A Single Architecture for Representing Invariance Under Any Space Group
Cindy Y. Zhang, Elif Ertekin, Peter Orbanz, Ryan P. Adams
TL;DR
This paper introduces a universal neural network architecture that automatically adapts to enforce invariance under any of the 230 three-dimensional space groups, improving generalization and transferability in modeling crystalline materials.
Contribution
The authors develop a single, adaptable architecture using symmetry-adapted Fourier bases that can enforce invariance to any space group without designing group-specific models.
Findings
Achieves competitive accuracy in material property prediction.
Enables zero-shot learning for unseen space groups.
Leverages structural similarities across groups for improved generalization.
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, 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…
Peer Reviews
Decision·ICLR 2026 Poster
1. The paper's primary strength is its rigorous theoretical derivation. It provides a clear analytical treatment of how space-group operations constrain Fourier coefficients. 2. The method avoids the complex graph construction required by GNNs. Its Transformer architecture is highly parallelizable, aligns with mainstream models. 3. The use of the $M_G$ matrix as a "group-conditional router" to explicitly inject invariance is novel and provides an elegant solution for weight sharing across all gr
1. In the primary property prediction task (Table 1), the model only achieves performance on par with, or marginally better than, outdated baselines. The lack of comparison to current SOTA (2024-2025) models makes the "competitive" claim unconvincing. 2. The "zero-shot" task compares CFT only against itself and completely omits baselines under the same zero-shot conditions. This leaves the central claim of generalization entirely unvalidated. 3. The paper's core selling point—generalization to u
- The setup for the use of Fourier basis and the explanation of how they are computed and the associated images make it very clear what the process is and were very helpful. - The inclusion of training and inference performance is very important as one of the key use cases of these models is a replacement for DFT calculations. - The comparison against many state of the art models in terms of both performance and computation time showcases the key benefits of the model and how other state of the
- You make a claim that the performance is only marginally higher on the held out groups, yet groups 71 and 140 in the Bulk Modulus test have a MAE of almost double the seen groups, I don't think that counts as marginally and would warrant further explanation on why their performance is so low. - The model is not state of the art in any specific task showcasing that while their method does have benefits in computational performance, it does not significantly boost performance on any tasks. Seei
The paper tackles an important problem of finding invariant neural representations of crystallographic observations that generalize across all point groups. The work is thoroughly motivated and theoretically sound. The paper presentation is exceptional, aside from several grammatical errors and points that need further clarification. The numerical results are promising, particularly those on the improved computational performance.
The paper experiments are limited in the following aspects 1) the orbit distance results are only reported for the CFT architecture, making it unclear if the technique has improved the learning of the representations. 2) The material property prediction results demonstrate marginally equivalent or worse results in comparison with SOTA techniques. 3) It's unclear the effect of the choice of the basis dimension and resulting encoding dimensions and the role/benefits of using the pre-trained encode
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Taxonomy
TopicsMachine Learning in Materials Science · Model Reduction and Neural Networks · Quantum many-body systems