A Single Architecture for Representing Invariance Under Any Space Group
Authors: Cindy Zhang
Elif Ertekin
Peter Orbanz
Ryan P Adams
Venue: ICLR 2026 Poster (2026)
Keywords: symmetry
group invariance
space groups
crystallographic groups
Fourier series
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: A single neural architecture that automatically adapts to be invariant under any of the 230 space groups given the group identity as input.

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 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.

Published as a conference paper at ICLR 2026
A SINGLE ARCHITECTURE FOR REPRESENTING INVARIANCE UNDER ANY SPACE GROUP
Cindy Y. Zhang
Elif Ertekin
Department of Computer Science Princeton University
Department of Mechanical Science & Engineering University of Illinois Urbana-Champaige
Sindyz@prine wwwpzeeepiinetoooomo ince on eddu
erteking alinois.ede
Peter Orbanz
Ryan P. Adams
Gatsby Computational Neuroscience Unit University College London
Department of Computer Science
Princeton University
P. P.orbanzeucl, ac..uk
rpa@pri nceton. edu
ABSTRACT
Incorporating known symmetries in data into machine learning models has con- sistently improved predictive accuracy, robustness, and generalization. However, achieving exact invariance to specific symmetries typically requires designing be- spoke architectures for each group, limiting scalability and preventing knowledge transfer across related symmetries. In the case of the space Proups-symmeete critical 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 crystallographic symme- tries by developing single machine learning architecture that is capable of adapt- ing weights automatically to enforce invariance to any input space group. Our approach is based on constructing symmetry symmetry-adapted Fourier bases through plicit characterization constraints that group operations impose on Fourier co- efficients. Encoding these constraints into 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 measure- ments 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.
INTRODUCTION
Many physical systems are governed by known symmetrics, from the arrangement of atoms a crystal to the laws of motion. By encoding these symmetries-typicllyy described by mathematical groups-direcdd into a model's architecture, we can enforce that its predictions respect the under- lying physical constraints. These group symmetrization techniques act as a powerful inductive bias, leveraging a priori knowledge of geometric and physical invariances to enhance model efficiency and reduce training complexity (Thomas et al. 2018 Jing al. 2020 Satorras et al., 2021 Liu
et al. 2023 Du et al. 2023). However, each new type of desired group invariance has generally required a new neural network architecture. In this work we do something different: we develop single architecture that can adapt and shure structure across the crystallographic groups.
Crystallographic groups are important several different domains (Ten Eyek] 1973) Hinuma al., 2017 Watanabe & Lu 2018), but most commonly arise in materials science and condensed matter physics, where they describe the transformations that preserve structure in a crystalline solid. The groups are discrete sets of transformations on Euclidean space. They each include translations to capture periodicity, but also include other isometries such as rotations, reflections, glides, inversions, and screws. It is desirable capture these symmetries in machine learning models for, e.g., building generative models of crystalline solids (Chang et al., 2025) Zeni et al. (025) and neural ansätze for solving the associated electronic Schrödinger equation (L. al. 2022 Huang et al. 2025).

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A challenge of symmetrized machine learning models for crystallographic groups, however, is data sparsity: there are 230 groups in three dimensions and even the most commonly-used materials benchmark (Jain et al.) 2013) has only about 200,000 data points, averaging fewer than 1,000 amples per group. In practice, the data distribution is heavily skewed, with many groups having few no examples for certain properties. Thus developing separate machine learning architecture for each group is unlikely to be successful; rather we must find a way to inform our models with group symmetry while also sharing parameters across the different groups. We develop such an architecture in this work and show that it can generalize even to groups it has not been trained on.
Our approach centers on a new algorithm for constructing symmetry-adaptee Fourier bases for the crystallographic groups. We analytically derive the explicit constraints imposed by crystallographic symmetries on the Fourier coefficients of functions defined over Fuclidean space. We prove that this symmetry-adee -adatedd Fourier basis spans the space of continuous, qquare-integraal functions invariant to the given crystallographic group. This allows us to represent erystal structures linearly using atom positions expanded in the symmetry-adapteed Fourier basis.
Crystalline materials are typically represented as graphs within ML pipelines, which need to be augmented with additional edges or features to encode the periodic lattice structure explicitly (Xie) & Grossman, 2018: Yan et al. 2022). In our framework, rather than separately augmenting each graph with information about the crystal lattice, we have a layer that effectively encodes positions in the standard Fourier basis and then multiplies the encodings by a precomputed, group-dependent adjacency matrix that defines the constraints between Fourier coefficients. This symmetry-adaptee encoding can serve as input to further network layers to produce a crystallographic group-invariant output, allowing the parameters of the neural network be shared across all space groups
We empirically verify the effectiveness of this approach in learning positional encodings within crystal structures that accurately reflect orbit distances. We further validate our framework by using these positional encodings in a Transformer model, improving performance over standard positional encodings and achieving competitive performance across several benchmark tasks for material prop- erty prediction. We also show empirically that the approach can perform zero-shot learning and generalize across groups for which it has never seen data. Our framework can be flexibly used as an encoding module that integrates with existing ML models to capture the exact symmetries of crystallographic data.
The paper is structured as follows. We start by defining crystallographic groups and presenting the intuition from micddimnniional Fourier series that extends to a Fourier basis for functions invariant to crystallographic groups. We then derive the exact constraints
Fourier coefficients in Section 3.1 and their dual graph representation in Section 3.2 The adjacency matrices of these graphs are key the architecture introduced in Section 3.3 that adapts its invariance properties to different crystallographic groups. We contextualize our findings with a discussion of related work in Section The details of our experiments and results are Section 5
2 BACKGROUND
In this section, we describe invariance under crystallographic groups and define the G-invariant functions of interest. We then present a key result from Adams & Orbanz (2023) proving that a Fourier representation exists for continuous G-invariant functions.
2.1 CRYSTALLOGRAPHIC GROUP INVARIANCE
Crystallographic groups are discrete groups of isometries that tile Fuclidean space with a repeating fundamental region. We consider the fundamental region to be convex polytope. Copies of Il under the group's 's isometries fill R without gaps or overlaps. Only finite number of crystal- lographic groups exist in each dimension; there are 17 such groups on I², called the wallpaper groups, and 230 such groups RR3, called the space groups, which classify all possible symmetries of crystals. The tiling patterns of the wallpaper groups are shown in Appendix A.J
Each isometry é in crystallographic group G takes the form d(x) Axx +t, where A R is an orthonormal matrix and E R" is a translation vector. By definition, G contains translations by any integer combination of n linearly independent basis vectors 7 T Ta. Ta. making it both discrete
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and infinite. A function
R + R is G-invariant if it satisfies:
f = (x:
for all Φ G (nd a E IR...
G-invariance is crucial in modeling physical phenomena in crystals. According to Neumann's 's prin- ciple, if a physical system is invariant with respect to the symmetries of group G, then any physical observables of that system must also be invariant to the same symmetries (Bradley & Cracknell] 2009). Formally, this invariance implies that can be expressed solely in terms of equivalence classes of points, known orbits, defined by the group action. The orbis of point æ E R" is the set (ó(x) has continuous bijective mapping the fundamental region II, which contains exactly point from each orbit. Because an invariant function fully determined by its values on the fundamental region II, this region serves as natural domain for analysis.
