Space Group Equivariant Crystal Diffusion
Authors: Rees Chang
Angela Pak
Alex Guerra
Ni Zhan
Nick Richardson
Elif Ertekin
Ryan P Adams
Venue: NeurIPS 2025 poster ()
Keywords: crystals
generative
diffusion
space group
equivariant
wyckoff
asymmetric unit
transformer
score matching
Primary area: machine_learning_for_sciences
TL;DR: We enforced space group symmetry during denoising diffusion of crystals with a space group equivariant score model.

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
Rees Chang' Angela Pak', Alex Guerra², Ni Zhan', Nick Richardson 2
Elif Ertekin', Ryan P. Adams
Deparrment Materials Science Engineering, University of Illinois at Urbana-Champaign Department of Computer Science, Princeton University
Materials Research Laboratory, University of Illinois at Urbana-Champaign
Abstract
Accelerating inverse design of crystalline materials with generative models has significant implications for range of technologies. Unlike other atomic systems, 3D erystals are invariant to discrete groups of isometries called the space groups. Crucially, these space group symmetries are known to heavily influence materials propertics. We propose SGEquiDiff, a crystal generative model which naturally handles space group constraints with space group invariant likelihoods. SGEquiD- consists of an SE(3)-invariant, telescoping discrete sampler of crystal lattices; transformer-bast based autoregressive sampling of Wyckoff po- sitions, 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. SGEquilDiff achieves state-of-the-arr performance on standard benchmark datasets as assessed by quantitative proxy metrics and quantum mechanical calculations. Our code is available at httpps: https:///ttthbb htw.hhu...oomeeeeeoemeeeeeeeeeee rtt.:reeesscceeeee ctrreessss/equuii.f...
Introduction
Crystals comprise critical technologies like batteries (66) topological materials [90], electronic devices (96), photovoltaics (28] and more. Materials scientists have catalogued O(10²) crystals
experimentally 77] and (110)) in silico with density functional theory (DFT) simulation [1] 39 79].
In contrast, the number of stable crystalline materials with five elements or less is estimated to exceed 1003, and even higher-order compositions offer a promising path to rapidly explore the vast space of crystals (100, 40 62 10].
common in real materials [6] Generative models
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 point into itself is called a stabilizer group. When a set of points in 3 have conjugate stabilizer groups, the set is called Wyckoff position. Importantly, as shown in Figure [ Wyckoff positions can have zero volume, comprising points, lines, planes. These zero volume are referred as special Wyckoff positions. We give more formal treatment of space groups and Wyckoff positions in Sec.
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 's principle states that all crystal properties share the same invariances as the crystal itself 655 Thus even slightly perturbing atoms out of special Wyckoff positions will reduce the crystal's space group symmetry and subsequently cause significant (even discontinuous) changes to its macroscopic properties [14 11 82 9.. Secondly, we
"Correspondence reesuc20illinoiss
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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14
20
Percent of crystals
12
Pereent of crystals
60
10
30
40
30
20
50
200
Space group number
Wyckoff dimension
Figure (a) The asymmetric unit (II) and special Wyckoff positions labeled by letter in the conventional unit cell of space group 10. (b-c) Histograms of occupied space groups and Wyckoff dimensionalities by crystals in the MP20 [100 39 7 training dataset. Space groups and Wyckoff positions were determined by the Saccerroupnnayyeer module in pymatgen [70 using toler- ances of 0.1 Á and 5 These tolerances help account for the moderate convergence criteria of the Materials Project DFT relaxations.
show empirically in Figure ! that known materials are usually invariant to high symmetry space groups with atoms zero-vol vero-volume Wyckoff positions. An intuitive explanation for this phenomenon is that atoms of the same species in given crystal are most stable in 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 be relaxed into high symmetry positions with DFT machine learned force fields. However, besides the apparent difficulty of generating atoms sufficiently close 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, propose SGEquiDiff, which enforces space group constraints during generation and yields explicit space group invariant model likelihoods. SGEquiDiff combines SE(3)-invariant dis- cretized lattice sampling: permutation invarriant,, transformer- based generation of occupied Wyckoff positions and elements; and space group equivariant diffusion of atomic coordinates. For the latter, built 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 against scores from 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 [4] or a discontinuous generation process [55] 78
SGEquiDiff proposes an explicit-likelibe likelihooi autoregressive alternative to diffusion-based sampling of space group- constraind 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 [10 19]. commonly used for semiconductors and magnetic devices.
We verified the efficacy of model on standard crystal datasets with proxy metrics and rigorous DFT calculations.
Related Work
Early crystal generative models represented crystals as voxelized images 667
34 or padded tensors
of 3D coordinates 49 75 to train variational autoencoders (VAE) [18 generative adversarial
networks (66). Several works have enforced the SF(3) and periodic translational invariance of crystals by leveraging graph neural networks. One popular approach is use diffusion models [83 84
33

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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 [2]; the stochastic
interpolant interpolants framework [I] expanding the model design space 35; and the Bayesian Flow Networks
framework [27 977 which iteratively refines model parameters instead of samples. Other works
have aimed to learn the SE(3) and periodic invariances through data augmentations [104] [1 or
data canonicalization [0I]] with large language models (LLMs) [O1] or with diffusion 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] [03] 77T. We view
many of these existing works 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- conditoond generation without hard constraints. The graph diffusion model MatterGen [07] 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 generous SpaceGroupAnalyzerr tolerance of 0.2Á.
91 70
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 [08] trained 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 [04 considered space group constraints for the task of distribution matching under the GelowNet framework [6] but did not address how to sample atom coordinates. DiffCSP++ [41 trained 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, DifiCSP++ 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. Wyckoff types and Wyckoff positions but relies on DiffCSP++ or MLIPPs to refine atoms from uniform random coordinates in the Wyckoff positions. We note that MI.IPs 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). WyckofIDiff [47] generates Wyckoff positions and elements with D3PM-based diffusion 33 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.
45 also autoregressively predicts atom
Preliminaries
Space groups Formally, a space group G E G. where G denotes the set of 230 space groups, is group of isometries that tiles R In particular, G is generated by an infinite subgroup of discrete lattice translations Tr- = {n, L1, n22L2, na La)n; E E, L, E R3j as well as a collection of other symmetry operations g(-) (R₂()) + ug E 0(3), U E R$) E G, where Ry is point group operation (rotation, reflection, or identity) and Ug is a translation.
Wyckoff positions Given a space group and point x E R3, the stabilizer group G2 = [و r) C G (also called the site symmetry group in materials science) is the finite subgroup of that leaves x invariant. A Wyckoff position is then defined as the set of points with conjugate stabilizer groups, i.., (a' /3 g E G Gz = gGZzg" -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 change
3

