Title: Space Group Equivariant Crystal Diffusion
Venue: NeurIPS 2025
Accepted: yes
Keywords: crystals

generative

diffusion

space group

equivariant

wyckoff

asymmetric unit

transformer

score matching
Abstract: Accelerating inverse design of crystalline materials with generative models has significant implications for a range of technologies. Unlike other atomic systems, 3D crystals are invariant to discrete groups of isometries called the space groups. Crucially, these space group symmetries are known to heavily influence materials properties. We propose SGEquiDiff, a crystal generative model which naturally handles space group constraints with space group invariant likelihoods. SGEquiDiff consists of an SE(3)-invariant, telescoping discrete sampler of crystal lattices; permutation-invariant, transformer-based autoregressive sampling of Wyckoff positions, elements, and numbers of symmetrically unique atoms; and space group equivariant diffusion of atomic coordinates. We show that space group equivariant vector fields automatically live in the tangent spaces of the Wyckoff positions. SGEquiDiff achieves state-of-the-art performance on standard benchmark datasets as assessed by quantitative proxy metrics and quantum mechanical calculations. Our code is available at https://github.com/rees-c/sgequidiff.

Space Group Equivariant Crystal Diffusion

Abstract

Accelerating inverse design of crystalline materials with generative models has significant implications for a range of technologies. Unlike other atomic systems, 3D crystals are invariant to discrete groups of isometries called the space groups. Crucially, these space group symmetries are known to heavily influence materials properties. We propose SGEquiDiff, a crystal generative model which naturally handles space group constraints with space group invariant likelihoods. SGEquiDiff consists of an SE(3)-invariant, telescoping discrete sampler of crystal lattices; permutation-invariant, transformer-based autoregressive sampling of Wyckoff positions, elements, and numbers of symmetrically unique atoms; and space group equivariant diffusion of atomic coordinates. We show that space group equivariant vector fields automatically live in the tangent spaces of the Wyckoff positions. SGEquiDiff achieves state-of-the-art performance on standard benchmark datasets as assessed by quantitative proxy metrics and quantum mechanical calculations. Our code is available at https://github.com/rees-c/sgequidiff .

1 Introduction

Crystals comprise critical technologies like batteries [66], topological materials [90], electronic devices [96], photovoltaics [28], and more. Materials scientists have catalogued O (10 5 ) crystals experimentally [7] and O (10 6 ) in silico with density functional theory (DFT) simulation [15, 39, 79]. In contrast, the number of stable crystalline materials with five elements or less is estimated to exceed 10 13 , and even higher-order compositions are common in real materials [16]. Generative models offer a promising path to rapidly explore the vast space of crystals [100, 40, 62, 10].

Unlike molecules, crystals span the periodic table and exhibit discrete spatial symmetries according to one of 230 space groups [2]. Specifically, crystals have invariances to discrete translations, rotations, reflections, and sequences thereof that transform atoms into themselves or into identical atoms. The list of space group actions which map a point into itself is called a stabilizer group . When a set of points in R 3 have conjugate stabilizer groups, the set is called a Wyckoff position . Importantly, as shown in Figure 1, Wyckoff positions can have zero volume, comprising points, lines, or planes. These zero volume sets are referred to as special Wyckoff positions . We give a more formal treatment of space groups and Wyckoff positions in Sec. 3.

Despite the fact that most existing crystal generative models ignore space groups and Wyckoff positions, they are critical for modeling real materials. Firstly, space groups and Wyckoff positions correlate strongly with materials properties; Neumann's principle states that all crystal properties share the same invariances as the crystal itself [65]. Thus even slightly perturbing atoms out of special Wyckoff positions will reduce the crystal's space group symmetry and can subsequently cause significant (even discontinuous) changes to its macroscopic properties [14, 11, 82, 9]. Secondly, we

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show empirically in Figure 1 that known materials are usually invariant to high symmetry space groups with atoms in zero-volume Wyckoff positions. An intuitive explanation for this phenomenon is that atoms of the same species in a given crystal are most stable in a specific neighboring environment, and the tendency to attain this environment naturally leads to symmetry [32]. Yet, most existing crystal generative models learn continuous distributions over all three spatial dimensions of atom positions, assigning zero probability measure to placing atoms in special Wyckoff positions. One might argue that crystals from these models can be relaxed into high symmetry positions with DFT or machine learned force fields. However, besides the apparent difficulty of generating atoms sufficiently close to high symmetry positions for relaxation [107, 31, 42], such a framework assigns different model probabilities to crystals that relax into the same structure, obfuscating training and evaluation.

In this paper, we propose SGEquiDiff, which enforces space group constraints during generation and yields explicit space group invariant model likelihoods. SGEquiDiff combines SE(3)-invariant, discretized lattice sampling; permutation-invariant, transformer-based generation of occupied Wyckoff positions and elements; and space group equivariant diffusion of atomic coordinates. For the latter, we built a space group equivariant graph neural network and trained it with space group equivariant scores from a Space Group Wrapped Normal distribution.

The main contributions of our work are summarized as follows:

We built a space group equivariant denoising graph neural network and regressed it against scores from a novel Space Group Wrapped Normal distribution.

We prove that space group equivariant vector fields live in the tangent space of each Wyckoff position, ensuring atoms never leave their Wyckoff positions during diffusion. This result obviates the need to project atoms onto Wyckoff positions, which can lead to indeterminate probability distributions [41] or a discontinuous generation process [55, 78].

SGEquiDiff proposes an explicit-likelihood autoregressive alternative to diffusion-based sampling of space group-constrained lattice parameters. The autoregressive sampling also enables masked in-filling. This may be useful, e.g., to generate optimal substrate materials with low lattice mismatch for epitaxial crystal growth [110, 19], commonly used for semiconductors and magnetic devices.

We verified the efficacy of our model on standard crystal datasets with proxy metrics and rigorous DFT calculations.

2 Related Work

Early crystal generative models represented crystals as voxelized images [67, 34] or padded tensors of 3D coordinates [49, 75] to train variational autoencoders (VAE) [18] or generative adversarial networks [26]. Several works have enforced the SE(3) and periodic translational invariance of crystals by leveraging graph neural networks. One popular approach is to use diffusion models [83, 84, 33]

on crystal lattices, atom types, and atom positions [100, 40, 107, 57]. These models have also been extended with the flow matching framework [58], accelerating sampling [62]; the stochastic interpolants framework [1], expanding the model design space [35]; and the Bayesian Flow Networks framework [27, 97] which iteratively refines model parameters instead of samples. Other works have aimed to learn the SE(3) and periodic invariances through data augmentations [104, 31] or data canonicalization [101] with large language models (LLMs) [31, 101] or with diffusion of raw attributes [104] or of latents [42]. Recent works have used LLMs to generate noisy crystals or textual context which is passed as input to graph-based diffusion or flow matching [86, 103, 71]. We view many of these existing works as orthogonal to ours since our handling of space group symmetries can be combined with their modeling frameworks.

Two of the aforementioned works attempted to learn space group-conditioned generation without hard constraints. The graph diffusion model MatterGen [107] was fine-tuned on 14 space groups and used ground truth numbers of atoms per unit cell per space group to initialize generation. However, they could only generate stable, unique, and novel crystals in target space groups with 16% accuracy. Similarly, CrystalLLM [31] only managed 24% accuracy despite a generous SpaceGroupAnalyzer [91, 70] tolerance of 0.2Å.