G} of all points mapping to it under the group's symmetries. The quotient set R" /
2.2 FOURIER REPRESENTATIONS
In order to construct G-invariant functions, want set of basis functions that periodic. One natural choice a Fourier series. In the one mne-dimensionll setting, a smooth function f R - of period L admits the well-known expansion
f(x) = ap +
Σ cos(2 + by. sin(2 k=1
These sine and cosine modes are exactly the eigenfunctions of the one-dimensional Laplacian,
d²
dx e de subject
c(x -)= c()), =)
with eigenvalues À = 12)2 Two key insights carry forward to crystallographic groups: (1)
Translation invariance imposes boundary conditions that yield discrete spectrum of Laplace eigen- functions, and (2) the resulting eigenfunctions form an orthonormal basis for all square-integrable L-periodic functions.
Adams & Orbanz (2033) show that for any crystallographic group G, there exists a sequence of G- invariant functions 1 62,, on Rn such that any G-invariant function in L2ll)) can be represented as linear combination of these basis functions. This basis generalizes the standard Fourier series to account for all symmetries crystallographic group, not just translations. The work characterizes these basis functions, e;, as the solutions to the constrained partial differential equation
- Ae
subject
e o for all G.
While this result establishes the existence of a G-invariant Fourier basis, the construction in Adams & Orbanz (2023) relies on numerical methods to approximate the eigenfunctions of the Laplace operator. We introduce an alternative, analytical approach in the following section.
METHODS
In this section, we present the theoretical and algorithmic foundations for our adaptive framework. First, we derive the analytical constraints that invariance to crystallographic group imposes function's Fourier series coefficients. Second, we show how these constraints define a complete, G-invariant basis, which admits dual representation in terms of graphs. These graphs allow to formalize a constructive algorithm for the G-invariant basis. Finally, we use these basis functions to define the Crystal Fourier Transformer (CFT), space group adaptive architecture.
3.1 DERIVING GROUP CONSTRAINTS IN FOURIER SPACE
Our goal is find universal representation for the constraints imposed by invariance to crys- tallographic group G. We achieve this by analyzing the effect group operations on the Fourier transform of a G-invariant function f R R.
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Translation Invariance induces points which Fourier coefficients are bonzero.
Other isumetries induce constraints between Fourier coefficients, which represeuted weighted edges.
Each connected component corresponds basts funesion that weighted over Fourier modes.
(o(ae) 201-1 12
P
7
4
c((a) c 1
P8
We construct the Fourier basis for G-invariant functions using a dual representation in terms of graphs. The translations a crystallographic group G give rise to a discrete lattice of points corresponding the Fourier series. We represent constraints induced by additional isometries as weighted edges between these points. For wallpaper group pg, which contains the translations and glide reflections present in the tiling pattern, get edges of weight -1 (red) and +1 (blue). Each clique in the resulting graph can be expanded as one of the pictured Fourier basis functions.
Figure 1:
The translation symmetries present in any crystal structure require that f be periodic. This period- icity constrains the support of its Fourier transform, F(w), to discrete set of frequencies known as reciprocal lattice, L For any frequency w not on this lattice ((w) must zero.
The non-translation isometries in G (e.g., rotations, reflections, glides) impose additional constraints that relate the values of the Fourier coefficients at different points on the reciprocal lattice. The following proposition makes this relationship precise.
R be a G- G-invariant function and let F(w) be its Fourier transform. All symmetry operations & E G have the form 6(x) Ax + t, where A is orthogonal matrix and t is a translation vector. For any & E G, the Fourier coefficients must satisfy the following relation for all w L*
Proposition 3.1. Let f: R +
F(w)) = T (
(3)
The proof is in Appendix A.2 Equation 3 states that the Fourier coefficients at frequencies w and Aw are coupled by phase factor e determined by the specific isometry d. These constraints partition the reciprocal lattice into orbits of related frequencies and define the exact structure of the invariant basis functions, as formalized in main result.
Theorem 3.2. Let be a crystallographic group and L. be its reciprocal lattice. The linear com- ponents (A&) of the group's 's transformations (Ø(x) Agx tt) partition C into disjoint orbits O, where frequencies W1, W2 E EO if w2 = AgW1 for some & E G.
Assume orbit O is phase-consistent, meaning for any w E O and any transformation o that maps w to üself (i.e., Agw = w), the corresponding phase factor e "uow A7 is equal to 1.
Then, corresponding basis function (2)) be constructed as:
(a = Eweo We-rw "22x To
where the complex coefficients we-ww relative to an arbitrary reference frequency & E O for which We- = Every continuous, invariant function f admits uniformly convergent expansion
-w are uniquely determined by the phase constraint in equation
Eo coeo.
where the family {eo} is constructed over all phase-consistent orbits in L
The proof in Appendix functions from continuous problem I to discrete problem on the reciprocal lattice. Each basis function is a specific linear combination of Fourier modes, where the coefficients are uniquely determined the group's 's symmetry constraints. The phase-consistency condition filters out fre- quencies that cannot support a globally consistent invariant function.
Theorem 3.2 transforms the problem of constructing G-invariant
While Theorem 3.2 provides the analytical form of the basis, it does not specify computational procedure for identifying these orbits and their coefficients. To operationalize this, we introduce a dual graph representation that makes the group constraints concrete and computationally tractable.

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Algorithm Symmetry-AAapted Fourier Basis Construction
Require: Space group G. reciprocal lattice L
1: Construct constraint graph 9 with nodes L and edges w Aw weighted by el2m T for each d(x) Az + t G
2: Remove nodes with inconsistent self-loops (weight # 1) from g
3: Identify phase-consistent orbits {Ox} connected components of
4: Construct basis functions c02(z) = Eweo, Wg-nu e where -e-w is the product of edge weights along any path from node & to node
5: return Basis {
3.2 DUAL GRAPHS OF CRYSTALLOGRAPHIC FOURIER SERIES
The algebraic relationships and phase constraints from Proposition 3.1 can be captured directed graph on the reciprocal lattice. In this graph, nodes represent frequencies E L" and each symme- try operation <xx) Ax t induces set of directed edges. For each w, an edge is drawn from w to Aw with a complex weight equal to the phase factor e'2m A
This graph structure is a key computational tool for encoding the basis functions. The phase- consistent orbits of Theorem 3.2 correspond exactly to the connected components of this graph after nodes with inconsistent self-loops are removed. The basis function coefficients wg-sw are the product of edge weights along any path from a reference node & to another node w within a com- ponent. This framework leads directly Algorithm for constructing the full symmetry-adapler basis for any space group G. Since the reciprocal lattice L is infinite, we in practice construct the reciprocal lattice up to maximum frequency radius, yielding finite set of basis functions that form practical approximation to the complete basis.
Example of basis construction. We describe the construction of the constraint graph and the corre- sponding basis for the wallpaper group pg. which consists of translations and glide reflections. For simplicity, we'll use the standard basis vectors e₁ = [] 0] and e₂ [0 1] for the translations. Invariance to these standard basis shifts induces the reciprocal lattice z² where Fourier coefficients are nonzero. Next we incorporate glide reflections, given by the form
0 1/2
(xx) 0
which are a combination of a reflection A and half-shift t. We calculate the weight for the edge from frequency to its reflected counterpart Aw using the phase formula from Proposition 3.1
o 1
0 1/2
exp
i2
[wi
exp(inw2).