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of basis. By convention, when x: is described with respect to the lattice basis {L1, Ly, Ls}. the size of the orbit of in the unit cell, kgælg E G, gz È [0, 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 increasing multiplicity.
Unit cells The infinite translational periodicity of a crystal can be represented with parallelipiped T called the unit cell. The unit cell reduces infinite crystals by removing redundancy induced by Tr, the group of discrete lattice translations. In this way, crystals are represented by the tuple M = (A, X, L.), where A = (a', .... d'y) E A & are the atom types, A is the set of all chemical elements, and N is the number of atoms in the unit cell; X
{(2{ .... )aa [T} are the Cartesian atom coordinates; and L = (L1, L2, La) E 123x3 are the unit cell basis vectors. Given M, the infinite periodic structure can be reconstructed by applying the actions of Tr {(a!, 11 + n22 + nl Lyy L Ing Z}. Alternatively, the atom coordinates can be given in the lattice basis instead of the Cartesian basis. Coordinates in the lattice basis commonly referred to fractional coordinates. In fractional N coordinates, the infinite crystal is reconstructed as {(4,, xt + 0161 + ng22 + ngCa), j=1 Ing E Z). For the rest of this paper, 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 with a minimum-volune 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 convex polytope IT E R3, the asymmetric unit (ASU), which maximally reduces infinite crystals by removing all redundancies consider atoms in the ASU with induced by the space group G 2 Ty, Under this formulation,
fractional coordinates X = (,1,,-., .... In) E II} E Raxs atom types A = (aI,..,,,,, E A", and Wyckoff positions W { Wn)] [w; Gaa} J E W" where
N. The infinite periodic
structure of a crystal can be reconstructed by applying the actions of G to II, i.e.,
(1yz)) | 2 E II, 9. G/w,, ie nI)....n)..
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
et al. [30] to ensure that the ASUs exact, i... that II tiles R without overlaps at the boundaries ÒII.
construct samples Xg ~ Pt=0 from a noisy prior XT P:=T by evolving xy through time with a learned denoising process. The training process generally involves iteratively adding noise to data samples Xo via the forward stochastic differential equation (SDE),
Diffusion models Diffusion- generative models [33 84
dx = f(x, t)dt + (t)
where f(x, t) R + Rd is the drift coefficient, w is the standard Wiener process, and r7(t) E IR is the diffusion coefficient. Samples are generated by running the process backwards in time from T to O via the reverse SDE.
dx [f(x, t) (t) log px + nf(f)dw,
(2)
where w is another standard Wiener process. The generative model aims to learn the score of the marginal distribution Vx log.p(()) and then simulate Eq. 2 to sample from Pp. 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 3 - R is space group equivariant if it satisfies Rgf(x) fgz) for all y= (Rg, "g)) G, where Rg O(3) is point group element and vg RR3 is translation. Symmetrization via summing over group elements has become popular method build invariant
and equivariant neural networks in the last few years [105] 74). Recently, Mirramezani et al. [63

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Space group
3. Wyckoff positions and elements
G P(G)
Discretized conventional lattice parameters Sample parameters G-constraints
Transformer
G(l poller G)
4. Space group equivariant denoising diffusion
High telescoping sampler
Wyckoft position stop
Ma ()
pa G) where =
Stopped?
Vee
GNN
póh Ney,
X;
X
Transformer
A6,16%
Figure 2: Illustration of our crystal generation process.
showed how symmetrization be applied to space groups despite their countably infinite size. In particular, given the finite quotient group G/Tz and Tz-invariant function f RR3 + R3, the following function is space group equivariant:
f(x)
1
Σ gEG/TI.
fgz))
(3)
G//TE
Besides faithfully modeling the space group symmetries of crystals, we discovered that space group cquivariant vector fields conveniently live in the tangent spaces of the Wyckoff positions. We formalize this in the following theorem.
Theorem 1
Let G be space group, RS be point residing in Wyckoff position with stabilizer group G., and f T + R be space group equivariant function. Then, for arbitrary constant IR and any space group element g G..
g(x + cf(x)) = x x ef(x) = G2 S Gxtef(a)
(4)
See proof in Sec. A.I. 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, OD Wyckoff position (a point) lies in a ID Wyckoff position (a line), then space group equivariant vector field cannot move atom along the line through the point. This is problematic because we observed examples where traversing between symmetrically unique regions of ID Wyckoff position requires moving through OD Wyckoff position (c.g.. .. in space group 192, the ID Wyckoff position 12k split by the OD 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 OD Wyckoff positions while maintaining that the marginal distribution atom coordinates is still space group invariant.
4 Proposed method: SGEquiDiff
4.1 Crystal representation
Using sympy 61 and PyXtal (22). removed redundancy induced by space group symmetry from every Wyckoff position by intersecting each one with its exact asymmetric unit. Each zero- dimensional Wyckoff position was reduced to single point, each one-dimensional position of line segments, each two-dimensional position to a set of convex polygonal ASU facets, and each dinee-dimensional position to the ASU interior. We will refer these convex sets 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 will be described in the following sections.
set
We decomposed each crystal into M (G, T,, A, W, X) E (G, R3x3 A", Bm IRax3), where n the number of atoms in the asymmetric unit, G G is the space group, L E 33x3 is the
5

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conventional lattice basis, A E A" are elements, W E w are Wyckoff positions, and X E R 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:
p(M) p(G) x p(L[G) x p(W, X p(X
(5)
Structuring the generation process in this way rather than jointly modeling all crystal components avoids tuning per loss weights [1OO], wemory-intensiie mask states [T04]. 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 lattice lengths (a, b, c) and 3 angles between them (a, () 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 | (a, b, c, a, Y),, the model learned
P[[|G) = II p(4l G).
(6)
where p(lileci: G) has support 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;
15 require 7 = 90"; 16 to 74 require a = B = = 90" 75 to 142 require a = B = 7 = 90 and a b; B 90", and 7 120%; and 195 to 230 require a b c and
143 194 require a = b, = = y = 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 O eqqual to 1. We enforce positive volume by dynamically setting the support p(7)a, a, b, c, a, 3, G) to satisfy
Volume abc /1 + 2 cos cos 3 cos y cos² cos B cos 0.
Besides physical constraints, lattice generation requires the flexibility to learn highly peaked distri- butions since small perturbations to a crystal lattice can significantly alter materials properties. In BaTiO; for example, 0.03 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 0.01 Å. Naively, this resolution requires a softmax over N; = O(10)) classes per lattice parameter to achieve 100À 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 was sampled from p(b [[ G). Then, the selected class b1 was further coarsely binned and one of these higher resolution bins was selected from p(by) bb, Les, G). This process was repeated K times to achieve higher levels of resolution as
K
p(lillei, G) II p(bulba-1,,es G) k=1
where 4, = bx and by V. Choosing p(bylbx-1,,ee G)) to be categorical distribution over a small number of classes ny < Nj achieves (1/n (1/N)) resolution with (nz < O(N)) memory. We used K = 2 and nı 100 in our experiments. We represented conditioning min-max normalization and Gaussian random Fourier features [9). Logits for pbb G) were produced with an MLP. Space group constraints were enforced with hard-coded masking of lattice parameters and their logits.
I and by by using
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 0.3A and 5 ranges, respectively. Noising was not applied at inference time.