Relevant to our model, a growing number of works have considered hard space group constraints during generation; however, to our knowledge, they either produce crystals without space group invariant likelihoods, thus assigning different likelihoods to symmetrically equivalent atoms, or rely on local structure relaxations using machine learning interatomic potentials (MLIPs) or DFT. WyCryst [108] trained a VAE to generate elements and Wyckoff position occupations, but rely on expensive DFT relaxations of atoms from uniform random locations in the Wyckoff positions. Crystal-GFN [64] considered space group constraints for the task of distribution matching under the GFlowNet framework [6] but did not address how to sample atom coordinates. DiffCSP++ [41] trained a graph-based diffusion model with masked diffusion of the unit cell lattice, continuous element diffusion with a post-hoc argmax , and projected diffusion of atom positions on the Wyckoff subspaces. However, DiffCSP++ is not space group equivariant to the best of our knowledge and relies on fixed templates from the training data of space group, number of atoms per unit cell, and occupied Wyckoff positions. SymmCD [55] is a non-equivariant diffusion model which leverages asymmetric units to reduce memory footprints; they use discrete diffusion of Wyckoff positions and elements and post-hoc projections of atomic coordinates onto Wyckoff positions. SymmBFN [78] recently extended SymmCD to the Bayesian Flow Networks framework [27], also requiring post-hoc projections of atom positions to the Wyckoff subspaces. CrystalFormer [10] trained an autoregressive transformer, canonicalizing crystals as a sequence of atoms ordered lexicographically by Wyckoff letter and then fractional coordinates. However, they rely on von Mises distributions which are not space group invariant. WyckoffTransformer [45] also autoregressively predicts atom types and Wyckoff positions but relies on DiffCSP++ or MLIPs to refine atoms from uniform random coordinates in the Wyckoff positions. We note that MLIPs are hindered by kinetic barriers and only preserve space group symmetry if they predict conservative forces of an invariant energy, generally requiring an expensive backward pass for inference [24, 77, 51]. WyckoffDiff [47] generates Wyckoff positions and elements with D3PM-based diffusion [3] but similarly relies on MLIPs to determine atom coordinates. Unlike these existing works, our model learns to generate complete crystals from scratch and produces space group invariant likelihoods via space group equivariant diffusion.

3 Preliminaries

Space groups Formally, a space group G ∈ G , where G denotes the set of 230 space groups, is a group of isometries that tiles R 3 . In particular, G is generated by an infinite subgroup of discrete lattice translations T L = { n 1 L 1 , n 2 L 2 , n 3 L 3 | n i ∈ Z , L i ∈ R 3 } as well as a collection of other symmetry operations g ( · ) = { R g ( · ) + v g | R g ∈ O(3) , v g ∈ R 3 } ∈ G , where R g is a point group operation (rotation, reflection, or identity) and v g is a translation.

Wyckoff positions Given a space group and a point x ∈ R 3 , the stabilizer group G x ≡ { g | gx = x } ⊂ G (also called the site symmetry group in materials science) is the finite subgroup of G that leaves x invariant. A Wyckoff position is then defined as the set of points with conjugate stabilizer groups, i.e., { x ′ |∃ g ∈ G : G x ′ = gG x g -1 } . Conceptually, if g is a point group operation, this means that all points in a Wyckoff position are invariant to the same space group operations up to a change

of basis. By convention, when x is described with respect to the lattice basis { L 1 , L 2 , L 3 } , the size of the orbit of x in the unit cell, |{ gx | g ∈ G,gx ∈ [0 , 1) 3 }| , is called the Wyckoff multiplicity . Wyckoff positions whose stabilizer groups are non-trivial, i.e., include more than the identity operation, are referred to as special Wyckoff positions as opposed to the general Wyckoff position defined by the identity stabilizer group. Wyckoff positions are labeled by multiplicity and Wyckoff letter , where the lexicographic ordering gives the Wyckoff positions in (partial) order of increasing multiplicity.

Unit cells The infinite translational periodicity of a crystal can be represented with a parallelipiped Γ called the unit cell . The unit cell reduces infinite crystals by removing redundancy induced by T L , the group of discrete lattice translations. In this way, crystals are represented by the tuple M = ( A,X,L ) , where A = ( a ′ 1 , ..., a ′ N ) ∈ A N are the atom types, A is the set of all chemical elements, and N is the number of atoms in the unit cell; X = { ( x ′ 1 , ..., x ′ N ) | x ′ i ∈ Γ } are the Cartesian atom coordinates; and L = ( L 1 , L 2 , L 3 ) ∈ R 3 × 3 are the unit cell basis vectors. Given M , the infinite periodic structure can be reconstructed by applying the actions of T L as { ( a ′ i , x ′ i + n 1 L 1 + n 2 L 2 + n 3 L 3 ) N i =1 | n j ∈ Z } . Alternatively, the atom coordinates can be given in the lattice basis instead of the Cartesian basis. Coordinates in the lattice basis are commonly referred to as fractional coordinates. In fractional coordinates, the infinite crystal is reconstructed as { ( a ′ i , x ′ i + n 1 e 1 + n 2 e 2 + n 3 e 3 ) N i =1 | n j ∈ Z } . For the rest of this paper, we assume atom coordinates are always in fractional coordinates. While the choice of unit cell is not unique, prior crystal generative models [100, 40, 62] either canonicalize it with a minimum-volume primitive cell determined by the Niggli algorithm [29] or a conventional cell [2] which contains all the symmetries of the space group.

Asymmetric units In our work, we represented crystals with a convex polytope Π ∈ R 3 , the asymmetric unit (ASU), which maximally reduces infinite crystals by removing all redundancies induced by the space group G ⊇ T L . Under this formulation, we consider atoms in the ASU with fractional coordinates X = { ( x 1 , ..., x n ) | x i ∈ Π } ∈ R n × 3 , atom types A = ( a 1 , ..., a n ) ∈ A n , and Wyckoff positions W = { ( w 1 , ..., w n ) | w i = G x i } ∈ W n where n ≤ N . The infinite periodic structure of a crystal can be reconstructed by applying the actions of G to Π , i.e.,

By focusing on these n symmetrically inequivalent atoms, we reduced our model's memory footprint and minimized the dimensionality of the generative modeling task. We canonicalized the non-unique choice of ASU using those listed in the International Tables for Crystallography [2] with additional conditions on faces, edges, and vertices from Grosse-Kunstleve et al. [30] to ensure that the ASUs are exact , i.e., that Π tiles R 3 without overlaps at the boundaries ∂ Π .

Diffusion models Diffusion-based generative models [33, 84] construct samples x 0 ∼ p t =0 from a noisy prior x T ∼ p t = T by evolving x T through time t with a learned denoising process. The training process generally involves iteratively adding noise to data samples x 0 via the forward stochastic differential equation (SDE),

where f ( x , t ) : R d → R d is the drift coefficient, w is the standard Wiener process, and η ( t ) ∈ R is the diffusion coefficient. Samples are generated by running the process backwards in time from T to 0 via the reverse SDE,

where ¯ w is another standard Wiener process. The generative model aims to learn the score of the marginal distribution ∇ x log p t ( x ) , and then simulate Eq. 2 to sample from p 0 . In our model, we have f ( x , t ) = 0 .

Space group equivariant functions For a group of isometries G ′ , Rezende et al. [76] and Köhler et al. [52] showed that equivariant flows preserve invariant probability densities. Conversely, the gradient of a G ′ -invariant function is equivariant [44]. We are interested in using this result for the space groups. A function f : R 3 → R 3 is space group equivariant if it satisfies R g f ( x ) = f ( gx ) for all g ≡ ( R g , v g ) ∈ G , where R g ∈ O (3) is a point group element and v g ∈ R 3 is a translation. Symmetrization via summing over group elements has become a popular method to build invariant and equivariant neural networks in the last few years [105, 74]. Recently, Mirramezani et al. [63]

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showed how symmetrization can be applied to space groups despite their countably infinite size. In particular, given the finite quotient group G/T L and a T L -invariant function ˆ f : R 3 → R 3 , the following function is space group equivariant:

Besides faithfully modeling the space group symmetries of crystals, we discovered that space group equivariant vector fields conveniently live in the tangent spaces of the Wyckoff positions. We formalize this in the following theorem.