This result provides simple rule for the edge weights: the weight is +1 if we is even and -1 if we odd. This pattern of alternating positive (blue) and negative (red) edge weights is exactly what visualized in Figure
Each w corresponds to the Fourier mode 2mm x0 We form the basis functions by taking weighted sum of the Fourier modes each connected component. For example, the component contain- ing (-1,0) and (1, 0) has an edge of weight +1 in both directions between the two points. This corresponds to the basis function
(x())= fxx(-1 0fæ 2 0j
The component containing (-1, 1) and (1, has an edge of -1 in both directions. This corresponds
the basis function
e2 (w) = 1 1/m 2n11 1/
This G-invariant basis construction continues for each connected component in the constraint graph, up until some finite radius cutoff. The information needed to write out the analytic form of the basis functions lies in the edge weights, i.e., the adjacency matrix, of the graph.
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Inputs
G-Invariant Encoding
Shared-Weight Transformer
Standard Fourier
Adjacency matrix
Atom positious [
Multi-head self-acteution
MLP
Mc
Multi-head self-siteution
MLP
Multi headi- sclf-aitentionn
MLP
Pooling
Coustraint graph
MLP
Atomic element emabedidigg: (+,)
Space group
Profiction
Figure 2: Diagram of the Crystal Fourier Transformer architecture. Atom positions are first encoded into standard Fourier modes. A group-conditional routing matrix, Mg transforms these modes into a provably invariant basis. These adaptive positional encodings, combined with other invariant features, then processed by Transformer whose weights are shared across all space groups to predict material properties.
3.3 CRYSTAL FOURIER TRANSFORMER
The construction of the constraint graph for the symmetry-adapee Fourier basis provides the back- bone for novel neural architecture that learns group-invarian representations adaptively. We intro- duce the Crystal Fourier Transformer (CFT), model designed to process crystallographic data by explicitly building in invariance to its space group symmetry. The core of CFT is an encoding mod- ule that transforms atomic coordinates into feature representation that is invariant to the specified group G.
This transformation is two-step process, visualized in Figure First, for input coordinate and set of reciprocal lattice frequencies {wx ) we compute a vector standard Fourier modes, (a:) C where [v(=)]} key architectural innovation: a linear transformation defined by the pre-computed adjacency matrix Mg of the dual graph for the specified group G. The result is a vector of canmetry--daptee basis functions evaluated at æ::
(2mw) = This vector itself is not invariant. Second, we apply our
(x = MGv(a.).
The matrix Mg acts as a group-conditional routing mechanism. It linearly combines the standard Fourier modes according to the exact constraints dictated by the group G, effectively projecting the input representation onto the G-invariant basis. This mechanism allows single, fixed down- stream architecture achieve invariance any of the 230 space groups simply by swapping the corresponding routing matrix Mg.
CFT leverages this encoding module within standard encoder-only Transformer architecture Vaswani et al.) 2017), where the self-attention mechanism allows for the modeling of complex interactions between atoms crystal. For each atom in unit cell, we pass the atom position and the space group of the crystal into the encoding module, the only group-dependent component of the model, and we produce an output embedding that is G-invariant and contains symmetry-avee aware positional information.
The G-invariant positional encoding is directly added to a standard embedding of the atom's 's chemi- cal element to form the input token for each atom. The resulting sequence of tokens is processed by the Transformer encoder. Because the input features are already G-invariant, any function applied to them, including the self-attention mechanism, will produce G-invariant output. This design allows the core parameters of the Transformer to be shared across all 230 space groups, enabling the model generalize effectively even for groups with sparse data. More generally, the output of the G-invariant encoding layer can be used as input to any ML model, and the model will become invariant to the input space group while keeping its weights shared across all space groups.

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4 RELATED WORK
Group Equivariance and Invariance in ML. Symmetries have a long history in machine learning, dating back to work by Minsky & Papert (1888) discussing invariance in single-layer perceptrons when summing over finite group of symmetries. This was then extended to multi-layer perceptrons by SSawe-TTyloo (194),, who showed that invariant MLPs can be constructed by partitioning the connections between layers into weight-sharing orbits. The introduction of translation invariance in Convolutional Neural Networks (CNNs) (LeCun et al. 1989) was a breakthrough in computer vi- sion, capturing the symmetries of image data. Many works (Cohen & Welling, 201 Kondor al.) 2018 Cohen et al. 2019, Finzi et al. 2020 (Zhdanov et al., (024)) have generalized the CNN archi- tecture to be equivariant under additional group symmetries, and it was proven in KKoddor & Trivedi 20188 that convolutional structure is necessary for equivariance to actions of compact groups.
Symmetries in AI for Science. For tasks dealing with the symmetries of the physical world,
incorporating equivariance to SE(3) and E(3) symmetries has consistently improved performance and data efficiency (Cohen & Welling, 2016b Cohen et al.] 2018:) Fuchs et al. 2020; Geiger &
Smidi, 2022, Du et al. 2023). These architectures have found applications across geometry, physics, and chemistry, including protein structure classification (Weiler et al. 2018), learning interatomic
potentials and force fields (Batatia et al.] 2022; Batzner et 2022). 3D point cloud segmentation
(Guo et al. 2020), and molecular conformer generation (Jing et al. 2020). These architectures are tailored to specific compact symmetry groups, leveraging spherical harmonics and tensor products of the irreducible representations (Worrall et al. [011 Thomas et al., 2018). While these models capture the symmetries of 3D space, they are unable to capture the infinite discrete symmetries inherent to crystallographic structures.
Learning Crystal Representations. Crystalline materials are typically represented graphs within ML pipelines. Since initial work on crystal graph representations by Xie & Grossman (2018)
and Chen et al. (2019), which use atoms as nodes and bonds as edges, many follow-up works have explored different methods of incorporating additional spatial information and symmetries in the
graph representations (Choudhary & DeCost 202T Yan et al. 2022 Ruff et al. 2024 Wang al.
2024
Yan et al. 2). A separate line of work focuses on creating canomical, unambiguous
crystal representations to ensure that models learn from consistent data inputs (Widdowson Kurlin
2022
Nigam et al. (022). These methods differ from our work in a key aspect. While they implicilly handle or are designed for specific symmetries, the Crystal Fourier Transformer a single, adaptive architecture that enforces exact G-invariance for any of the 230 space groups.
5 EXPERIMENTS
Our experiments investigate three central questions: (1) Does our symmetry-adapp encoding mod- ule learn geometrically meaningful representations that accurately capture the underlying orbit space of crystal? (2) How does CFT perform on standard, large-scus scale material property prediction bench- marks against state-of-the-art graph-based models? (3) Does the adaptive nature of our architecture enable effective zero-shot generalization to space groups unseen during training? We demonstrate that CFT learns correct, symmetry-awen distance metrics, achieves competitive performance on benchmark tasks, and generalizes robustly. Full implementation details for the experiments can be found in Appendix A.5
5.1 EXPERIMENTAL SETUP
Dataset and Task. For our primary benchmark experiments, we use data from the Materials Project, one of the largest databases of computational material properties (Jain et al. 2013). The task is to predict four key material properties from the crystal structure: total energy (e (eV/atom), band gap (),), bulk modulus (log GPa), and shear modulus (log GPa). For each material, the model receives the atomic numbers and fractional coordinates of atoms within the unit cell, the 3 x 3 lattice vectors, and the corresponding space group identifier (1-230). At the time of training, there were 152,149 crystals with known total energies and band gaps, and 1 11,997 crystals with known bulk and shear moduli in the Materials Project database.
