The parity gap in crystal tensor prediction
cond-mat.mtrl-sci, cs.LG, physics.comp-ph, quant-ph
Submitted: 2026-08-19
Updated: 2026-09-16
Code: https://github.com/KurbanIntelligenceLab/equiparity
License: http://creativecommons.org/licenses/by/4.0/
The gist: Crystal symmetry dictates whether a physical response tensor must vanish, establishing a direct test for machine learning predictions independent of property calculations.
Terminology
Abstract
Crystal symmetry dictates whether a physical response tensor must vanish, establishing a direct test for machine learning predictions independent of property calculations. We derive the parity gap, a group-theoretic metric quantifying the piezoelectric tensor freedom permitted by a crystal's proper rotation subgroup SO(3) but eliminated by inversion symmetry in O(3). Across state-of-the-art equivariant neural network architectures, unconstrained SO(3) models systematically predict forbidden non-zero responses matching the parity gap of each centrosymmetric crystal class, while polar distortion paths dynamically map output responses to the loss of inversion symmetry. Regression controls confirm that enforcing full O(3) parity incurs no consistent accuracy cost across predictive tasks. Crucially, while training interventions using explicit zero labels reduce violation magnitudes, they leave residual forbidden outputs. Exact physical compliance instead requires structural enforcement through O(3) representation design or explicit output antisymmetrization. The parity gap thus provides a unified framework to distinguish empirical error reduction from exact structural compliance with physical law.
Sources
- A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems
- Meta-Learning Symmetries by Reparameterization
- SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks
- Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs
- EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations
- Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction
- EquiformerV3: Scaling Efficient, Expressive, and General SE(3)-Equivariant Graph Attention Transformers
- MLIPAudit: A benchmarking tool for Machine Learned Interatomic Potentials
- Symmetry Breaking and Equivariant Neural Networks
- MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields
- Geometric and Physical Quantities Improve E(3) Equivariant Message Passing
- Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passing
- Equivariant message passing for the prediction of tensorial properties and molecular spectra
- TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials
- SchNet: A continuous-filter convolutional neural network for modeling quantum interactions
- Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules
- A Space Group Symmetry Informed Network for O(3) Equivariant Crystal Tensor Prediction
- Efficient Equivariant High-Order Crystal Tensor Prediction via Cartesian Local-Environment Many-Body Coupling
- A foundation model for atomistic materials chemistry
- Open Materials 2024 (OMat24) Inorganic Materials Dataset and Models
Related papers
- AES-Debye: an Accurate, Efficient, and Scalable Engine for Debye Scattering Calculations
- Cooperative Quantum Optical Effects of Moir'e Exciton Superlattices
- Imaging Surface Magnetization in Altermagnetic MnTe Films
- Accidental accuracy and formal consistency in GW +BSE: Exact benchmarks and regime-dependent error cancellation
- Modifying van der Waals Materials via Cavity Vacuum Fluctuations
- Linear dichroic soft X-ray microscopy of ferroelectric stripe domains in epitaxial K 0.6 Na 0.4 NbO 3