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4.3 Transformer-based sampling of W, AA, and n
Similarly to Cao al. Kazeev et al. 46), and Kelvinius et al. 47 we leveraged a transformer architecture [92] to autoregressively predict atomic Wyckoff positions W E W and elemental species A E A as
p(W, A G) II p(w,lues Ocis Rc G)p(a, aci L, (.sppsptppp asi L, G.].
(7)
The number of atoms the ASU n is sampled implicitly via p(stop/wgss agi, 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- sero-dimensional Wyckoff positions which are already occupied to negative infinity to prevent fit with the cross-entropy loss.
sampling overlapping atoms at those locations. The transformer 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. Learnabl 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 1 initial features. For details and 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-invariann Wrapped Normal (WN) distribution 107 40). we learned scores of a G-invariant Space Group Wrapped Normal (SGWN) distribution:
Σ exp (-1|xz gvoll 9EG
20
Theorem 2
The SGWN is G-invariant, i.e., Vgg = (Rg vn) E G, [g0) p(x)( Addition- ally, the score is space group equivariant, i.e., Va, log RRy ( + Ug (m)) R. v., log p( (see Sec. A. for proof).
Following Jiao et al. 440) we set the noise scale o, with the exponential scheduler: do 0 and 1-1
o, = 01 ( ( if > O. As de gets large, the SGWN approaches the uniform distribution.
Noisy samples di were created by sampling Gaussian noise ~ ( 7) and reparameterizing as I, = To + Pap (ote) where (z)(-) (-) is an orthogonal projection the langent space of zo's Wyckoff shape. For the backward process, we constructed the uniform prior P(IT) on each Wyckoff position. To do without redundantly assigning probability mass to symmetrieally equivalent points, we placed uniform distributions on the relevant Wyckoff shapes. See Sec. A.3 for details. We used predictor-correcto sampling [85] sample Zo.
To learn the space group equivariant scores of the SGWN distribution, we built a space group equivariant graph neural network so (see Sec. A.4 for architectural details) and trained it with the following score matching loss:
Lx (e (0200 [150 () - > Vz, log p(a [x])
where Ae- = E V a. log of X and V x, log p( (ro) is described in Sec. A.2
was employed for training stability. Approximate computation
To memory when symmetrizing 89 with Equation 3, note that only |G|/|G2| | elements in the set of points {gølg E G/Tr.} are unique. Thus we only predicted fo(gx) on unique points and subsequently indexed into these predictions to compute the full sum over the | summands in Equation 3 For points x E R3 in special Wyckoff positions with large stabilizer groups Gz, this significantly reduced the number of forward passes through the model.
7

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Table 1: MP-20 dataset. We report metrics of validity for 10,000 sampled crystals; uniqueness and novelty for 1, 1,000 valid crystals; diversity and goodness-of-ft distributional distances for valid, unique, and novel crystals; and stability for valid, unique, and novel crystals. Stability is defined as having a DFT-predicter
Enull < 0.1 eV/atom with respect to the Materials Project v2022.8 8..3. 23. T represents the inference-time temperature of discrete distributions in SGEquiDiff. Unless noted otherwise, T=1.0. Best result is bolded, second best is underlined. Standard deviations over three training runs are reported in parentheses as errors of rightmost digits.
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5 Experiments
Evaluations. We evaluated SGEquiDiff on two benchmark datasets: MP20 100), containing 45,23 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 's StructureMatcher (70) with stol=0.3 3, angle_tol=5, and 1tol=0.2 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 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 p and number of unique elements Nei well as Jensen-Shannon divergences (JSD) for space group G and occupied Wyckoff dimensionalities dwyckoff; (2) Central Moment Discrepancy (CMD) 106 up 50 moments between ground truth test and generated CrystaINN structural fingerprints [109: and (3) structural and compositional diversity as measured by average pairwise L2-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) caleulations to assess thermodynamic stability. Aligned with analyses of DFT-calculated energies for experimentally observed crystals [8]] and of DFT errors relative experiment [8 5) stability was defined as having predicted energy above the hull less than 0.1 e//atom with respect to crystals in the Materials Project v2022.8.23 2. (39.. Further
details the DFT calculations are in Sec. A. 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
68 48) Due to the computational expense of DFT calculations, we only computed S.U.N. rates on subset of baseline models. While prior works pre-relax generated crystals with foundational machine learning interatomic potentials (MLIPs) before conducting DFT relaxations 662, 86 [07], we opted not to use MLIPs to avoid conflating their biases
area of research in the materials science community [60] 25 43
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 DP with the flow matching framework. DiffCSP++ (12.3M parameters)
41 extends DiffCSP by enforcing space group constraints with projected diffusion of the lattice and
8

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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 DifICSP and DiffCSP++, we used the same diffusion corrector step sizes as for MP-20, and the + 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.
Time 6-/buch
Validay (9)
UN nie [93]
Diaribation
distance
CMD Structre
Divertyy
Strcture
Compoition
Template
Structure
W2
Wx.
JSDg
RSD besourt
Scuucure
Composition
DIOCSP
1250
1.8
0.8244
0.3992
0,3759
0.114
0.3512
0.547
15,69
Symml
210
87:11
26.18
14.8
W8.2
1,126
0.1505
0,3730
0.1146
0.4775
07643
15.36
SUllguDiit
530p
90.74m 79
g8.7g
89.8
6,1756.a)
0.310
0.130
0.4180g
0,8013g0
16.3
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 SGE was competitive on all metrics and achieved the highest S.U.N. rate (Table D. While MatterGen achieved a comparable S. 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 metries 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-constraieed models (DiffCSP++, SymmCD) strongly outperformed baselines on the JSD metric for space groups and Wyckoff dimensions. We also show that by increasing the inference- time temperature of SGFquiDiff'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 SiFquiDiff 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 SGFquiDiff was able to scale to more atoms while maintaining high rate of valid, unique, and novel crystals. SGFquiDiff 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 DifICSP 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 DDPMM-baee [39 sampling of E(3)-invariant lattice matrix representations proposed in DiffCSP++ and used in SymmCD (T We used the same cosine noise scheduler as DiffCSP++, jointly diffusing lattices +LDiff. We observed that +Lifff
and atom coordinates. We term this variant of SGEquiDiff improved the Wasserstein distance for number of elements (possibly because conditioning of the Syyckof-EElemen transformer lattices was removed) and CMD of structural fingerprints. However, our autoregressive lattice sampler yielded better validity, JSD of Wyckoff dimensionallitiss, and S.U.A 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 +I.ifff in Fig.
We also replaced our space group and Wyckoff position features with those introduced in SymmCD (55) (5). We found our featurizations yielded better structure validity, composition validity, Wasserstein distance for atomic density, structural divergence, and validation licelioosssoods 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 FAENNttinspired (20] graph neural network (GNN) with the EGNN-inspired [80 model (CSPNet) used by other crystal generative models [40 41 62 Under fixed number of trainable parameters (2.2M), we found our GNN yielded better metries for structural validity,
composition validity, and CMD of structural fingerprints (T