Baselines. We compare CFT against state-of-the-a art graph neural network (GNN) architectures designed for crystal structures. These models represent fundamentally different approach, relying
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on graph representations of the unit cell and its neighborhood instead of our Fourier basis encoding.
Our baselines include CGCNN (Xie Grossman 2018). ALIGNN (Choudhary & DeCost [021), )
and Matformer Yan et al. 2022). We also compare against the same Transtormer architecture but with the standard sine and cosine positional encodings from (Vaswani et al. (017)) for each dimension, rather than the proposed G-invariant encoding, to isolate the effect of our proposed encoding module. All baseline models were re-trained and evaluated under identical conditions for fair comparison. The numbers may differ from the results reported in the original papers due the Materials Project dataset having greatly increased in size in recent years.
5.2 SYMMETRY-ADAPTER ENCODINGS CAPTURE ORBIT DISTANCE
fundamental assumption of our work is that the G-invariant encoding module can learn represen- tations that capture the true geometric structure of the orbit space. We first validate this capability in a controlled, self-supervised setting before applying the model to downstream tasks. The goal is to train the encoder to produce positional encodings where the Euclidean distance between embed- dings directly corresponds to the true orbit distance dz( x2) between atoms. The orbit distance is defined as
da( x2) == min #G |l (x1) (222)))
(8)
It measures the shortest Euclidean distance between any two points in the respective orbits of the atoms, which stays invariant to applications of group actions to either atom.
Setup. We construct synthetic dataset containing 100,000 samples for each of the 230 space groups. Each sample consists of a pair of random atomic positions as fractional coordinates in (0, space group, and randomly generated lattice vectors that satisfy the constraints of the Bravais lattice for the group. The model takes as input the atomic position and lattice vectors in parallel branches. The position is passed through the G-invariant encoding module and subsequent ResNet (He et al.] 2016) to encode symmetry information, while the concatenated lattice vectors are passed through ResNet to encode scale information. The element wise product of the outputs of the two ResNets is the final positional encoding. The model is trained to minimize the Mean Squared Error (MSE) between the 1.2 distance of the output embeddings and the ground-truth orbit distance.
The trained model achieves a test Mean Absolute Error (MAE) of 0. A against an average orbit distance of 2.724 A in the synthetic dataset. This low error means that the embeddings accurately capture the complex, Non-Euclidean orbit distance metric across all 230 space groups We include a qualitative case study of the embeddings for the 2D wallpaper group p 6m in Appendix A.6 for which the embedding map is provably correct.
Results.
5.3
MATERIAL PROPERTY PREDICTION
Having validated our encoding module, we now evaluate the full CFT architecture on the task of predicting material properties from the Materials Project dataset. This experiment assesses CFT's ability to compete with specialized GNNs that have been the dominant paradigm for this task.
Setup. We use the pre-trained positional encoding model from Section 5.2 to initialize the positional encodings in CFT. This initialization captures the physical scale of the crystal in this feature space, which is important because the bourier-based encoding dimensionless. The positional encoding is directly added to learned embedding for the atom's 's chemical element, which forms the input tokens to standard Transformer encoder with three multi-head self-attention layers with eight heads each, followed by mean pooling and a MLP that outputs the final prediction. Because all inputs are G-invariant, the Transformer's weights are shared across all 230 space groups, allowing the model to leverage data from common groups to improve predictions for rare ones.
Results. Table [ shows the test MAE for CFT and baseline models. Our CFT model achieves competitive performance across all four properties and outperforms all baselines on Total Energy and Shear Moduli prediction. Notably, it significantly surpasses the Transformer baseline that uses standard positional encodings, demonstrating the usefulness of the commetry-adapter encoding. CFT outperforms CGCNN across all properties and is on par with ALIGNN and Matformer.
CFT uses fundamentally different approach from the graph neural networks, which explicitly encode distances and periodic boundary conditions in the edges (Yan et al., 2022) and encode ad- ditional geometric features such as bond angles (Choudhary & DeCost 2021 CFT relies on the
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Total Energy
Band Gap
Bulk Moduli
Shear Moduli
Method
cV/atom
cV L
log GPa
log GPa
CFT
0.197 + 0.009
0.306 + 0.006
0.082 + 0.008
0.158 + 0.011
Transformer
0.220 + 0.019
0.440 + 0.014
0.142 + 0.012
0.226 + 0.016
CGCNN
0.453 + 0.014
0.343 + 0.074
0.093 0.012
0.178 = 0.007
ALIGNN
0.203 + 0.005
0.235 + 0.002
0.076 + 0.002
0.160 + 0.009
Matformer
0.371 + 0.011
0.213 0.003
0.074 0.002
0.179 + 0.003
Table I: Test Mean Absolute Error (MAE) comparisons on the Materials Project dataset. Lower values indicate better performance. Results are reported as mean + one standard deviation over runs. Bold indicates the best performance for each property.
positional encoding module to capture meaningful geometric information and on the multi-head attention layers to capture complex interactions between atoms.
Efficiency. A key advantage of our framework is its computational efficiency. Staph--aser models typically require constructing neighborhood graphs and performing iterative message passing, which can be computationally intensive and difficult to parallelize. CFT's core symmetry operation is a single matrix- vector product with the precomputed routing matrix Mg. This, combined with the inherent parallelizability of the Transformer architecture, leads to significant speedups. This performance advantage holds even with a larger parameter budget. The empirical results in Table _ show CFT to be substantially faster than ALIGNN and Matformer in both training and inference.
Method
Parameters
Training time (per epoch)
Inference time (total)
CFT ALIGNN
5.34M
91
60
4.03M
592 sec
451 sec
Matformer 2.78M
266 see
222 sec
Table 2: Training and inference time comparison. Training time is measured per epoch dataset of 120k crystals. Total inference time is measured on a dataset of 10k crystals. Amortized pre- processing costs, including the pretraining of CFT's positional encoding module and the one-time graph construction for ALIGNN and Matformer, are not included. All models were benchmarked on single NVIDIA L40 GPU.
5.4 ZERO-SHOT GENERALIZATION TO UNSEEN SPACE GROUPS
The core hypothesis behind our adaptive architecture that, by explicitly parameterizing the group constraints, CFT can generalize to materials from previously unseen space groups. We test this hypothesis a zero-shot setting by holding out all space groups containing inversion symmetry from the training set.
Setup. Let Givv denote the set of space groups with inversion symmetry (centrosymmetric groups). This subset accounts for approximately 49% of the bulk/shear modulus data and 25% of the total- energy data. For each architecture we train two models: (1) Zero-shot: trained only on the 138 non-centrosymmeerie space groups \ Gavv and (2) All-data: trained the full dataset of 230 space groups. Both models are then evaluated on the same dataset of held-out inversion groups. For CFT in the zero-shot regime, we simply provide inference time the precomputed routing matrix Mg corresponding to the unseen group G; no parameter updates or fine-tuning are performed.