Theorem 1 Let G be a space group, x ∈ R 3 be a point residing in a Wyckoff position with stabilizer group G x , and f : R 3 → R 3 be a space group equivariant function. Then, for arbitrary constant c ∈ R and any space group element g ∈ G x ,

See proof in Sec. A.1. This result implies that a space group equivariant flow is automatically restricted to the manifold of its Wyckoff position, preventing the need for projections that lead to indeterminate probability distributions [41] or a discontinuous generation process [55, 78]. The result also implies that if, for example, a 0D Wyckoff position (a point) lies in a 1D Wyckoff position (a line), then a space group equivariant vector field cannot move an atom along the line through the point. This is problematic because we observed examples where traversing between symmetrically unique regions of a 1D Wyckoff position requires moving through a 0D Wyckoff position (e.g., in space group 192, the 1D Wyckoff position 12k is split by the 0D Wyckoff position 4c. See Fig. 3). This issue motivates SDE-based sampling since Gaussian noise (projected onto the tangent space of the Wyckoff position) can stochastically perturb atoms through 0D Wyckoff positions while maintaining that the marginal distribution over atom coordinates is still space group invariant.

4 Proposed method: SGEquiDiff

4.1 Crystal representation

Using sympy [61] and PyXtal [22], we removed redundancy induced by space group symmetry from every Wyckoff position by intersecting each one with its exact asymmetric unit. Each zerodimensional Wyckoff position was reduced to a single point, each one-dimensional position to a set of line segments, each two-dimensional position to a set of convex polygonal ASU facets, and each three-dimensional position to the ASU interior. We will refer to these convex sets as Wyckoff shapes . Intersections for space group 192 are shown as an example in Figure 3. We used Wyckoff shapes for memory-efficient crystal representation and sampling as will be described in the following sections.

We decomposed each crystal into M = ( G,L,A,W,X ) ∈ ( G , R 3 × 3 , A n , W n , R n × 3 ) , where n is the number of atoms in the asymmetric unit, G ∈ G is the space group, L ∈ R 3 × 3 is the

conventional lattice basis, A ∈ A n are elements, W ∈ W n are Wyckoff positions, and X ∈ R n × 3 are fractional atom coordinates. Several variables in M impose hard constraints on each other. To handle these dependencies, the model sequentially samples each constrained variable after its constraining variable(s). We factorized the generation process as follows:

Structuring the generation process in this way rather than jointly modeling all crystal components avoids tuning per-component loss weights [100], memory-intensive mask states [104], and post-hoc projections that are intractable to marginalize over [55]. We set p ( G ) to the empirical training distribution. Now we describe our method for the remaining conditional distributions.

4.2 Telescoping discrete sampling of L

We parameterized univariate conditionals to autoregressively sample the 3 lattice lengths ( a, b, c ) and 3 angles between them ( α, β, γ ) for conventional unit cells, applying physical constraints to each conditional. Canonicalizing the lattice with the conventional lattice parameters makes the lattice sampling automatically SE(3) invariant. Denoting l = ( a, b, c, α, β, γ ) , the model learned

where p ( l i | l <i , G ) has support over positive values with finite range determined by the data. Under the space group constraints, crystal lattices can be binned into 6 crystal families, each putting unique constraints on the lattice parameters. Space groups 1 to 2 impose no constraints; 3 to 15 require α = γ = 90 ◦ ; 16 to 74 require α = β = γ = 90 ◦ ; 75 to 142 require α = β = γ = 90 ◦ and a = b ; 143 to 194 require a = b , α = β = 90 ◦ , and γ = 120 ◦ ; and 195 to 230 require a = b = c and α = β = γ = 90 ◦ . Furthermore, the crystal lattice must have non-zero volume. To impose these constraints, the model only learns univariate conditionals for the lattice parameters unconstrained by the crystal families, leaving constrained terms in the product of Equation 6 equal to 1. We enforce positive volume by dynamically setting the support of p ( γ | a, b, c, α, β, G ) to satisfy

Besides physical constraints, lattice generation requires the flexibility to learn highly peaked distributions since small perturbations to a crystal lattice can significantly alter materials properties. In BaTiO 3 for example, 0.03Å strain was found to increase the ferroelectric transition temperature by 500 ◦ C and the remnant polarization by 250% [13]. In contrast, the range of conventional lattice lengths in the MP20 dataset is over 100Å. To address this challenge, we chose to discretize the lattice parameters to a resolution of 0.01Å. Naively, this resolution requires a softmax over N l = O (10 4 ) classes per lattice parameter to achieve a 100Å range. We overcame this poor scaling by telescoping the categorical distribution. See Figure 2 for a visual explanation. At a high level, the range of lattice parameters was first binned very coarsely, and a class b 1 was sampled from p ( b 1 | l <i , G ) . Then, the selected class b 1 was further coarsely binned and one of these higher resolution bins was selected from p ( b 2 | b 1 , l <i , G ) . This process was repeated K times to achieve higher levels of resolution as

where l i = b K and b 0 = ∅ . Choosing p ( b k | b k -1 , l <i , G ) to be a categorical distribution over a small number of classes n l ≪ N l achieves (1 /n K l ) = (1 /N l ) resolution with O ( n l K ) ≪O ( N l ) memory. We used K = 2 and n l = 100 in our experiments. We represented conditioning on l i and b k by using min-max normalization and Gaussian random Fourier features [89]. Logits for p ( b k | b k -1 , l <i , G ) were produced with an MLP. Space group constraints were enforced with hard-coded masking of lattice parameters and their logits.

To better align the distributions of partially complete lattices seen during training and inference, we employed noisy teacher forcing during training. Specifically, we minimized the negative log likelihood of the next ground truth lattice parameter conditioned on noisy versions of previous ground truth lattice parameters, where noise was masked to respect space group constraints. Lattice lengths and angles were augmented with uniform random noise with 0.3Å and 5 ◦ ranges, respectively. Noising was not applied at inference time.

4.3 Transformer-based sampling of W , A , and n

Similarly to Cao et al. [10], Kazeev et al. [46], and Kelvinius et al. [47], we leveraged a transformer architecture [92] to autoregressively predict atomic Wyckoff positions W ∈ W n and elemental species A ∈ A n as

The number of atoms in the ASU n is sampled implicitly via p (stop | w ≤ i , a ≤ i , L, G ) . We used attention masking to enforce unique token orderings per crystal during training and inference. Specifically, atoms were ordered lexicographically first by Wyckoff letter and then by atomic number. The transformer input is invariant to permutations of any remaining ties. We additionally set logits of zero-dimensional Wyckoff positions which are already occupied to negative infinity to prevent sampling overlapping atoms at those locations. The transformer was fit with the cross-entropy loss. See section A.4 for details of our transformer architecture. We did not use a positional encoding since the model does not need to distinguish between different orderings of the same atoms.

To aid generalization across the conditioning variables, we initialized embeddings from physical descriptors. Element embeddings were initialized as those introduced by Xie and Grossman [99]. Similarly to the lattice sampler, min-max normalized lattice parameters were embedded with Gaussian random Fourier features. Learnable stop tokens were initialized from Gaussian noise. We represented each of the 230 space groups with 62 initial features and each of the 1731 Wyckoff positions across space groups with 231 initial features. For details and an ablation against the symmetry features used in SymmCD [55], see Sec. A.4.3.