---

Crystal structure prediction (CSP). Consistent with previous works 40 41
we evaluated
SGEquiDiff on its ability to predict crystals from a test set using elemental composition as input. Per- formance was assessed by calculating match rate (MR) and root mean squared error (RMSE) of atomic fractional coordinates based on 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, 5, angle_tol= the pretrained +LDiff variant of SGEquiDiff with the metric bearning-based approach proposed in + (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. orr by supplying composition as conditioning during training. While more challenging, learning instead of strictly enforcing composition may be desirable since arbitrary composition is not guaranteed to host stable erystal (accordingly, experimentally observed compositions can be weighted averages of multiple phases).
and 1to1=0.3. To adapt SGEquiDiff to this task, we combined
We highlight that the current CSP task is slightly mis- aligned 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, single composition can host multiple stable crystal structures depending on exper- imental conditions. For example, iron can exist as body-centered cubic (2 atoms per conventional cell), face-centered cubic (4 atoms per conventional cell), 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].
Table 3: Crystal structure prediction. Leveraging the template selection and el- ement substitution approach developed in
DiffCSP++ [41], we evaluated SGEquiD-
ifff on the crystal structure prediction
task introduced by DiffCSP 40).
MP-20
MR (%) 1
RMSE 1
CDVAE' 100
33,90
0.1045
DinCSP
40
51.49
0.0631
FlowMM 62
61.39
0.0566
DinCSP++ (+CSPML) 4
70.58
0.0272
SGEquiDiff (-LDiff.CPMML)
69.42
0.0416
Uses ground truch number of atoms per unit cell.
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 the-art generation rates of stable, unique, and novel crystals 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-tine 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. SGFquiDiff'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 non- scalar 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 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, are used for military applications.
10

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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 resouree (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 supported by the National Science Foundation under Grant Nos. DGE 21 21-46756 and 21 2118201.
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19

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A Appendix
Symmetry reduction
Space group orbiting
Special Wyckoff positions and the asymmetric unit in the conventional unit cell of hexagonal space group 192. Closed asymmetric unit boundary edges and facets (0II) are in orange.
Figure 3:
Table 4: Normalized root mean square displacements (RMSD) in Angstroms and their standard deviations calculated with pymatgen StructureMatcherr between as as-generated and DFT-relaxed crystals. RMSD is normalized by (V/N)1/3 for unit cell volume V and number atoms per cell N.
CDVAE [TO
DiffCSP++ 41
MatterGen [107
SGEquiDiff
SGEquiDiff=LLDiff 0.0860+0.0010
0.0642++00056
0.0543-0.0812
0.0270++00087
0.0619+000804
A.1 Proofs
Theorem Let G be a space group, x R be point residing in Wyckoff position with stabilizer group Ga, and f R + R be a space group equivariant function. Then, for arbitrary constant and space group element g E Gz.
g(x + cf(x)) = + cf(x),
or equivalently, G. S Gztef(u).
Proof. Let {Rg., Ug. } = 92 E G2 be stabilizer group element where Rg, E 03) is a point group operation, ug E [0, 1)3 is fractional lattice translation, and the action of gr is Rg (-) + Uga. The definition of stabilizer group requires that
Rgg. x + Ug. =
Ugg O.
Thus, we have
[(Rg, = f(x)).
(10)
The definition of space group equivariance requires that for any R G,
[(Rx) R()
(11)
Combining equations 10 and I1
we sec
fg) = f(x),
(12)
i.e., f(x) is also stabilized by G.. Finally, we have
Rg (x + cf(x))
= Rg x + e.g
(13)
= + cf(x).
(14)
Theorem The score of the SGWN distribution is space group equivariant, i.c. given = {R, v) E G.
V n, log,((R, + v|z0) = RV x, log pog,((;)0))
20

---

Proof. The score of the SGWN is given as
Vx, logg(a))))) Eg,- (Rgdy)E exp
-=lls ( 20 21)))
(Rz30 + 12j vxt
20
o ((p(1))20)
Now, we transform the input with [R,
Va, log ((r; + v o/00) = (B, exp
|R +u- mR 203
(Ry20 + +ts -Rx v)
pip( Ra, + o/
We will prove the equivariance of the numerator, which we denote of (RR, vl20). For the denominator, it can be similarly proved that the SGWN is G-invariant, i.e., pl Rx, + =l(0)) = p(rssoo))
REx + (R2 + 4)1e (Ryz0 + +t, Rez v)
6 Re, + v/
Σ (R
exp
20
||( RT (Ryz0 1 u))l
Σ
[ (Ryar + - v) - xz)
exp
20²
(Rid4))
= Σ
1
RD (Ryz0 + 1 - w)1le
+ - -0) mn)] (R
cxp
20²
(RyAt
| 9- (R, + t,))lle))
Σ
+ (()z - xτ)
exp
207
(R
||ate (Ry ty:)113))
Σ (Ryig))
[ +ly mz)
exp
20
A.2 Implementation details
We approximated the SGWN distribution with truncated sum over lattice translations as
|
(Rz20 + L + Lr))
(x2))00) x
Σ
Σ exp
20
ty LzEzzn] mmm EGG/T,
where m E E.. We have dropped the dependence of q on G and or from the notation for clarity. The space group-equivariant score of g(2)(z0) is given as
Va, log g( (ro) Va. log
Σ
Σ
exp
[2
(Ryz0 + + +tx) tt)
207
t Ezen mm (Ry155
Euezrn EjRy,e cxp
(R 20
(Ry20 + tú + 1t - xz)
of (20)
(15)
For each Wyckoff position w, pre-computed approximate values of A
Po V., log p(r) with Monte Carlo sampling. We de- note Pwr (Uniform(-)) as the uniform distribution on the Wyckoff shapes (see Sec. 4.1 and A.3) corresponding to Wyckoff position w. First, we sampled points { from P... Uniform(-) Then, for each sample z/j we sampled Gaussian noise A'0 of) and reparameterized as x; = z + Pag (c). The approximation of x for Wyckoff position w in space group G was then computed as
log ((2ila/: G, w, o1)
In our experiments, set 2500, 3, 01 0.002, σT 0.5, and T = 1000.
21