Results. For each held-out inversion group G Ginve compute the group-wist wise MAE under the zero-shot and all-data training regimes, denoted MAE and MAEall-dan. performance gap for group G as
We define the
AMAEG = MAE zeero MAE
aall-da
Figure plots these signed gaps AMAEG for all held-out groups with at least 20 data points for both shear modulus and total energy; values close to zero indicate that the zero-shot model matches the fully supervised model on that group. We summarize these results with the group-bulanced absolute performance gap
GB-Gap
[ EGEGIn |AMAEG]
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Total Energy Performance Gap (Zero shot MAE All data MAE)
1.50
CFT lours)
AUINN
Matformer
AMНE (eV/ttmm)
1.00
0.50
-0.25
Shear Modulus Performance (Zere-she MAE data MAE)
AMKE (log GPS)
0.4
0.2
113
139
121
166
167
157
189
190
26
Figure 3: We evaluate the zero-shot generalization capability of CFT in predicting the shear modulus and total energy of materials from groups containing inversion symmetry. For each held-out group, we plot the performance gap AMAE (Eq. 9). the difference between the zero-shot MAE (trained without any data from inversion groups) and the all-data MAE (trained all 230 space groups). Smaller values are better; AMAF = O corresponds to perfect zero-shot generalization to that group.
This metric first computes MAE per space group and then averages across groups, each held- out group contributes equally regardless its size. For shear modulus, the GB-Gap is 0.042 log GPa for CFT, compared to 0.120 for ALIGNN and 0.080 for Matformer. For total energy, the GB- Gap is 0.141 eVVatom for CFT, versus 0.309 for ALIGNN and 0.197 for Matformer. CFT thus exhibits the smallest average per-group degradation when moving from the all-data regime zero- shot generalization.
As shown in Figure 3 CFT maintains small gaps across all held-out groups, whereas ALIGNN and Matformer exhibit pronounced failure modes on several groups. In particular, for groups 113, 162, 200, and 216 the zero-shot shear-modulus MAE of ALIGNN and Matformer exceeds the all- data MAE by more than factor of five. These failures occur primarily for groups that combine inversion with serews or glides and high cubic or hexagonal symmetry, patterns that are rare among the non-ine ann-inversion groups used for training. We hypothesize that, in such cases, the GNNs struggle to extrapolate these geometric motifs from related groups, while CFT's explicit symmetry enforcement enables more reliable performance the unseen groups.
6 CONCLUSION
In this work, we introduce the Crystal Fourier Transformer, a single, adaptive neural architecture capable of enforcing invariance to any of the 230 crystallographic space groups. Our approach is built on a novel method for constructing symmetry-adapted Fourier bases by analytically deriving the linear constraints that group operations impose Fourier coefficients, which are then encoded in a group-conditional routing matrix. We demonstrate that this weight-sharing model learns geo- metrically meaningful representations of the crystal orbit space, achieves competitive performance on material property prediction benchmarks, and generalizes in a zero-shot setting to unseen space groups, illustrating its potential to overcome the data sparsity inherent materials science.
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ACKNOWLEDGMENTS
The authors would like to thank Keqiang Yan, Kevin Han Huang. Alex Guerra, Drew Novick, Ash- win Sah, and Michael Zhang for helpful discussions and feedback on earlier versions of this work. This work was supported by NSF OAC 21 18201 and the Gatsby Charitable Foundation.
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A APPENDIX
A.1
EXAMPLES
There 17 crystallographic groups in IR2 also called wallpaper groups, which describe the possi- ble symmetries when tiling all of two-dimensional space with convex shape. The tiling behavior described by each of these groups is displayed in Figure 4
gan
F
F
F
T
P
P
P
7
P
F
F
d
b
F
T
F
d
F
T
F
7
2
F
F
F'
F
F
F
F
E
d
UT La LAIT LUTITI
d
d
d
i
d
E
4
d
E
F
F
44 M4
H
T
F
b.
1
P
M
R
CA
Figure 4: Tiling behavior of the 17 wallpaper groups. The fundamental region II, highlighted red, is the smallest unit that can be repeated to form the entire tiling.
39.48
78.96
78.96
157.91
157.91
157.91
197.39
197.39
197.39
197.39
Figure 5: The first 10 non-constant basis functions for the wallpaper group pg. The number dis- played above each function the corresponding eigenvalue from equation
Following the procedure described in Figure we can derive the exact analytic form of the basis functions invariant any given crystallographic group G. By construction, each basis function is of the form
cx(x)
Σ wi -we
WEO
where O the orbit of canonical point This an eigenfunction of the constrained PDE
- Xe
(11)
subject to
e = = ò for all ø E G
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52.64
92.12 12
157.91
157.91
210.55
210.55
250.83
315. 368.
368.47
Figure 6: The first 10 non-constant basis functions for space group 193, p6g/mem. This group contains 6-fold rotational symmetries, 2-fold rotational symmetries, and mirror symmetries visible along different faces.
with corresponding cigenvalue 111 The first 10 basis functions for the wallpaper group pg, ordered by corresponding eigenvalue, are shown Figure 5 Our procedure naturally extends to nigher-dimensional crystallographic groups; the first 10 basis functions for the space group p63/mem are shown in Figure
A.2 PROOF OF PROPOSITION
Consider function f R + IR and its Fourier transform:
F(w) = = f(z))e(12 :2n dx.
(12)
Given crystallographic group G with translations that are integer combinations of linearly inde- pendent basis vectors a1...., an a G-invariant function must be invariant to all translations of the form Σ, Gaa; where z", iie., we require:
<+ cai = f(x) for all c Z".
(13)
Shifting function introduces phase factor into the Fourier transform. Satisfying the equation above requires:
F(w) "2 csa, F(w) for all
(14)
For simplicity, let us take a1,.. a.n to the standard basis R", Then, in the case where ei and all other gi2mw = 1 if and only if W1 E 2, this simple shift invariance requires that F(w) = everywhere that W1 Z.
0, equation 14 is satisfied if and only if eaos: = 1 F(w) 0. Because
Applying the same logic to each dimension of w, we see that this shift invariance implies F(w) = when & Z", i.e., it only be nonzero on this discrete grid of points. These points correspond to the Fourier series and the values of F(w) at those points are the coefficients of the Fourier series. This grid of points is typically referred to as the reciprocal lattice.
More generally, our group actions are the form Ax + t. where A is an orthonormal matrix. We first observe how Sfange-of-variahhe of the form T(z) Bz + e alters the Fourier transform.
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Taking (z c) and dz -d=
| f( (Bz c)e =2 = =dz=
f(z)ee 12T (B" (a c)) dæ
B R
[ f(z)e
"2mm T ' (2 BBB da
12Tw
B
22 B
B
f(z)e-2x((B
wTB--T)Too dx
B
B
22w B
l
F(B-Tw)
w)
B
Given f(Ax + +) f(x), where |A| and A
A, this requires
F(w) = 22m A'eF(Aw) for all w.
(15)
A.3 PROOF OF THEOREM 3.2
Let be a crystallographic group and L its reciprocal lattice. By Proposition 3.1 if f G-invariant with Fourier coefficients F Fww), then for any Φ (A,t) t) G and w E L',
F(w) = 22m A F(Aw).
(16)
This relation implies that the point-group factors A partition L into disjoint orbits O C*. Definition A.1 (Phase consistency). Fix an orbit O and w E O. The stabilizer of in G is G ((A,t) E G: Aw = w). We call O phase-consistent if
(2 T A =1 for all (A,t) Gw-
Equivalently (via concatenation derived below), every closed path in the constraint graph on O has edge-weight product 1.