4.4 Space group equivariant diffusion of X

We leveraged the score matching framework [84, 36] to generate atomic fractional coordinates X . Unlike prior works that fit to scores of the translation-invariant Wrapped Normal (WN) distribution [107, 41, 40], we learned scores of a G -invariant Space Group Wrapped Normal (SGWN) distribution:

Theorem 2 The SGWN is G -invariant, i.e., ∀ g = ( R g , v g ) ∈ G, p ( gx t | x 0 ) = p ( x t | x 0 ) . Additionally, the score is space group equivariant, i.e., ∇ x t log p ( R g x t + v g | x 0 ) = R g ∇ x t log p ( x t | x 0 ) (see Sec. A.1 for proof).

Following Jiao et al. [40], we set the noise scale σ t with the exponential scheduler: σ 0 = 0 and σ t = σ 1 ( σ T σ 1 ) t -1 T -1 if t > 0 . As σ t gets large, the SGWN approaches the uniform distribution. Noisy samples x t were created by sampling Gaussian noise ϵ ∼ N (0 , I ) and reparameterizing as x t = x 0 + P x 0 ( σ t ϵ ) where P x 0 ( · ) is an orthogonal projection to the tangent space of x 0 's Wyckoff shape. For the backward process, we constructed the uniform prior p ( x T ) on each Wyckoff position. To do so without redundantly assigning probability mass to symmetrically equivalent points, we placed uniform distributions on the relevant Wyckoff shapes. See Sec. A.3 for details. We used predictor-corrector sampling [85] to sample x 0 .

To learn the space group equivariant scores of the SGWN distribution, we built a space group equivariant graph neural network s θ (see Sec. A.4 for architectural details) and trained it with the following score matching loss:

where λ t = E [ ||∇ x t log p ( x t | x 0 ) || ] -1 was employed for training stability. Approximate computation of λ t and ∇ x t log p ( x t | x 0 ) is described in Sec. A.2.

To save memory when symmetrizing s θ with Equation 3, we note that only | G | / | G x | ≤ | G | / | T L | elements in the set of points { gx | g ∈ G/T L } are unique. Thus we only predicted ˆ f θ ( gx ) on unique points and subsequently indexed into these predictions to compute the full sum over the | G | / | T L | summands in Equation 3. For points x ∈ R 3 in special Wyckoff positions with large stabilizer groups G x , this significantly reduced the number of forward passes through the model.

* Uses fixed templates from the training data. † Values reported by Miller et al. [62].

5 Experiments

Evaluations. Weevaluated SGEquiDiff on two benchmark datasets: MP20 [100], containing 45,231 experimentally known crystals with up to 20 atoms per unit cell, and the more challenging MPTS52 dataset [4], containing 40,476 experimentally known crystals with up to 52 atoms per unit cell. Evaluations were conducted on 10,000 generated crystals. Following Xie et al. [100], we computed structural and compositional validity percentages using heuristics of interatomic distances and charge, respectively. We note that only ∼ 90% of crystals in MP20 pass the composition validity checker based on SMACT [16]. Of the 10,000 generated crystals, we randomly sampled 1,000 determined to be both structurally and compositionally valid. Of these 1,000 crystals, we determined how many were unique and novel with respect to the training data using pymatgen 's StructureMatcher [70] with stol=0.3 , angle_tol=5 , and ltol=0.2 ( U.N. structures ). We also followed SymmCD and calculated the fraction of unique and novel templates ( U.N. templates ), where a template is defined as a space group and multiset of occupied Wyckoff positions. U.N. structures were used to compute (1) distribution distances between ground truth test and generated materials properties, including Wasserstein distances for atomic density ρ and number of unique elements N el as well as Jensen-Shannon divergences (JSD) for space group G and occupied Wyckoff dimensionalities d Wyckoff ; (2) Central Moment Discrepancy (CMD) [106] up to 50 moments between ground truth test and generated CrystalNN structural fingerprints [109]; and (3) structural and compositional diversity as measured by average pairwise L 2 -distances between CrystalNN and Magpie [93] fingerprints, respectively. Average sampling times per batch of 500 crystals were measured on an NVIDIA A40 GPU. We list hyperparameters and training times for SGEquiDiff in Sec. A.7.

Finally, for all U.N. structures out of 1,000 random valid crystals, we conducted structure relaxations with expensive density functional theory (DFT) calculations to assess thermodynamic stability. Aligned with analyses of DFT-calculated energies for experimentally observed crystals [88] and of DFT errors relative to experiment [87, 5], stability was defined as having a predicted energy above the hull less than 0.1 eV/atom with respect to crystals in the Materials Project v2022.8.23 [39]. Further details on the DFT calculations are in Sec. A.8. The fraction of stable materials was multiplied by the fraction of U.N. structures to calculate the stable, unique, and novel (S.U.N.) rate. We qualify that a low energy above the hull as predicted by DFT has been found to be a necessary but insufficient condition for synthesizability [88]; developing physical theories for predictive synthesis is an active area of research in the materials science community [60, 25, 43, 12, 68, 48]. Due to the computational expense of DFT calculations, we only computed S.U.N. rates on a subset of baseline models. While prior works pre-relax generated crystals with foundational machine learning interatomic potentials (MLIPs) before conducting DFT relaxations [62, 86, 107], we opted not to use MLIPs to avoid conflating their biases [17, 53] with our evaluations of the generative models.

Baselines. We compared our model to several prior methods. CDVAE (4.9M parameters) [100] predicts lattices and numbers of atoms per unit cell with a VAE and then samples elements and coordinates with denoising diffusion. DiffCSP (12.3M parameters) [40] jointly diffuses the lattice with the atoms using fractional atom coordinates. MatterGen (44.6M parameters) [107] similarly diffuses the lattice, atom types, and atom positions, but leverages Cartesian atom coordinates. FlowMM [62] extends DiffCSP with the flow matching framework. DiffCSP++ (12.3M parameters) [41] extends DiffCSP by enforcing space group constraints with projected diffusion of the lattice and

Table 2: MPTS-52 dataset. We report metrics of validity for 10,000 sampled crystals; uniqueness and novelty for 1,000 valid crystals; and diversity and distributional distances for valid, unique, and novel crystals. Best result is bolded, second best is underlined. For DiffCSP and DiffCSP++, we used the same diffusion corrector step sizes as for MP-20, and the DiffCSP++ batch size was decreased to 96 to avoid OOMs. CDVAE failed to train due to graph construction errors from isolated atoms. Standard deviations over three training runs are reported in parentheses as errors of rightmost digits.

atom coordinates; however, they do so without space group equivariant scores. SymmCD (60.4M parameters) [55] enforces space group constraints with non-equivariant diffusion of atoms in the asymmetric unit followed by post-hoc projections to the Wyckoff positions.

Results. On MP20, we found SGEquiDiff was competitive on all metrics and achieved the highest S.U.N. rate (Table 1). While MatterGen achieved a comparable S.U.N. rate, we note that it is a significantly larger model (44.6M parameters) than SGEquiDiff (5.5M parameters), sampled crystals ∼ 16 times slower, and yielded worse JSD metrics for space groups and Wyckoff dimensions. Compared to DiffCSP++, SGEquiDiff achieved a slightly higher S.U.N. rate and significantly higher U.N. template rate with ∼ 2.2 times fewer model parameters. Unsurprisingly, SGEquiDiff and other space group-constrained models (DiffCSP++, SymmCD) strongly outperformed baselines on the JSD metric for space groups and Wyckoff dimensions. We also show that by increasing the inferencetime temperature of SGEquiDiff's categorical distributions over space group, Wyckoff position, element, and the stop token, the U.N. rates and composition diversity metric can be significantly improved at the expense of slightly lower validity rates and higher distributional distances to the test set. Inspecting normalized root mean square Cartesian displacements between as-generated and DFT-relaxed structures, we found all models performed similarly, with MatterGen being the best on average (Table 4). Examples of S.U.N. crystals from SGEquiDiff are shown in Fig. 4.

