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A Single Architecture for Representing Invariance Under Any Space Group

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
symmetrygroup invariancespace groupscrystallographic groupsFourier series
Abstract:

Incorporating known symmetries in data into machine learning models has consistently improved predictive accuracy, robustness, and generalization. However, achieving exact invariance to specific symmetries typically requires designing bespoke architectures for each group of symmetries, limiting scalability and preventing knowledge transfer across related symmetries. In the case of the space groups—symmetries critical to modeling crystalline solids in materials science and condensed matter physics—this challenge is particularly salient as there are 230 such groups in three dimensions. In this work we present a new approach to such crystallographic symmetries by developing a single machine learning architecture that is capable of adapting its weights automatically to enforce invariance to any input space group. Our approach is based on constructing symmetry-adapted Fourier bases through an explicit characterization of constraints that group operations impose on Fourier coefficients. Encoding these constraints into a neural network layer enables weight sharing across different space groups, allowing the model to leverage structural similarities between groups and overcome data sparsity when limited measurements are available for specific groups. We demonstrate the effectiveness of this approach in achieving competitive performance on material property prediction tasks and performing zero-shot learning to generalize to unseen groups.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper proposes a single neural network architecture that can adapt its weights to enforce invariance to any of the 230 three-dimensional crystallographic space groups, rather than requiring bespoke designs for each group. It sits within the 'Tensor Property Prediction with O(3) and Space Group Equivariance' leaf, which contains four papers including the original work. This leaf focuses on predicting rank-2 or higher tensors (elastic, piezoelectric, dielectric) with combined rotation equivariance and space group invariance. The taxonomy reveals this is a moderately populated research direction within the broader property prediction category, suggesting active but not overcrowded exploration of tensor-valued predictions under crystallographic constraints.

The taxonomy tree shows the paper's leaf is part of the 'Space Group Equivariant Architectures for Property Prediction' branch, which also includes scalar property prediction and chemical disorder modeling. Neighboring branches address generative models (diffusion, flow matching, autoregressive generation) and theoretical foundations (expressiveness, general equivariance frameworks). The scope note for the paper's leaf explicitly excludes generative models and scalar-only predictions, positioning this work at the intersection of rigorous symmetry enforcement and tensor-valued output prediction. The broader taxonomy reveals parallel efforts in representation learning and symmetry breaking, indicating the field explores both strict equivariance and relaxed variants depending on application needs.

Among 15 candidates examined across three contributions, the analytical construction of symmetry-adapted Fourier bases shows one refutable candidate out of 10 examined, suggesting some prior work on Fourier-based symmetry encoding exists within the limited search scope. The single adaptive architecture contribution examined 2 candidates with no refutations, and the Crystal Fourier Transformer architecture examined 3 candidates with no refutations. The statistics indicate that within the top-15 semantic matches, the Fourier basis construction has the most substantial prior overlap, while the adaptive architecture and transformer components appear more distinctive. However, the limited search scope (15 candidates, not exhaustive) means these findings reflect only the most semantically similar work retrieved.

Based on the limited literature search of 15 candidates, the work appears to occupy a moderately novel position within tensor property prediction under space group constraints. The adaptive weight-sharing mechanism across all 230 space groups distinguishes it from sibling papers that may target specific groups or tensor types. The single refutable pair among 15 candidates suggests the Fourier basis approach has some precedent, but the overall architecture combining adaptive invariance with Fourier constraints may represent a synthesis not fully captured by prior work within the examined scope.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
15
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: Enforcing invariance to crystallographic space groups in neural networks. The field has organized itself around several complementary directions. Space Group Equivariant Architectures for Property Prediction focuses on building models that respect crystallographic symmetries when predicting material properties, often leveraging O(3) equivariance and tensor representations (e.g., Crystal Tensor Network[3], StrainTensorNet[7]). Space Group Constrained Generative Models tackle the inverse problem of designing new crystal structures while maintaining symmetry constraints (e.g., Space Group Diffusion[2], Space Group Flow[4]). Theoretical Foundations and General Equivariance Frameworks provide the mathematical underpinnings for these approaches, while Representation Learning and Encoding Strategies explore how to efficiently encode crystallographic information. Symmetry Breaking and Relaxed Equivariance address scenarios where strict invariance may be too restrictive, and Benchmarking, Evaluation, and Computational Methods ensure reproducibility and scalability across the field. Within property prediction, a central tension exists between strict equivariance and computational efficiency. Works like Completeness Invariant Models[5] emphasize theoretical guarantees of completeness, while others prioritize scalability for large-scale screening. Invariant Space Group[0] sits within the tensor property prediction cluster, focusing specifically on O(3) and space group equivariance for predicting tensorial material properties. Its emphasis on rigorous symmetry enforcement aligns closely with Crystal Tensor Network[3] and StrainTensorNet[7], which similarly target tensor-valued outputs like elastic or piezoelectric tensors (Piezoelectric Tensor Prediction[12]). Compared to these neighbors, Invariant Space Group[0] appears to push further on the theoretical side of ensuring complete invariance, contrasting with approaches that might relax symmetry constraints for practical gains. The broader challenge remains balancing expressiveness, computational cost, and the physical correctness guaranteed by strict equivariance.

Claimed Contributions

Single adaptive architecture for any crystallographic space group invariance

The authors introduce a unified neural network architecture that can adapt to enforce exact invariance to any of the 230 three-dimensional space groups by conditioning on the input group, rather than requiring separate architectures for each group. This enables weight sharing across different space groups and allows the model to leverage structural similarities between groups.

2 retrieved papers
Analytical construction of symmetry-adapted Fourier bases via constraint characterization

The authors derive analytical constraints that crystallographic group operations impose on Fourier coefficients and prove these constraints define a complete basis for group-invariant functions. They introduce a dual graph representation where nodes are reciprocal lattice frequencies and edges encode phase relationships, enabling algorithmic construction of the symmetry-adapted basis.

10 retrieved papers
Can Refute
Crystal Fourier Transformer architecture with group-conditional encoding

The authors present a Transformer-based architecture that uses a group-conditional routing matrix to transform standard Fourier modes into provably invariant positional encodings. This encoding module can be integrated with existing ML models to capture exact symmetries while sharing weights across all 230 space groups.

3 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Single adaptive architecture for any crystallographic space group invariance

The authors introduce a unified neural network architecture that can adapt to enforce exact invariance to any of the 230 three-dimensional space groups by conditioning on the input group, rather than requiring separate architectures for each group. This enables weight sharing across different space groups and allows the model to leverage structural similarities between groups.

Contribution

Analytical construction of symmetry-adapted Fourier bases via constraint characterization

The authors derive analytical constraints that crystallographic group operations impose on Fourier coefficients and prove these constraints define a complete basis for group-invariant functions. They introduce a dual graph representation where nodes are reciprocal lattice frequencies and edges encode phase relationships, enabling algorithmic construction of the symmetry-adapted basis.

Contribution

Crystal Fourier Transformer architecture with group-conditional encoding

The authors present a Transformer-based architecture that uses a group-conditional routing matrix to transform standard Fourier modes into provably invariant positional encodings. This encoding module can be integrated with existing ML models to capture exact symmetries while sharing weights across all 230 space groups.

Table of Contents