---

A.3 Uniform priors on Wyckoff positions
To sample from the uniform distribution in given Wyckoff position, sampled uniformly on the Wyckoff shapes in the ASU. The likelihoods of these uniform priors are simply given as 1 for OD 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 OD Wyckoff position, coordinates are fully determined and 'sampling' simply returns point. For a ID Wyckoff position, we sampled line segment with probability proportional to its length, and then sampled uniformly on the line segment. For 2D Wyckoff position, we pre-computed Delaunay triangulations of the polygonal facets belonging to the Wyckoff position, sampled triangle proportionally to its area, and then sampled from a uniform Dirichlet distribution 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:
z² +- MLP(esallew(lla)
2/1 EncoderLayer( (an Meansal)
2w; F4 Split(=
24 + AILP(zA.CW)
Attention K |ew [stopp V |e [stop] 2w, mask mw)
PWstop
=eall, PA Attention cx V all Q 2,4 mask m.a
where esG, ew, and C are predicted embeddings of the crystal's space group, occupied Wyckoff positions, and atomic elements, respectively; Imax is the number of encoder layers; Meausal is sall causal attention mask enforcing atom orderings; ew Estop and all are predicted embeddings of all sampleable Wyckoff positions, the stop token, and all sampleable elements, respectively; mw and m are attention masks enforcing lexicographic atom orderings; at.stop and PA are the model probabilities over Wyckoff positions, the stop token, and elements; and LncoderLayerrx,, m) is module summarized by the following:
norm LN(x)
x + x + MulthheadSefffAttnttitttoommm.s m))
x Dropout(:)
x Dropout (xr + (inea((Dropaut((MM[(())))
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 striance-preservii 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 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 without replacement with probabilities drawn from
sampled frequencies Uy 2 n [--112, 512]
22

---

discretized standard normal distribution. The architecture is summarized as follows:
tmmb (t)
Plane Waneeeaiaen - (2 [ccee ( (in(2
sin
hg < MLP [cc]//PPaneaaaaemmbdddddng(]]]])
Cjj < MIP (PlancaaveEmbodddngggs -x,x |RFF Norm())
fij MLP (eny))
1/+1 hy
MLP GraphNorm
VINI jEM
Mmmax 11
MLP(1.
where t is the diffusion timestep; I; E [0, is the fractional coordinate of atom i; dy is the minimum-image image pairwise Cartesian distance between atoms and j. i.e, for lattice matrix R3x3
dg-
min 02.
[Lx; Liæ + n2 nn)ll;
h is the node feature of the ith atom after I rounds of message passing; E R are the conventional lattice parameters; mmax the number of message passing layers; a learnable scalar initialized to zero; ea, the embedding of atom i's element; and fi E I& is the non- non-symmetrized score prediction (see Eq. 3) for atom i. For the LDiff variant of our model, additionally had
<- MLP
kmase
N
where E R is the predicted noise for the lattice k-vector (see
41
We see that Si is invariant to lattice translations since the model's spatial inputs are the minimum- image distances dı under periodic lattice translations and translation invariant plane wave embeddings of relative and absolute atomic fractional coordinates (zz - 24 and 2;, respectively). We symmetrized
fe with space group equivariance according Equation 3
A.4.3 Space group and Wyckoff position features
To represent space group, we created one-hot features indicating the lattice centering types, 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, 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 + + + 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 lattice systems, totaling + 26 x 15 = 397 features.
To featurize 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, 2, or 3), and the 17 unique Wyckoff multiplicities; and 48 Fourier features of Wyckoff shapes' midpoints and vertices (see 1.. This yielded 62 + 100 + 4 17 + 48 = 231 Wyckoff position features. To compute Fourier features v E R18 we collected the midpoint and vertices of each Wyckoff shape belonging to the Wyckoff position. We used these points
23

---

{pr E p= to compute the following:
2n linspace(start 1, end = steps 8) E R1x8
4 flatten(p:f) E I
1 The
(sin())), E IR:48
We confirmed 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 x 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 (+ CSPNer). For fair comparison in the latter experiment, edge frequencies (96), and the number of trainable parameters (2.2M) by reducing CSPNet's s hidden dimension from 512 224. Standard deviations over three training runs reported in parentheses as errors of rightmost digits.
fixed the number of message passing steps (5), the number of Gaussian
Valichaton Wyckeet
Validry (8) Compostion
U.N. mie (8) Strcctue
Distmbution distrcee
CMD Stuciute
Doveritty
log Hielbbol
Strustuee
Templat
W;
Wx...
Composition
Stttquibit
18
0.171% 0.0395
15,4k
00.4925p
98.55
0.17062M
-(5P%e
30.00
0.122
Table 6: LDiff ablation on MP-20. We replaced our autoregressive lattice sampling method with
joint diffusion of atom coordinates and E(3)-invariant lattice matrix representations proposed in DiffCSP++ [41] (+LDiif)). We report metrics of validity for 10,000 sampled crystals (including the percentage of generated lattices with positive determinant, a requirement enforce symmetry of chiral space groups); uniqueness and novelty for 1,000 valid crystals; diversity and goodness-off offfft distributional distances for valid, unique, and novel crystals; and stability for valid, unique, and novel crystals. Stability is defined as having DFT-predicte Emull < 0.1 eVaatom with respect to the Materials Project v2022. 8.23. Best result is bolded, second best is underlined. Standard deviations over three training runs are reported in parentheses as eIrors of rightmost digits.
Bax
Volidyy (R1]
L. mac 1931
Dannbatton dHane
CMD
Diarady Corpostion
SUN (31
Composiice
Torglan
387
SGE
22
100
9925
26.6 $4.00
18
30.bu 90.5ps
0,3538 à 04855ar
BL7333
0. 171kk 0.154
03150gm 0.344.mm
1564
25.00
96.05
97.02am
20
0,55%
15.8.m
13,65
24

---

A.6 Examples of S.U.N. crystals
LalmTl:
NaNdTl:
TIPPts
Ca:Sn,Pti
Nd:CuzAs:
Y2AgIr
BeCoSi
Ta:Cr
LasCe,Ir:
EryFe;Co:
Figure S.U.N. crystals generated by SGEquiDiff trained MP20.
Li:ThBia
TisNh:Mo
FuAs:Pta
Y2Tmaln:
LuPuRuz
YbAgaP:
Figure 5: S.U.N. crystals generated by the +LDiff variant of SGEquiDiff trained MP20.
25

---

A.7 Training and hyperparameters
Our code was written with Py Torch 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 al. [00] and Baird et al. A. Hyperparameters were tuned manually.
We adopted modular architecture, simultaneously but independently training the space group sampler, lattice sampler, Wyckoff-Elemen Elemmett transformer, and denoising graph neural network, Early stopping based validation performance was applied to each module separately, eliminating the need for gradient balancing between modules.
Hyperparameter
Value
Batch size
256
Number epochs
1000
Teacher forced lattice length noise range
0.3
Teacher forced lattice angle noise range
5.0
Lattice sampler hidden dimension
256
Lattice length Fourier scale
5.0
Lattice angle Fourier scale
1.0
Lattice length bin edges Fourier scale Lattice angle bin edges Fourier scale Transformer dropout rate Transformer hidden dimension Transformer number hidden layers Transformer number of heads
2.0
1.0
0.1
256
2
SGWN lattice translations
3
SGWN Monte Carlo samples
2500
Time embedding dimension
128
Number plane wave frequencies
96
Number of Cartesian distance radial basis functions
96
Radial basis functions cutoff distance
10.0
Edge embedder hidden dimension
128
Node embedder hidden dimension
256
Number of message passing steps
5
Space group learning rate
1 x 10-3
Lattice learning rate
1 10-4
Transformer learning rate
1 10-1
GNN learning rate
x 10-3
Weight decay
0.0
Learning rate scheduler
ReduceLRRnPllteauu
Scheduler factor
0.6
Scheduler patience
30
Minimum learning rate
1 x 10-5
Gradient clipping by value
0.5
Number of parameters
5,513,396
26