We encode equation 16 as directed, edge-weighted graph on L for each (A,t) E Gaand E L", place an edge w + Aw of weight w(w + Aw) == e (2 w At Lemma A.2 (Path concatenation). If there are edges (wi د w2) of weight wi and (w2 - ws) of weight w2, then there is edge (wi + wa) of weight wiw2.
Proof. Let 4i (x)) = A + t and (2((x)) = Azæ + l2 The edge w - A has weight W1 e 22T t,t, and Ai AzAlw has weight wa (( Ag Since so (x(())
A²A + (Agt1 + t2), the composed edge w + AgAlw has weight
2 (Aatittt)) c (2n A A7ta) wiW2,
as claimed.
Fix phase-consistent orbit O and reference & E O. For any ws E O, let Pe-tw a path in the constraint graph from & to and define
W
w(e).
<EPe-w
By Lemma A.2 edge weights multiply under path concatenation. The definition of We-tw is in- dependent of the chosen path iff every closed cycle has unit product, i.e., satisfies the phase- consistency condition. If an orbit fails phase-consistency, then traversing nontrivial cycle gives non-unit factor, and iterating equation 16 around the cycle forces F(w) = 0 for all E O; hence only phase-consistent orbits can support nonzero coefficients.
For phase-consistent orbit O, choose an arbitrary scalar dg E C and set
F(w) cg We--wa
O.
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Then equation I6 holds identically O (shifting w + Aw multiplies ug-w by the appropriate edge weight), so the inverse Fourier series yields the G- G-invariant function
(a(
-E we - 2r
WEO
Distinct orbits O o O' have disjoint frequency supports, whence eo 1 Eor in (2II)) by Fourier orthogonality on II.
Now let f be any G-invariant function. Its Fourier coefficients vanish off L and satisfy equation I6 on each orbit. For any orbit O that is not phase-consistent, the cycle constraint forces F 0 O; for phase-consistent O the coefficients are proportional to (we-wwweo Therefore
f(x) =
Σ O phase-consistent
co (o))),
ie., {eo} spans the subspace G- G-inaariant functions in (II This construction an explicit realization of the representation theorem of Adams & Orbanz (2023), which guarantees complete orthogonal basis of constrained Laplace eigenfunctions for continuous G-invariant functions.
A.4 DERIVING FOURIER COFFFICIENTS FOR GAUSSIAN DENSITY
We show one way in which the symmetry-adapted Fourier basis can encode atom positions crystal in a way that captures the atom's 's environment. Consider crystal from space group G with unit cell is defined by B [b1 b2 by] R3x3 (rows of B are basis vectors). Given atom in the crystal at position summation of isotropic Gaussians centered at each atom in the orbit of æ. Recall that each isometry G takes the form <xx) Aæ where A an orthonormal matrix and is a translation vector. The density of interest is
R33 want to encode the environment of this atom given by the
Ily [-Ily-(Aa) (Ax + t)|??)
ply) = Σ exp
(17)
202
(A.t)G
We can approximate this density p with Fourier series. We begin by taking the Fourier transform in terms of where
p(w) / p(y)e 2x dy.
o(y)e-2xiwyy
Recall that for an isotropic Gaussian g(y: ,4, .) = -1l--111 1(20t) its Fourier transform is given by ((w) = (2T) 3/2 3/2 6 0
(18)
Our density can be rewritten as
ply) Σ
g(y; Ax + tt)) (A,D)
By linearity,
p(w)
Σ (2m))3/2 3 2x 2 -2 (Aott)
(A,t)GG
(27)3/2203 2x*2001 |||!
e (Ax+t
(A,t)EG
2 211"
(2 -
Σ
-2rii
Σ 2mii-
(A.t)e
EEL
where the lattice L is generated by all linear combinations of the basis vectors in b bR, b3. and & is the finite subset of G satisfying
G = ((A,t) G 4,b + l2b2 labs, 1 E (0,
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Applying Poisson summation the lattice comb, have
Σ
2
B
Σ le L
8(w - k)
(19)
REL
where I.* denotes the reciprocal lattice. Hence
((w) (2m) 3/2
Ble
1wol?
Σ KEL*
Σ ((A,))G
e
8(w - k))
(20)
Taking the inverse Fourier transform, we see that the target density p(y) can be written Fourier series where the coefficient of the term e el for l E L", is
k (2T)) 3/2 03 2
Σ
2Tik -
(21)
(A.t)eū
Note that when working converting from fractional to Cartesian coordinates, replace ky E Z with
ke B B-k,
A.5
EXPERIMENT DETAILS
In this section, we describe our experimental setup in more detail. We start with the pretraining setup for the symmetry-adapted positional encodings in Section 5.2 then give configuration details for the material property prediction experiments in Section 5.3 and the zero-shot learning experiments in Section 5.4
A.5.1 PRETRAINING POSITIONAL ENCODINGS
The central objective of our pretraining is to map atomic positions into embedding space where the Euclidean distance between embeddings corresponds to the real-space orbit distance between the atoms. The orbit distance
do(0), == min (x)) d2 (222))
(22)
@1 P1.
the minimum Euclidean distance between any two atoms in the orbits of the two initial atoms, considering all symmetry operations of the crystal's space group and periodic boundary conditions.
Our model produces embeddings = f(pi G, L) and e₂ f(p2, G. The pretraining loss is the mean squared error (MSE) between the orbit distance and the L2 distance between the corre- sponding embeddings:
Spectrii (do (p1, p2) | - 22
(23)
The model is designed to explicitly handle crystallographic symmetries and lattice geometry through a two-branch architecture:
Symmetry-Adape Position Branch: The model first generates cymmetry-adaptee repre- sentation from the fractional coordinates. The input fractional positions are converted into of Fourier features, exp(i 2n p k), where k represents a of reciprocal lattice vectors. This Fourier representation is then processed by series of residual blocks to produce a position-baat based feature vector.
Lattice Geometry Branch: The 3 x lattice vectors are flattened into a 9-dimensional vector and fed into separate network of 3 residual blocks. This branch learns capture the scale and geometry the unit cell, which is essential for converting dimensionless fractional distances into real-space distances.
Each residual block consists of two dense layers, with LayerNorm and ReLU after each layer, that expands the feature vector to length 512 then projects back to length 256. The outputs from the two branches combined via element-wii wise multiplication. The resulting vector is then passed through a final MLP to produce the -28-dimensi positional embedding. The model was trained for 200 epochs using the Adam optimizer with learning rate of 0.0002 and batch size of 2000.
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A.5.2 MATERIAL PROPERTY PREDICTION
We use 80% training. 10% validation, and 10% test data splits across all property prediction experi- ments. All models are trained for 500 epochs.
Crystal Fourier Transformer. Each atom in crystal is represented by input token, which is the sum of 128-dimensional standard learnable atom embedding based on the atomic number, and a 128-dimensional positional encoding from the pretrained encoding module.
The sequence of input tokens is processed by a series of 3 Transformer blocks. Each block consists of multi-head self-attention layer and feed-forward network. These blocks have embedding di- mension 128 and 8 attention heads per block. The feed-forward network contains two dense layers, the first one expanding the embedding dimension 512 before projecting back to dimension 128, using SiLU activation and LayerNorm after each layer. After the final Transformer block, we per- form masked mean pooling over the atom tokens to produce single fixed-size vector representation for the entire crystal. This vector is then passed through a two-layer MLP with hidden dimensions of 2048 and 256 to produce the final scalar prediction.