We also benchmarked SGEquiDiff on the more difficult MPTS52 dataset (Table 2), which most other models do not present results on. Using the same hyperparameters as used on MP20, we found that SGEquiDiff was able to scale to more atoms while maintaining a high rate of valid, unique, and novel crystals. SGEquiDiff notably achieved the highest U.N. template rate, composition validity, and Wasserstein distances for density and number of elements. Interestingly, all space group-constrained models significantly outperformed DiffCSP in the validity metrics, highlighting the advantage of enforcing nontrivial space group symmetries as an inductive bias.

Ablations. To isolate effects of various components of SGEquiDiff, we performed ablations on the MP-20 dataset. First, we replaced our autoregressive lattice sampler with the DDPM-based [33] sampling of E(3)-invariant lattice matrix representations proposed in DiffCSP++ and used in SymmCD (Table 6). We used the same cosine noise scheduler as DiffCSP++, jointly diffusing lattices and atom coordinates. We term this variant of SGEquiDiff as +LDiff . We observed that +LDiff improved the Wasserstein distance for number of elements (possibly because conditioning of the Wyckoff-Element transformer on lattices was removed) and CMD of structural fingerprints. However, our autoregressive lattice sampler yielded better validity, JSD of Wyckoff dimensionalities, and S.U.N. rate. We also observed that in roughly 1% of samples, the +LDiff model produced lattice matrices with negative determinant, resulting in a spurious inversion that does not preserve the symmetry of the 22 chiral space groups. We show examples of S.U.N. crystals from +LDiff in Fig. 5

We also replaced our space group and Wyckoff position features with those introduced in SymmCD [55] (Table 5). We found our featurizations yielded better structure validity, composition validity, Wasserstein distance for atomic density, structural divergence, and validation log-likelihoods of ground truth Wyckoff positions. In contrast, SymmCD features yielded more diverse samples, evidenced by higher U.N. rates and composition diversity.

Since space group symmetrization via Eq. 3 can be applied to any periodic translation invariant model, we replaced our FAENet-inspired [20] graph neural network (GNN) with the EGNN-inspired [80] model ( CSPNet ) used by other crystal generative models [40, 41, 62]. Under a fixed number of trainable parameters (2.2M), we found our GNN yielded better metrics for structural validity, composition validity, and CMD of structural fingerprints (Table 5).

Crystal structure prediction (CSP). Consistent with previous works [40, 41, 62], we evaluated SGEquiDiff on its ability to predict crystals from a test set using elemental composition as input. Performance was assessed by calculating match rate (MR) and root mean squared error (RMSE) of atomic fractional coordinates based on a single generated crystal per composition. RMSE was normalized by the average free length per atom. Matching was done with pymatgen StructureMatcher [70] using stol=0.5 , angle_tol=10 , and ltol=0.3 . To adapt SGEquiDiff to this task, we combined the pretrained +LDiff variant of SGEquiDiff with the metric learning-based approach proposed in DiffCSP++ ( CSPML ) for selecting templates from training data and performing element substitutions to satisfy the given composition. Without CSP-specific training (unlike baselines), SGEquiDiff+LDiff achieved competitive performance (Table 3). We note SGEquiDiff could alternatively be adapted for CSP by reordering the factorization in Eq. 5 or by supplying composition as conditioning during training. While more challenging, learning instead of strictly enforcing composition may be desirable since an arbitrary composition is not guaranteed to host a stable crystal (accordingly, experimentally observed compositions can be weighted averages of multiple phases).

We highlight that the current CSP task is slightly misaligned with real-world application. Many existing models assume the number of atoms per unit cell will be known beforehand, which is not necessarily true in practice. Additionally, a single composition can host multiple stable crystal structures depending on experimental conditions. For example, iron can exist as a body-centered cubic (2 atoms per conventional cell), face-centered cubic (4 atoms per conventional cell), or at high pressures, hexagonal close-packed structure (6 atoms per conventional cell). Furthermore, materials discovery campaigns are often aimed at composition spaces with sparse training data, violating the i.i.d. assumption of the MP20 dataset splits [100].

6 Conclusion

In this paper, we proposed SGEquiDiff to generate crystals with space group invariant likelihoods by leveraging equivariant diffusion. We showed that space group equivariant flows automatically live on the manifolds of the Wyckoff positions. Significantly, SGEquiDiff achieved state-of-the-art generation rates of stable, unique, and novel crystals as evaluated by rigorous quantum mechanical simulations. We proposed an efficient autoregressive method for sampling space group-constrained lattices. We also showed that tuning the inference-time temperature for autoregressive sampling of discrete crystal attributes provides control over the novelty of generated crystals.

Limitations and future directions. Due to computational constraints, we were only able to calculate S.U.N. rates on 1,000 valid crystals per model, adding unknown variance to reported metrics. SGEquiDiff's training data was pre-processed to assign space groups to crystals with nonzero tolerances over atom positions and lattice angles; it is possible that some materials were incorrectly symmetrized to higher symmetry space groups than representative of reality. SGEquiDiff also does not provide guidance for how to synthesize generated materials. Other symmetry groups relevant to crystals were ignored, including magnetic space groups [94], spin space groups [98], and layer groups [23]. Practically relevant deviations of real materials from perfect crystals were not considered, including defects, compositional disorder, surface effects, and interfacial effects. Future work might include application to a broader set of tasks such as those derived from nonscalar properties [102, 59, 95] or materials spectroscopy [56]. Furthermore, training non-equivariant models with space group invariant target distributions and/or applying inference-time space group symmetrization may enable a broader diversity of model architectures to generate high symmetry crystals.

Broader impacts. This work has the potential to accelerate discovery of advanced materials for energy, electronics, optics, catalysis, aerospace, and more. Possible negative impacts include development of materials that are toxic, require energy-intensive processing, lead to depletion of raw minerals, or are used for military applications.

Table 3: Crystal structure prediction. Leveraging the template selection and element substitution approach developed in DiffCSP++ [41], we evaluated SGEquiDiff+LDiff on the crystal structure prediction task introduced by DiffCSP [40].

* Uses ground truth number of atoms per unit cell.

Acknowledgments and Disclosure of Funding

We acknowledge helpful discussions with Cindy Zhang, Barry Bradlyn, Alex M. Ganose, and Eric Toberer. This research used the Delta advanced computing and data resource (award OAC 2005572) and the Illinois Campus Cluster, operated by the Illinois Campus Cluster Program in conjunction with the National Center for Supercomputing Applications. The research was supported by the National Science Foundation under Grant Nos. DGE 21-46756 and 2118201.

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A Appendix

Symmetry reduction

Space group orbiting

an

B

A.1 Proofs

Theorem 1. Let G be a space group, x ∈ R 3 be a point residing in a Wyckoff position with stabilizer group G x , and f : R 3 → R 3 be a space group equivariant function. Then, for arbitrary constant c ∈ R and space group element g ∈ G x ,

or equivalently, G x ⊆ G x + cf ( x ) .

