learn correct rotation symmetry.
Limitations
- Requires specifying the largest possible symmetry group beforehand.
- May fail under spontaneous symmetry breaking when the data contain multiple equally probable outputs.
- Convergence analysis of mini-batch training relies on convexity and Lipschitz assumptions.
Experimental setup
Datasets: Johns Hopkins Turbulence Database 2D channel flow DNS Materials Project BaTiO3 structures pendulum simulations
Benchmarks: Johns Hopkins Turbulence Database super-resolution (channel and isotropic flow)
Metrics: equivariance error mean absolute error
Baselines: MSR LieGAN Trilinear upsampling non-equivariant CNN strict equivariant model
Builds on: relaxed group convolution group convolution MSR LieGAN
Closest papers by meaning
- Learning Infinitesimal Generators of Continuous Symmetries from Data (NeurIPS 2024 · similarity 0.87)
- AtlasD: Automatic Local Symmetry Discovery (ICML 2025 · similarity 0.86)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.86)
- Symmetry Discovery Beyond Affine Transformations (NeurIPS 2024 · similarity 0.85)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.85)
- Symmetry-Informed Governing Equation Discovery (NeurIPS 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.