The model was trained using the AdamW optimizer. The training runs were done with learning rate 0.0001, weight decay 0. 0.0001, and batch size 128.
Matformer. We trained Matformer models on the 2025 Materials Project dataset using the code provided by the original authors Yan et al. (2022). We use the Adam optimizer with a learning rate of 5e-4 and weight decay of 1e-5, coupled with a OneCycleI.R scheduler for learning rate adjustments. The models were trained with a batch size of 64, the maximum given the memory constraints of an NVIDIA 1.0 GPU. For graph construction, k-nearest neighbor strategy used to identify the 12 nearest neighbors within an 8 Á cutoff radius. Each atomic number is mapped to a 92-dimensional embedding using the CGCNN (Xie & Grossman, 2018) atomic features, before a linear transformation maps to 128-dimensional feature vector used input to the first Matformer message- meessge-ppssingg layer. The crystal lattice vectors are used in the graph construction, but information about bond angles is not.
ALIGNN We also retrain the ALIGNN models using the code provided by the original authors (Choudhary & DeCost 2021). The model comprises of ALIGNN layers and subsequent GCN layers, both utilizing 256 hidden features. The model processes 92-dimensional CGCNN-style atomic number embeddings for nodes, while bond distances and triplet bond angles were expanded using 80 and 40 radial basis functions, respectively, and embedded into 64-dimensional feature space. Crystal graphs were also constructed with a k-nearest neighbor strategy to identify the 12 nearest neighbors within 8 Á cutoff radius. We use the AdamW optimizer, a batch size of 32, learning rate of Se-4, and weight decay of le-5, coupled with a OneCycleLR scheduler.
A.5.3 WHY PRETRAIN?
The standard Fourier encoding seen in Figure 2 is dimensionless, so we need a way to capture the different scales of the crystals in the Materials Project. We pretrain the G-invariant encoding module to learn to scale the coefficients of the Fourier basis in a way that captures the geometry and scale of the crystal structures, using the orbit distance setup described in Section 5.2 Table shows that pretraining the positional encodings improves property prediction performance across the board, in comparison end-to-end training with random initialization of the positional encodings.
When pretraining the positional encoding module, we must choose a cutoff radius for the set of Fourier features +uo= to include, where includes all reciprocal lattice vectors up to certain radius R. In practice we have found that R = 5, corresponding to 514 Fourier modes, performs the best on downstream prediction tasks while maintaining tractable model size. Figure ( shows the trade-off between the number of Fourier modes used and performance on the orbit distance regression task. This error should go to increase the number of basis functions.
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Total Energy eVattom
Band Gap
Bulk Moduli
Shear Moduli log GPa
Method
eV
log GPa
CFT with pretraining CFT without pretraining
0.197 + 0.009 0.306 + 0.006
0.082 + 0.008 0.158 + 0.011
0.207 + 0.003 0.338 0.009
0.134 + 0.010 0.215 + 0.008
Table 3: Test Mean Absolute Error (MAE) comparisons on the Materials Project dataset. Lower values indicate better performance. Results are reported as mean - one standard deviation over runs.
Orbit Distance Test MAE
0.2
6:17
646
Number of basis functions
Figure 7: Comparison of the number of Fourier modes used and the resulting test MAE for the orbit distance regression task.
A.5.4 ZERO-SHOT GENERALIZATION TO UNSEEN GROUPS
In these experiments, we define the hold-out set to be all space groups containing inversion symme- tries. This subset represents a significant portion of the data: approximately 49% of the Materials Project dataset for bulk/shear modulus and 25% of the dataset for total energy. The zero-shot models are trained exclusively on data from the 138 space groups that not contain inversion symmetries, whereas the all-data models are trained data from all 230 space groups. Models from both regimes are evaluated on the same dataset of held-out inversion groups to calculate the performance gap.
We use the same CFT architecture as in the previous experiments: Transformer blocks, each consisting of multi-head self-attention layer and feed-forward network. These blocks have em- bedding dimension 128 and attention heads per block. After the final Transformer block, we perform masked mean pooling over the atom tokens and then pass through two-layer MIP with hidden dimensions 2048 and 256 to produce the final scalar prediction. Similarly, we use the same Matformer and ALIGNN architectures we did for the property prediction task, described in Section A.5.2
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Published as a conference paper at ICLR 2026
A.6 ORBIT DISTANCES IN P6M
We can visualize the learned embeddings for the 2D wallpaper group p6m. Given a tiling pattern with d 6m symmetries and choice of a canonical fundamental region (e-g., the one highlighted in red on the left in Figure B). the mapping of each point æ to the unique point in its group orbit {Ø(x) Φ E G} that lies in the fundamental region is known isometric mapping where the standard Euclidean distance perfectly preserves orbit distance. As shown in Figure g our model's learned 2D embeddings, using the pretraining setup described in A.5.T] autonomously discover and reproduce this exact geometric mapping without any prior knowledge of it. Different random seeds all produce some rotation of this mapping. This provides strong evidence that our architecture learns the correct underlying metric of the orbit space.
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Posla EREA Pegice
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Unit coll group p6m
Figure 8: In the special case of wallpaper group pém, which consists of 6-fold rotational symmetry and mirror symmetries, mapping any point to its image in the fundamental region (right triangle bordered in red the left) exactly preserves the orbit distance between every pair of points. On the right, our G-invariant learned positional encodings discover the same isometric mapping in red, without any explicit encoding of this mapping.