Proof. Let { R g x , v g x } = g x ∈ G x be a stabilizer group element where R g x ∈ O (3) is a point group operation, v g x ∈ [0 , 1) 3 is a fractional lattice translation, and the action of g x is R g x ( · ) + v g x . The definition of a stabilizer group requires that

The definition of space group equivariance requires that for any R ∈ G ,

i.e., f ( x ) is also stabilized by G x . Finally, we have

Theorem 2. The score of the SGWN distribution is space group equivariant, i.e., given g = { R,v } ∈ G ,

Thus, we have

Combining equations 10 and 11, we see

Proof. The score of the SGWN is given as

Now, we transform the input with g = { R,v } .

We will prove the equivariance of the numerator, which we denote as ϕ ( Rx t + v | x 0 ) . For the denominator, it can be similarly proved that the SGWN is G -invariant, i.e., p ( Rx t + v | x 0 ) = p ( x t | x 0 ) .

A.2 Implementation details

We approximated the SGWN distribution with a truncated sum over lattice translations as

where m ∈ Z + . We have dropped the dependence of q on G and σ t from the notation for clarity. The space group-equivariant score of q ( x t | x 0 ) is given as

For each Wyckoff position w , we pre-computed approximate values of λ t = E x t ∼ p ( x t | x 0 ) ,x 0 ∼P w (Uniform( · )) [ ||∇ x t log p ( x t | x 0 ) || ] -1 with Monte Carlo sampling. We denote P w (Uniform( · )) as the uniform distribution on the Wyckoff shapes (see Sec. 4.1 and A.3) corresponding to Wyckoff position w . First, we sampled s points { x i 0 } s i =1 from P w (Uniform( · )) . Then, for each sample x i 0 , we sampled Gaussian noise ϵ ∼ N (0 , σ 2 t ) and reparameterized as x i t = x i 0 + P x i 0 ( ϵ ) . The approximation of λ t for Wyckoff position w in space group G was then computed as

In our experiments, we set s = 2500 , m = 3 , σ 1 = 0 . 002 , σ T = 0 . 5 , and T = 1000 .

A.3 Uniform priors on Wyckoff positions

To sample from the uniform distribution in a given Wyckoff position, we sampled uniformly on the Wyckoff shapes in the ASU. The likelihoods of these uniform priors are simply given as 1 for 0D Wyckoff positions, and the reciprocal of the total length, area, and volume of the relevant Wyckoff shapes for the 1-, 2-, and 3-D Wyckoff positions, respectively. For a 0D Wyckoff position, coordinates are fully determined and 'sampling' simply returns a point. For a 1D Wyckoff position, we sampled a line segment with probability proportional to its length, and then sampled uniformly on the line segment. For a 2D Wyckoff position, we pre-computed Delaunay triangulations of the polygonal facets belonging to the Wyckoff position, sampled a triangle proportionally to its area, and then sampled from a uniform Dirichlet distribution in the triangle. For a 3D Wyckoff position, we used rejection sampling from a bounding box around the ASU.

A.4 Architecture

A.4.1 Wyckoff-Element Transformer

Our encoder-decoder Transformer architecture can be summarized as follows:

where e SG , e W , and e A are predicted embeddings of the crystal's space group, occupied Wyckoff positions, and atomic elements, respectively; l max is the number of encoder layers; m causal is a causal attention mask enforcing atom orderings; e all W , e stop , and e all A are predicted embeddings of all sampleable Wyckoff positions, the stop token, and all sampleable elements, respectively; m W and m A are attention masks enforcing lexicographic atom orderings; p W, stop and p A are the model probabilities over Wyckoff positions, the stop token, and elements; and EncoderLayer(x , m) is a module summarized by the following:

A.4.2 Graph Neural Network

Our graph neural network architecture can be summarized as follows. Our GNN was a modified version of FAENet [20], replacing sum pooling with variance-preserving aggregation [81] and removing frame averaging since we trivially achieve SE(3) invariance by canonicalizing crystals with the ASU representation. We constructed fully connected atom graphs G wherein each atom in the conventional unit cell was connected to every other atom in the conventional unit cell by their minimum-length edge under periodic boundary conditions. If ties existed, all corresponding edges were included. At initialization, to build features which were invariant to lattice translations, we sampled frequencies ν k ∈ Z 3 ∩ [ -512 , 512] 3 \ 0 without replacement with probabilities drawn from

a discretized standard normal distribution. The architecture is summarized as follows:

where t is the diffusion timestep; x i ∈ [0 , 1) 3 is the fractional coordinate of atom i ; d ij is the minimum-image pairwise Cartesian distance between atoms i and j , i.e., for lattice matrix L ∈ R 3 × 3 ,

h l i is the node feature of the i th atom after l rounds of message passing; l ∈ R 6 are the conventional lattice parameters; l max is the number of message passing layers; a is a learnable scalar initialized to zero; e a i is the embedding of atom i 's element; and ˆ f i ∈ R 3 is the non-symmetrized score prediction (see Eq. 3) for atom i . For the LDiff variant of our model, we additionally had

where ϵ L ∈ R 6 is the predicted noise for the lattice k -vector (see [41]).

We see that ˆ f i is invariant to lattice translations since the model's spatial inputs are the minimumimage distances d ij under periodic lattice translations and translation invariant plane wave embeddings of relative and absolute atomic fractional coordinates ( x j -x i and x i , respectively). We symmetrized ˆ f i with space group equivariance according to Equation 3.

A.4.3 Space group and Wyckoff position features

To represent a space group, we created one-hot features indicating the 6 lattice centering types, 6 crystal families, 32 crystallographic point groups, whether the space group is chiral, and whether the space group is centrosymmetric. Since this representation ignores fractional translations from glide and screw symmetries, we counted the maximum number of collisions between space group representations (16), arbitrarily indexed colliding space groups, and created additional one-hot features to avoid collisions. This yielded 6 + 6 + 32 + 1 + 1 + 16 = 62 features. While the last 16 features are not physically motivated, they give the model the capacity to differentiate space groups and learn their relationships. These features are more abstracted than those of SymmCD [55], which used binary features indicating the 26 space group symmetry operations, 15 symmetry axes, and 7 lattice systems, totaling 7 + 26 × 15 = 397 features.

To featurize a Wyckoff position, we used our 62 space group features; one-hot features for the 100 unique site symmetry symbols across space groups, the 4 numbers of degrees of freedom in atom coordinates (0, 1, 2, or 3), and the 17 unique Wyckoff multiplicities; and 48 Fourier features of Wyckoff shapes' midpoints and vertices (see Sec 4.1). This yielded 62 + 100 + 4 + 17 + 48 = 231 Wyckoff position features. To compute Fourier features ν ∈ R 48 , we collected the midpoint and vertices of each Wyckoff shape belonging to the Wyckoff position. We used these points

{ p i ∈ R 3 × 1 } n w i =1 to compute the following: f = 2 π · linspace(start = 1 , end = 32 , steps = 8) ∈ R 1 × 8 ν i = flatten( p i f ) ∈ R 24 n

Weconfirmed that this representation yielded no collisions between featurizations of different Wyckoff positions. Our representation differs from that of SymmCD, which represents each Wyckoff position by one-hot features of the 15 symmetry axes and 13 symmetry operations (totaling 15 × 13 = 195 features) corresponding to the Wyckoff position's stabilizer group.

A.5 Ablation results

Table 5: SGEquiDiff Ablations on MP-20. We replaced our space group and Wyckoff position features with those of SymmCD [55] ( +SymmCD features ) and our denoising graph neural network with that of DiffCSP [40], DiffCSP++ [41], and FlowMM [62] ( +CSPNet ). For fair comparison in the latter experiment, we fixed the number of message passing steps (5), the number of Gaussian edge frequencies (96), and the number of trainable parameters (2.2M) by reducing CSPNet's hidden dimension from 512 to 224. Standard deviations over three training runs are reported in parentheses as errors of rightmost digits.