---

A.8 Density Functional Theory Calculations
Total energy calculations were performed with density functional theory (DFT) using the Vienna Ab Initio Simulation Package (VASP) [54] with Projector Augmented Wave (PAW) pseudopotentials [8] The Perdew-Butte relaxations. All input parameters, plane wave cutoffs, convergence criteria, and k-point densities
(PBE) 73 exchange-correlation functional was used for structural
were determined by pymatgen MPRelaxSet [69]
Relaxations which don't t converge with default settings from VASP (version 6.4) and MPRelaxSet do not necessarily indicate issues with generated crystal structures. Specifically, 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 commetry--indingg 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 O. 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 '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 de- termined through convex hull analyses using competing phases available in MP v2022.8.23
339
27

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Reviews:
[{"review_id": "kIsSb1BkDx", "reviewer": "Reviewer_YxwH", "summary": "This 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.", "questions": "See 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.", "limitations": "I thank the authors for discussing the weaknesses/limitations of their work in some parts of their paper.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "The author's adequately addressed my concerns in the rebuttal, and are working to add more DFT calculations in the next revision.", "rebuttal": "Thank you for the thoughtful review and encouraging comments. Below we try to address your concerns. \n\n> As 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\nIf we understand correctly, this comment refers to our statement in lines 375-376. To clarify, we believe there are interesting avenues for future work that explore the boundary between space group-constrained models and non-constrained ones to potentially combine the strengths of both. For example, one might train a non-equivariant model with the equivariant scores of our Space Group Wrapped Normal. Furthermore, one might use a non-equivariant model during training, but then symmetrize it with equation 3 to yield space group equivariant scores at inference time. Such approaches could potentially unlock a broader diversity of model architectures able to obtain high symmetry structures. We believe that this capacity to inspire new modeling approaches is a core strength of our work rather than a weakness.\n\n> The 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\nWe agree and are currently running several hundred (~800) more DFT calculations per model. Unfortunately we could not obtain the results by the end of the rebuttal period."}, {"review_id": "L8WCgSfMs9", "reviewer": "Reviewer_56AU", "summary": "The 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.", "questions": "1. 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.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "1. 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.", "was_revised": true, "final_justification": "After going through the author's rebuttals and the answers they have provided to the other reviewers, I feel like they solved all my doubts. I think that the additional ablations and the rewriting they have promised will make the paper more informative and will help reproducibility.", "rebuttal": "Thank you for the helpful feedback and questions. Below we aim to answer questions and address your concerns. Please let us know if we can clarify anything further.\n\n> The notation used for functions and group actions is sometimes overlapping, which can be misleading at times\n\nThank you for pointing this out. If we understand correctly, you are referring to the usage of $g(t)$ in equations 1 and 2 overlapping with the usage of $g$ as a group element. To avoid confusion, we will replace $g(t)$ in equations 1 and 2 with $\\omega (t)$.\n\n> Line 185 to 188: it's not completely clear what the message is there.\n\nWe agree this is a confusing point and will update the text for clarity. The intended message is that, for example, if 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 of space groups where traversing between symmetrically unique regions of a 1D Wyckoff position requires moving through a 0D Wyckoff position. In contrast to, e.g., ODE-based sampling, the use of Gaussian noise (projected onto the tangent space of the Wyckoff position) in SDE-based sampling allows models to stochastically overcome these barriers while maintaining that the marginal distribution over atom coordinates is still space group invariant.\n\n> 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\nDuring training, we maximize the log probability of the next ground truth lattice parameter conditioned on noisy versions of the previous ground truth lattice parameters (where noise is masked such that space group constraints are respected). At sampling time, noising is not applied. We will add this to the revised paper.\n\n> 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\nWe agree and will append the following text to the caption of table 1: “We report metrics of validity for 10,000 sampled crystals; uniqueness and novelty for 1,000 valid crystals; diversity and goodness-of-fit distributional distances for valid, unique, and novel crystals; and stability for 100 valid, unique, and novel crystals. Stability is defined as having a DFT-predicted $E_\\mathrm{hull}$ < 0.1 eV/atom with respect to the Materials Project v2024.11.14. Error bars are the standard deviation of three training runs. $T$ represents the inference-time temperature of discrete distributions in SGEquiDiff.” We will append similar text to the caption of table 2.\n\n> 1. 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.\n\nYes, as you mentioned, equations 10 and 11 hold because as in line 1133, we are only considering group actions from the stabilizer group. To avoid confusion, we will update the notation to read $R_{g_x}$ and $v_{g_x}$ instead of $R$ and $v$.\n\n> 2. 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?\n\nWe agree (although we applied early stopping based on validation performance as a soft regularizer). For simplicity, we have changed our code to sample space groups directly from the training distribution. We observed no changes to our experimental results.\n\n> 3. 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?\n\nThank you for the question and suggestion. FAENet is not required to enforce space group equivariance. From lines 174-175, any periodic translation invariant architecture suffices, including the EGNN backbone used in DiffCSP and others. We have conducted the ablation on MP20 as suggested using the aforementioned EGNN backbone (”CSPNet”). For fair comparison, we held constant 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. We observed our architecture performed slightly better than CSPNet on the structural validity and U.N. metrics (possibly due to our use of Cartesian distances as edge features) while performing the same on other structural metrics. We will include the results (below) in our appendix, where standard deviations over 3 training runs are reported and best result is bold.\n\n|  | Structural validity | U.N. rate | CMD | Structure diversity |\n| --- | --- | --- | --- | --- |\n| SGEquiDiff | **99.47 +/- 0.30** | **83.40 +/- 0.79** | 0.1968 +/- 0.0078 | 0.9149 +/- 0.0241 |\n| +CSPNet | 98.84 +/- 0.12 | 81.33 +/- 1.04 | 0.2052 +/- 0.0014 | 0.9286 +/- 0.0259 |\n\n> 4. Space groups define specific constraints to the lattice angles. How do you enforce those constraints during sampling? Are they hard-coded?