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Reviews:
[{"review_id": "y41jlKaUCf", "reviewer": "Reviewer_HXJZ", "summary": "This paper introduces the Crystal Fourier Transformer (CFT), an architecture for crystal property prediction. The core idea is to use a pre-computed, group-specific matrix, $M_G$, to explicitly project atomic coordinates into a G-invariant feature space. This design enables a single, weight-shared Transformer backbone to process all 230 space groups, aiming to address data sparsity and enable generalization to unseen groups.", "questions": "1. Why was it assumed that GNNs cannot generalize in this zero-shot setting? Baselines must be evaluated under the identical zero-shot setup (e.g., holding out the same 6 groups). It is plausible that neural networks can learn this generalization implicitly, and the authors have not proven their explicit $M_G$ method is superior or necessary.\n2. Why focus on the saturated task of property prediction rather than the current SOTA challenge of crystal structure generation? Has this G-invariant encoding scheme ($M_G$) been tested for generative tasks, and does it offer any advantage?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "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.\n2. The method avoids the complex graph construction required by GNNs. Its Transformer architecture is highly parallelizable, aligns with mainstream models.\n3. 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 groups.", "weaknesses": "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.\n2. 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.\n3. The paper's core selling point—generalization to unseen groups —is of limited practical value. \n\nMissing related work on similar projection into invariant features:\n1. A new perspective on building efficient and expressive 3D equivariant graph neural networks， Neurips 2024", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZQLGuXaahj", "reviewer": "Reviewer_MwQk", "summary": "The paper \"A Single Architecture for Representing Invariance Under Any Space Group\" presents a novel approach to producing space group invariant representations of crystal structures. The main innovation is Crystal Fourier Transformer (CFT) which uses a Fourier basis conversion to make a space-group invariant representation of the atomic coordinates. They showcase that CFT has comparable performance to other state of the art models while having considerably less training and inference time.", "questions": "1. Are the times listed on Table 2 inclusive of the pretraining time required for CFT positional encodings?\n2. I am not sure I fully understand the setup for the synthetic dataset in section 5.2. To my understanding, you can't select random positions for atoms in a crystal as they are locked to certain regions by the space group of the crystal. If my understanding is wrong, please let me know, but how did you place the atoms when creating the synthetic dataset so that it followed the space group rules?\n3. What is the explanation for why training on Shear Modulus sometimes increases MAE?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- 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.\n- 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.\n- 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 art methods have become quite slow in comparison due to the costly crystal graph representation. This clearly shows how their method can be used to create more efficient models and representation techniques that could be built upon in future work.", "weaknesses": "- 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. \n- 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. Seeing as how modern machine learning models are already much faster than DFT even with longer training and inference times compared to CFT, it would be better to see how their method could improve performance.\n- Seeing as how one of the main claims is that computation time is significantly reduced, I would like to see more clarity in terms of number of parameters of each model and how that may affect runtime as well. It would be interesting if the models were all similarly sized but had vastly different runtime or if the performance of CFT is comparable to other models which may be much larger (or smaller).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sQYiBmidcd", "reviewer": "Reviewer_vVEH", "summary": "The authors introduce a novel algorithm for constructing a symmetry adapted Fourier basis that generalizes invariance across the point groups. This basis provides an alternative to the more commonly used graph-based representation of crystal structures.", "questions": "**Q1. Orbit-distance evaluation is architecture-limited.** Clarify how inconsistent self-loops in the graph representation (line 212) are detected. My understanding from Figure 1 is that these correspond to single-node connected components by virtue of being self-looping nodes; confirm whether this is the intended general definition and provide the formal criterion used.\n\n**Q2. Property prediction is not yet compelling.** Clarify the role of the pre-trained positional encoder. In particular, is there a measurable performance benefit to pre-training the embedding rather than learning entirely end-to-end?\n\n**Q3. Basis choice and encoder role are under-specified.** Provide ablations on the dimensionality of the chosen basis, especially for the encoder training used in later experiments. My understanding is that the observed error in the orbit distance task is due to basis truncation, but it should be clarified. Also, please report the exact basis dimensions used (main text and Appendix)\n\n**Q4. Zero-shot comparisons.** Please compare to the zero-shot capabilities of other SOTA architectures.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "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.\n\nThe paper presentation is exceptional, aside from several grammatical errors and points that need further clarification.\n\nThe numerical results are promising, particularly those on the improved computational performance.", "weaknesses": "The paper experiments are limited in the following aspects\n1) the orbit distance results are only reported for the CFT architecture, making it unclear if the technique has improved the learning of the representations.\n2) The material property prediction results demonstrate marginally equivalent or worse results in comparison with SOTA techniques.\n3) 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 encoder.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "P4eaQmBehX", "reviewer": "Reviewer_JEwd", "summary": "This paper presents the Crystal Fourier Transformer (CFT), a single neural network architecture capable of enforcing invariance to any of the 230 crystallographic space groups without requiring separate architectures for each group. The key innovation is an analytical method for constructing symmetry-adapted Fourier bases by deriving explicit constraints that group operations impose on Fourier coefficients. These constraints are encoded in precomputed, group-specific adjacency matrices MG\\textbf{M}_G\nMG​ that act as routing mechanisms, allowing a single Transformer model to share weights across all space groups while adapting to different symmetries. The authors demonstrate that CFT learns geometrically meaningful orbit distance representations, achieves competitive performance on Materials Project benchmarks for predicting material properties (total energy, band gap, bulk/shear moduli), and successfully generalizes zero-shot to space groups unseen during training.", "questions": "Theoretical completeness: In Theorem 3.2, you filter out frequencies with inconsistent self-loops. Can you characterize which groups/frequencies are affected? What proportion of the reciprocal lattice typically survives this filtering for common space groups?\n\nGaussian distribution failure: You mention that \"datasets with Gaussian distributions are challenging for the current implementation\" and attribute this to covariance-based entropy approximation. Can you:\nProvide concrete failure modes or examples?\nExplain why this is problematic (Gaussians should be easy to model with covariance)?\n\nComparison with prior symmetry methods: Can you provide direct quantitative comparisons with on problems they can handle (low-dimensional representations)?\n\nArchitecture details:\nDuring training, how is the space group G provided to the model? As a one-hot encoding, or implicitly through the routing matrix?\nFor the orbit distance pretraining, why use ResNets rather than simpler MLPs? Were alternatives tested?\nHow was the frequency cutoff radius chosen?\n\nGeneralization breadth: The zero-shot experiment holds out 6 groups. Can you:\nTest on more aggressive holdouts (e.g., entire crystal systems)?\nShow how performance degrades as held-out groups become less similar to training groups?\nAnalyze which group properties enable transfer?\n\nBasis function validation: Beyond orbit distance, have you visualized or analyzed whether the learned basis functions match the theoretical predictions from Theorem 3.2? Can you show examples of learned vs. theoretical basis functions?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Mathematical rigor and elegance: The analytical approach to deriving Fourier constraints (Proposition 3.1, Theorem 3.2) is theoretically sound and provides exact rather than approximate invariance. The connection to graph theory through dual representations is intellectually satisfying.\n\nArchitecture innovation: Creating a single model that adapts to 230 different space groups through precomputed routing matrices $M_{G}$ is a clever solution to weight sharing across related symmetries. This is fundamentally different from prior work requiring separate architectures.\n\nComprehensive experimental validation: The paper validates three distinct aspects: (1) geometric correctness (orbit distance learning with MAE=0.102Å), (2) competitive performance on established benchmarks, and (3) zero-shot generalization capability. This multi-faceted evaluation is thorough.\n\nPractical efficiency: The 6-7× speedup over graph-based methods (Table 2) combined with competitive accuracy makes this practically attractive. The simple matrix-vector product for symmetry adaptation is elegant.\n\nStrong zero-shot results: Figure 3 shows minimal performance degradation on held-out groups, validating the weight-sharing hypothesis and demonstrating genuine transfer learning across symmetries.", "weaknesses": "Limited performance gains: While CFT is competitive, it doesn't consistently outperform baselines. It beats ALIGNN/Matformer on only 2/4 properties (Table 1), and the improvements are often within error bars. For Band Gap, Matformer achieves 0.213±0.003 vs CFT's 0.306±0.006. The practical advantage over existing methods is unclear.\n\nAbelian group restriction: The limitation to commutative Lie groups is significant. The shared eigenvector parametrization (Appendix C.2) that ensures commutativity is a fundamental architectural constraint. The paper doesn't discuss whether this could be relaxed or what barriers exist.\n\nTheory:\nWhat fraction of reciprocal lattice points satisfy phase-consistency?\nHow does approximation error scale with frequency cutoff radius?\nUnder what conditions does the Gaussian approximation for joint entropy fail?\nWhy specifically do Gaussian-distributed datasets cause problems (acknowledged but not investigated)?\n\nLimited baseline comparisons: The paper doesn't compare against other symmetry discovery methods even on small problems they can handle. Direct comparison on 2D wallpaper groups would strengthen claims.\n\nEvaluation scope:\nZero-shot experiment uses only 6 groups (~10% of data); testing on more held-out groups would be convincing.\nOnly Materials Project dataset evaluated; other crystal property databases would validate generalization.\nNo comparison of learned vs. ground-truth basis functions beyond the orbit distance task.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)