* For efficiency, stability calculations were conducted on 100 random valid, unique, and novel structures.

A.6 Examples of S.U.N. crystals

Er2Fe2Co2

Nd2Cu2As4

Y2AgIr

NaNdTl2

Ca4Sn4Pt4

La8Ce4Ir4

TlPPt5

Ta2Cr4

BeCoSi

Li2ThBi2

Y2Tm2In2

Ti5Nb3Mo

EuAs2Pt2

LuPuRu2

YbAg2P2

A.7 Training and hyperparameters

Our code was written with PyTorch [72] and PyTorch Geometric [21]. The model was trained with the Adam optimizer [50] on a single NVIDIA A40 GPU. Training SGEquiDiff took approximately 10 hours on MP20 and 12 hours on MPTS52. Data splits were the same as provided by Xie et al. [100] and Baird et al. [4]. Hyperparameters were tuned manually.

We adopted a modular architecture, simultaneously but independently training the space group sampler, lattice sampler, Wyckoff-Element transformer, and denoising graph neural network. Early stopping based on validation performance was applied to each module separately, eliminating the need for gradient balancing between modules.

A.8 Density Functional Theory Calculations

Total energy calculations were performed with density functional theory (DFT) using the Vienna Ab Initio Simulation Package (V ASP) [54] with Projector Augmented Wave (PAW) pseudopotentials [8]. The Perdew-Burke-Ernzerhof (PBE) [73] exchange-correlation functional was used for structural relaxations. All input parameters, plane wave cutoffs, convergence criteria, and k-point densities were determined by pymatgen MPRelaxSet [69].

Relaxations which don't converge with default settings from VASP (version 6.4) and MPRelaxSet do not necessarily indicate issues with generated crystal structures. Specifically, we encountered and attempted to resolve three failure modes during relaxations: errors related to symmetry-finding, issues converging energy and electronic charge density within DFT's self-consistent field (SCF) loops, and issues converging atomic positions. To resolve symmetry-finding errors (reported as SGRCON or INVGRP in VASP's output files), we tightened the tolerance of the symmetry detection algorithm by setting SYMPREC=1E-6 from the default 1E-5 . To resolve SCF convergence issues, we changed the electronic minimization algorithm to the slower but more stable ALGO=NORMAL from the default ALGO=FAST . If SCF-related issues persisted, we additionally set AMIX=0.2 from the default 0.4 , which results in slower but more stable convergence of the charge density, and increased the number of electronic optimization steps with NELM=200 (default 100 ). Finally, to resolve issues related to convergence of atomic positions, we changed the structure optimization algorithm to the slower but more precise IBRION=1 (default 2 ). We emphasize that these changes in VASP parameters will change the relaxation trajectory but not change the minimal energy structure or loosen convergence criteria relative to MPRelaxSet 's default settings.

For direct comparison with Materials Project (MP) data, we applied the MP's energy correction scheme for anions and mixing GGA/GGA+ U calculations [37, 38]. Crystal stabilities were determined through convex hull analyses using all competing phases available in MP v2022.8.23 [39].