\n\nConstraints are enforced in the same way during training and sampling with hard-coded masking and addition of any pre-determined angles. We will update lines 219-220 to reflect this.\n\n> 5. 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.\n\nYes, our space group featurization is different. SymmCD used binary features indicating the 26 space group symmetry operations, the 15 symmetry axes, and the 7 lattice systems. This yielded `7 + 26 x 15 = 397` features. We used more abstracted 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.\n\nAs an ablation, we replaced our space group and Wyckoff position features with those used in SymmCD. On MP20, we found our featurization yielded better structure validity, composition validity, Wasserstein distance for atomic density, structural divergence, and validation log-likelihoods of ground truth Wyckoff positions. The SymmCD features yielded a slightly better U.N. rate, Wasserstein distance for number of elements, and compositional diversity. The results (with standard deviations over 3 training runs) are summarized below where best result is bold. We believe the particularly strong result on Wyckoff log-likelihoods motivates our featurization.\n|  | Validation Wyckoff log-likelihood | Structural validity | Composition validity | U.N. rate | W_\\rho | W_{N_{el}} | JSD_{d_{Wyckoff}} | CMD | Structure diversity | Composition diversity |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| SGEquiDiff | **-0.3206 +/- 0.0083** | **99.47 +/- 0.30** | **85.01 +/- 0.88** | 83.40 +/- 0.79 | **0.5128 +/- 0.1030** | 0.2055 +/- 0.0067 | 0.0327 +/- 0.0078 | **0.1968 +/- 0.0078** | 0.9149 +/- 0.0241 | 15.58 +/- 0.06 |\n| +SymmCD features | -0.4925 +/- 0.0030 | 98.43 +/- 0.18 | 81.75 +/- 0.49 | **85.23 +/- 0.35** | 0.7565 +/- 0.0680 | **0.1654 +/- 0.0232** | 0.0229 +/- 0.0039 | 0.3470 +/- 0.0150 | 0.8960 +/- 0.0116 | **16.06 +/- 0.08** |\n\n> 6. The EquiCSP paper highlights that the score used by DiffCSP is not periodic translation invariant, meaning that the score for a position and a periodic translation of it are different. Does your score formulation address this issue or does it suffer still from this mismatch?\n\nYes, our score is periodic translation invariant. Following our paper’s notation, the space group wrapped normal (SGWN) distribution and its score can be written as\n\n\\[\n\\begin{aligned}\np_\\mathrm{SGWN}(x_t|x_0) &\\propto \\sum_{t_L \\in \\mathbb{Z}^3} \\sum_{\\\\{R_j, t_j\\\\} \\in G/T_L} \\exp\\left(\\frac{-\\lVert x_t - (R_j x_0 + t_j + t_L) \\rVert_F^2}{2\\sigma_t^2} \\right) \\\\\\\\\n\\nabla_{x_t}\\log p_\\mathrm{SGWN}(x_t|x_0) &= \\frac{\\sum_{t_L \\in \\mathbb{Z}^3} \\sum_{\\\\{R_j, t_j\\\\} \\in G/T_L} \\exp\\left(\\frac{-\\lVert x_t - (R_j x_0 + t_j + t_L) \\rVert_F^2}{2\\sigma_t^2} \\right) (R_jx_0 + t_j + t_L - x_t)}{\\sigma_t^2 p_\\mathrm{SGWN}(x_t|x_0)}.\n\\end{aligned}\n\\]\n\nApplying a lattice translation $t^\\prime_L \\in \\mathbb{Z}^3$ as $x_t \\rightarrow x_t + t^\\prime_L$ will leave the SGWN and its score invariant since $t^\\prime_L$ can be absorbed by the infinite sum $\\sum_{t_L \\in \\mathbb{Z}^3}$.\n\n> Missing the log in the score matching loss and also in line 280\n\nThank you for catching this!\n\n---\nWe hope this addresses your questions. If our responses are helpful, we would be grateful if you would consider raising your score."}, {"review_id": "Pi4IA4eiGY", "reviewer": "Reviewer_JT9f", "summary": "This 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.", "questions": "I 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?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## 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", "was_revised": true, "final_justification": "Most of my concerns have been addressed, and the clarifications help. I raise my score to 4.", "rebuttal": "Thank you for your helpful feedback and questions. Below we respond to your questions and concerns.\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).\n> \n\nWe note that Perov-5 and Carbon-24 are not used in more recent works [4,5,6,7,8] (including a followup work to [3]) because they are toy datasets containing unstable crystals; all crystals in Perov-5 have the same fractional coordinates and all crystals in Carbon-24 have only 1 element. In contrast, crystals in MP20 and MPTS52 have all been experimentally observed and span 89 elements of the periodic table with a large diversity of structures. These datasets are thus much more challenging and relevant for materials discovery, becoming the de facto benchmarks for crystal generative models.\n\n[4] A generative model for inorganic materials design. *Nature* **639**, 624–632 (2025)\n\n[5] SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Models (ICLR 2025)\n\n[6] All-atom Diffusion Transformers: Unified generative modelling of molecules and materials (ICML 2025)\n\n[7] Wyckoff Transformer: Generation of Symmetric Crystals (ICML 2025)\n\n[8] FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributions (NeurIPS 2024)\n\n> 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> \n\nWe agree. To strengthen our results, we modified the codes of DiffCSP and DiffCSP++ and benchmarked them on MPTS-52, using the same diffusion corrector step size as for MP20 and dropping the default batch size of DiffCSP++ to 96 to avoid OOMs. We additionally ran SymmCD on our own hardware. We failed to train CDVAE on MPTS-52 due to errors during graph construction arising from noised atoms becoming isolated within the model’s cutoff radius. As an additional diversity metric, we followed SymmCD and report the fraction of unique and novel templates with respect to the training data (where a template is defined as a space group and a multiset of occupied Wyckoff positions) out of 1000 structurally and compositionally valid sampled crystals. We found that SGEquiDiff had a significantly higher unique and novel template rate than baselines while maintaining high structural validity. Although DiffCSP++ yielded slightly higher structural validity, the model uses ground truth templates from training data which explains its extremely low rate of novel templates. The updated table 2 is below:\n\n|  | Sampling time | Structural validity | Composition validity | U.N. rate | W_\\rho | W_{N_{el}} | JSD_G | JSD_{d_{Wyckoff}} | CMD | Structure diversity | Composition diversity | Unique and novel template rate |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| DiffCSP | 467 | 67.47 | 55.8 | 79.8 | 1.189 | 0.5006 | 0.6900 | **0.0818** | 0.4836 | **0.8621** | **16.48** | 0.198 |\n| DiffCSP++ | 1230 | **99.87** | 77.52 | 86.9 | **0.8244** | 0.3692 | **0.2759** | 0.1154 | **0.3812** | **0.8457** | 15.69 | 0.010 |\n| SymmCD | **210** | 87.11 | **78.18** | **90.2** | 1.126 | **0.3506** | **0.2730** | 0.1146 | 0.4775 | 0.7843 | 15.36 | 0.148 |\n| SGEquiDiff | 678 +/- 58 | 97.19 +/- 0.43 | **78.21 +/- 0.18** | 86.5 +/- 0.7 | 1.405 +/- 0.274 | 0.4409 +/- 0.0472 | 0.3179 +/- 0.0252 | 0.1539 +/- 0.0393 | 0.4263 +/- 0.0412 | **0.8490 +/- 0.0363** | **16.21 +/- 0.32** | **0.333 +/- 0.021** |\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>\nWe believe that our inclusion of a comprehensive set of metrics (including new ones) is a strength of our work. We also respectfully disagree that SGEquiDiff underperforms on most other evaluation metrics - we believe its performance is on par with other SOTA models that we benchmarked. None of the baseline models achieved SOTA performance on more than 4 of the 12 metrics (although we don’t believe that disqualifies them from being useful and interesting). SGEquiDiff achieved SOTA (or comparable) performance on 4: Wasserstein distance for number of elements, Jensen-Shannon divergence to space groups and Wyckoff dimensions, and the S.U.N. metric. On several other metrics, SGEquiDiff’s performance is competitive in any practical sense (e.g., structure validity, composition validity, composition diversity). We highlight that SGEquiDiff (5.5M parameters) is the second smallest model (CDVAE has 4.9M, DiffCSP(++) has 12.3M, SymmCD has 60.4M, and Mattergen has 44.6M), which we will add to the paper.