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposed a novel methodology to leverage the symmetry constraints in materials to generate more accurate and symmetric crystals. In particular, the contributions include a novel autoregressive sampling of lattice parameters, transformer-based sampling of wyckoff positions and atoms, and finally, a space group equivariant diffusion of fractional coordinates. The authors benchmark their proposed model with relevant baselines in the MP20 and MPTS52 datasets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n\n- The paper is well-motivated, clearly structured, and effectively builds on prior work in the literature.\n- The authors make thoughtful design choices that are aligned with the structure of the underlying data. Notably, the design of the denoising network, avoiding atom overlaps in the transformer-based generative model, the use of telescoping discrete sampling of lattice parameters guided by the MP20 dataset distribution, and providing a compelling rationale for avoiding uniform distribution over atom positions.\n- The observation that equivariant vector fields naturally reside in the tangent spaces of Wyckoff positions is a valuable insight that could inform the future design of space group equivariant architectures.\n- The proposed model achieves competitive performance on the MP20 dataset and outperforms existing methods on the more challenging MPTS52 benchmark, demonstrating its capacity to accurately generate larger, more complex materials.\n\n\n\n\n\n\n\n\n**Weaknesses and Questions**:\n- Line 205: The paper states \"...p(G) as a simple vector of logits and trained it with the cross-entropy loss...,\" but it is unclear what the loss is computed between. Why not sample directly from the empirical distribution of space groups in the training dataset during generation?\n- While the factorization in Equation 5 helps structure the generative process, the challenges or limitations of jointly modelling all components of $M$ are not discussed. What specific difficulties would arise in learning a joint distribution?\n- The sampling of $n$ using the transformer in Section 4.3 is not described and should be detailed to ensure clarity and reproducibility.\n- The model imposes a specific ordering on atoms, even though materials are fundamentally unordered sets. Would it be more scientifically appropriate to avoid this imposed ordering altogether?\n- The Wyckoff position embeddings are not clearly explained. Specifically, how were the 62 features for 230 space groups and the 231-dimensional embeddings for 1731 Wyckoff positions derived? Additionally, is 1731 the total number of Wyckoff positions across all space groups?\n- Using the symmetry representation from SymmCD could be a viable alternative to reduce redundancy and minimize the token space. Was any ablation performed on token space design choices that led to the current setup in Section 4.3?\n- Is adding the LDiff functionally equivalent to using the invariant representation of lattice matrices introduced in DiffCSP++?\n- **Questions on Table 1**: \n  - What does $T$ represent in the table?\n  - What is the value of $T$ in the \"SGEquiDiff\" row?\n  - There appears to be an inverse relationship between composition validity and the unique and novel (U.N.) rate. Is there any explanation for this?\n  - It is incorrectly stated that SymmCD uses fixed templates.\n  - Please report the variance across multiple random seeds for SGEquiDiff (the best model per the S.U.N rate).\n  - Why is the Wasserstein distance for atomic density significantly higher for SGEquiDiff compared to the baselines?\n  - The table caption would benefit from being more descriptive and including relevant contextual details.\n- Since S.U.N. rates are not reported for all baselines, highlighting the percentage improvement over the \"next best\" model is misleading. Moreover, given the acknowledged variability in S.U.N. rates (as noted in the limitations), the Lines 334–335 statement is not well-supported.\n- Beyond computational efficiency, are there other advantages to using the asymmetric unit (ASU) over the conventional unit cell? Although more computationally demanding, the conventional cell may capture more structural information.\n\n\nI am willing to improve the scores if my questions are adequately addressed during the rebuttal.\n\nQUESTIONS:\nPlease refer to the **Strengths And Weaknesses** section for relevant questions alongside weaknesses.\n\nLIMITATIONS:\nYes. Please refer to the **Strengths And Weaknesses** section for more details.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-15"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces SGEquiDiff, a Space Group Equivariant Diffusion model for crystal structure generation. The method enforces space group constraints and provides explicit space group–invariant likelihoods. Experiments on standard crystal datasets demonstrate solid performance.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n* **S1** : The proposed model allows hard constraints to be incorporated through a space group–invariant likelihood, without requiring local structure relaxation.\n* **S2** : The paper presents a carefully constructed and comprehensive review of related work, clearly situating the proposed approach within the existing literature.\n* **S3** : The proposed method leverages the domain’s symmetry as an inductive bias. Given this, the improvement in S.U.N. rate is expected and not surprising.\n\n\n## Weaknesses\n\nI have several concerns regarding the experimental evaluation:\n\n* **W1** : The experiments are conducted on only two datasets, whereas related work such as [1] and [2] typically report results on at least three datasets (e.g., Perov-5, MP20, and Carbon-24). One of the selected datasets—though challenging—lacks thorough baseline comparisons. While the authors note that previous models have not reported results on this dataset, I would expect them to run those baselines using publicly available code to enable a fairer and more comprehensive comparison. Relying on results from just one baseline weakens the experimental support for the proposed method.\n* **W2** : Although the proposed method achieves the best performance on the S.U.N. metric, it underperforms on most of the other evaluation metrics compared to baselines.\n* **W3** : It is unclear how well the proposed model performs on the Crystal Structure Prediction (CSP) task—a central objective in crystal generation—unlike other models such as [1], [2], and [3], which explicitly evaluate this aspect.\n\n\n[1] Space Group Constrained Crystal Generation\n\n[2] Crystal Structure Prediction by Joint Equivariant Diffusion\n\n[3] FlowMM: Generating Materials with Riemannian Flow Matching\n\nQUESTIONS:\nI appreciate the motivation and design of the proposed method and believe I understand the core ideas. Given that the method explicitly enforces symmetry constraints, I would expect it to outperform approaches that do not incorporate such inductive biases.\n\nHowever, I find the experimental section lacking in several key areas: the dataset diversity is limited, comparisons with strong baselines are insufficient, and the model's potential for Crystal Structure Prediction—an important application—is not clearly demonstrated. Moreover, the performance gains are only evident in the S.U.N. metric, while no clear advantage is shown in other evaluation metrics. Strengthening these aspects would significantly enhance the credibility and impact of the work.\n\n## Questions for the authors:\n\n* **Q1** : Why does SGEquiDiff achieve better performance on the S.U.N. metric compared to baselines, yet underperform on most other evaluation metrics? How should the importance of the S.U.N. metric be interpreted relative to the others?\n\n* **Q2** : Can the proposed method be applied to Crystal Structure Prediction? If so, how does it perform compared to existing baselines on this task?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a new generative model for de-novo crystalline materials generation by modelling the most asymmetric unit of the conventional unit cell, rather than the primitive unit cell as done in most recent generative models. The generative process is split in an autoregressive way: first, they compute the Wyckoff \"description\" of the crystal (space group, Wyckoff position, lattice parameters) and atom type and then they run diffusion on the fractional coordinates (similar to crystal-structure prediction). In particular, they introduce the space-group wrapped normal distribution which is used to train the score network, which ensures that the resulting vector field is equivariant to the crystal’s space group symmetries. Results on MP-20 shows that they are able to improve S.U.N rate metric compared to Mattergen computed using DFT on 100 samples.\n\nSTRENGTHS AND WEAKNESSES:\n1. The idea of modelling the most asymmetric unit cell and the symmetry transformations (given by space group and Wyckoff positions) is an exciting research direction. Given the many symmetries constraints inherent in crystalline materials, restricting fractional coordinates to respect these constraints rather than modeling them over a continuum makes a lot of sense. In this context, the autoregressive approach considered in the paper is a novel contribution. Moreover, the ability to generate a Wyckoff description that rigorously obeys symmetry rules is a particularly strong result.\n\n2. The paper can benefit from more clarity as some parts were difficult to fully understand.Providing more details about the training procedures of the four different components of the model would likely help. I will highlight some observations here and add some additional question in the Question section\n\t- the notation used for functions and group actions is sometimes overlapping, which can be misleading at times\n\t- Line 185 to 188: it's not completely clear what the message is there. Is the main point just that space group equivariant vector field does not make any transition between different Wyckoff position dimensions (meaning that if the Wyckoff position is 1D the vector field will stay on that line), whereas the usual SDE approach does not respect this due to the Brownian motion? The use of the term *motivating* feels somewhat strange in this context.\n\t- Line 237 to 240: it's not completely clear how the model used to sample the lattice parameters is trained. Once one lattice parameter is generated, are you using a noisy version of it as conditioning to generate the others? Clarifying that process would be helpful\n\t- The table captions could be expanded to be more self-contained, for example by being be more specific about what is presented in them.\n\nQUESTIONS:\n1. I am bit confused about eq. 10 and eq. 11 in appendix. The two equations means that the function $f(x)$ is at the same time invariant and equivariant to $R$. But then this should hold just for trivial $R$, like if $R$ is the identity, right? Am I missing something here? Or is it just because we are considering group actions from the group stabilizer (which leaves the input unchanged and therefore the output is fixed) that this holds? Clarifying this would help avoid confusion.\n2. I'm curious about the decision to learn $p(G)$ (the distribution over the space groups), rather than setting it to the empirical distribution of the training set. Doesn't the optimization of the logits lead to the empirical distribution? \n3. I think it will be interesting and useful for the community to understand where the most gains in performance comes from. Although there are some differences in the input used across all the different generative models, many of them uses the same EGNN backbone architecture (from Satorras et al.). Is the FaeNet specifically needed to enforce some of the constraints you want to impose? Have you try to compare the performance using the same backbone architecture as the other baselines, to isolate the benefit of the different architecture?\n4. Space groups define specific constraints to the lattice angles. How do you enforce those constraints during sampling? Are they hard-coded?\n5. Is the featurization of the space group different from the one used in SymmCD? If so, I think it would be helpful to explicitly state that and highlight differences.\n6. The EquiCSP paper highlights that the score used by DiffCSP is not periodic translation invariant, meaning that the score for a position $x$ and a periodic translation of it are different. Does your score formulation address this issue or does it suffer still from this mismatch?\n\nMinor:\n\n1. Missing the log in the score matching loss $L_X$ and also in line 280\n\nI am willing to increase my score if my doubts are solved.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work proposes a diffusion model for crystals with explicit space-group equivariance. The authors do so by using a key-fact that space-group wrapped normal (SGWN) distributions are invariant with respect. A factorization of the generative distribution is proposed. Important physical constraints are enforced. For example, lattice cell volume positivity is imposed by updating the support of the conditional generative distribution for the lattice cell parameter $\\gamma$. An empirical evaluation on MP20 is carried out, and comparisons with other crystal generative models are made. A metric is proposed, involving the proportion of stable, unique, and novel (SUN) structures generated by the model. It is show that the proposed model improves upon others using this metric.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nSpace-groups are an important topic in the theory of crystals. Equivariance is an important topic in modern machine-learning. This work addresses the intersection of these two topics in a timely manner.\n\nGood concise intro on the immense size of crystal space. Good motivation with Neumann's principle, a classic reference in materials science.\n\n**Weaknesses**\n\nAs noted by the authors, one can relax the outputs of non-\"space group equivariant\" generative models to get high-symmetry structures. My interpretation of this is that the proposed modelling approach is not strictly necessary to achieve their stated outcomes.\n\nThe empirical advantage is not so clear. Only $N=100$ DFT relaxations were used in assessing their SUN metric. Therefore I have some doubts about the strength of the findings.\n\n\n**Quality**\n\nOverall the quality of this paper is high. I have confidence that the technical aspects of the paper are correct.\n\n\n**Clarity**\n\nThe writing is clear. Theoretical claims appear correct. Claims about improved empirical performance appear supported.\n\n\n\n\n**Significance**\n\nThis work could be impactful as it addresses the relevant problem of adding space groups to equivariant generative models for crystal generation.\n\n**Originality**\n\nThe work strikes me as original, well-motivated, and well-carried out.\n\nQUESTIONS:\nSee weaknesses. I am willing to place greater confidence in these findings if the number of structure used for the SUN metric was increased, e.g. by another order of magnitude.\n\nLIMITATIONS:\nI thank the authors for discussing the weaknesses/limitations of their work in some parts of their paper.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}