\n\nUltimately, the most important metric measures the ability to produce novel and synthesizable crystals. For this reason, the S.U.N. rate [4] is the most important metric as it leverages quantum chemistry calculations to assess stability.\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**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> \n\nCSP was not a focus of our work because we believe the current formulation of the task in ML papers is slightly misaligned with real world application. DiffCSP [2], FlowMM [3], and other models assume that the number of atoms per unit cell will be known beforehand, which is not true in practice. For example, iron exists in BCC (2 atoms per conventional cell), FCC (4 atoms per conventional cell), and at high pressures, HCP (6 atoms per conventional cell). DiffCSP++ [1] relies on training examples of templates which specify number of atoms, Wyckoff positions, and atom types; thus, performance will worsen in composition spaces with sparse training data. However, these sparse spaces are exactly where experimental materials discovery campaigns are aimed.\n\nNevertheless, we agree that CSP is an important problem for materials discovery and have done a preliminary evaluation of SGEquiDiff in the same way as prior crystal generative model papers, calculating match rate and RMSE based on a single generated crystal per composition. Due to the rebuttal time constraints, we only evaluated the CSP task on MP20 using the pre-trained SGEquiDiff+LDiff model from Table 1 with DiffCSP++’s CSPML method for template selection. Without any CSP-specific training or hyperparameter tuning (unlike the baselines), we observed competitive performance (see table below). We would be happy to revisit this experiment, performing full tuning to ensure representative and meaningful results for the revised paper.\n\nWe note that SGEquiDiff could be alternatively adapted for CSP by reordering the factorization in Eq 5 or by supplying composition as conditioning during training. Learning composition (rather than strictly enforcing it) may be desirable since an arbitrary composition is not guaranteed to host a stable crystal (accordingly, experimentally observed compositions can be averages of multiple phases).\n\n|  | MR (%) | RMSE |\n| --- | --- | --- |\n| CDVAE | 33.90 | 0.1045 |\n| DiffCSP | 51.49 | 0.0631 |\n| FlowMM | 61.39 | 0.0566 |\n| DiffCSP++ (+CSPML) | 70.58 | 0.0272 |\n| SGEquiDiff (+LDiff,CSPML) | 68.52 | 0.0459 |\n\n---\nWe hope this addresses your concerns. If our responses are helpful, we would be grateful if you would consider raising your score."}, {"review_id": "ZjE5bFQ0CL", "reviewer": "Reviewer_mXGs", "summary": "The 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.", "questions": "Please refer to the **Strengths And Weaknesses** section for relevant questions alongside weaknesses.", "limitations": "Yes. Please refer to the **Strengths And Weaknesses** section for more details.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "All my major concerns about missing information or experiments are addressed appropriately. The authors have also clarified the questions on design choices. Overall, I recommend acceptance of the paper due to its potential strong impact on the field of AI for crystal generation.", "rebuttal": "Thank you for the helpful feedback. Below we aim to address your concerns. Please let us know if we can clarify anything further.\n\n> Why not sample the training distribution of space groups?\n\nWe agree. For simplicity, we have changed our code to sample space groups directly from the training set, observing no changes to our results.\n\n> What difficulties would arise in learning a joint distribution?\n\nWe believe there are a few difficulties. (1) Joint modeling requires tuning per-component loss weights. (2) Jointly modeling number of atoms with atom attributes requires unnatural “mask states”. (3) Space groups impose hard constraints on lattices (lines 212-217), available Wyckoff positions (lines 120-130), and atom coordinates. Sampling these constrained and constraining variables jointly will yield constraint violations with high probability. Several works try learning space group constraints instead of enforcing them, but we explain in lines 21-45 and show empirically through the $JSD_G$ and $JSD_{d_{Wyckoff}}$ metrics why this might not be a good idea.\n\nWhile SymmCD diffuses atomic coordinates jointly with Wyckoff positions, the model requires post-hoc projections to correct constraint violations. This complicates the model distribution (since marginalizing out projections is intractable) and possibly contributes to the lower structural validity metric (Table 1) as the model cannot recover from projections to invalid structures.\n\n> sampling $n$ using the transformer in Section 4.3 is not described\n\nNumber of atoms $n$ is sampled implicitly via a stop token (p(stop | …) in Eq 7). We will state this in line 245.\n\n> Would it be more appropriate to avoid imposed atom orderings?\n\nWe agree, although “all models are wrong, but some are useful.” Autoregressive models with stop tokens can naturally handle variable numbers of atoms, unlike pure diffusion models. We highlight the strong precedent of applying sequential and non-permutation invariant architectures to unordered data across AI, often achieving SOTA results [9,10,11]. While one can train sequential models on all orderings, marginalizing out orderings is intractable for even 20 atoms.\n\n[9] An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale. ICLR 2021\n\n[10] Scaling Autoregressive Models for Content-Rich Text-to-Image Generation. TMLR 2022\n\n[11] Scalable Diffusion for Materials Generation. ICLR 2024\n\n> How were features for space groups and Wyckoff positions derived?\n\nWe will add the following to our appendix.\n\nFor the space groups, we used 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 translation symmetries and thus cannot differentiate certain space groups, we counted the max number of collisions between space group embeddings (16), arbitrarily indexed colliding embeddings, 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.\n\nTo featurize a Wyckoff position, we used our space group features (62); one-hot features for the site symmetry symbol (100), number of degrees of freedom in atom coordinates (4), and Wyckoff multiplicity (17); and 48 Fourier features of Wyckoff shapes’ midpoints and vertices (see Sec 4.1). This yielded `62+100+4+17+48=231` features. To compute Fourier features $\\nu \\in \\mathbb{R}^{48}$, we collected the midpoint and vertices of each Wyckoff shape belonging to the Wyckoff position. We used these points $\\\\{p_i\\in\\mathbb{R}^{3\\times1}\\\\}_{i=1}^{n_w}$ to compute the following:\n\n\\[\n\\begin{aligned}\nf &=2\\pi \\cdot \\operatorname{linspace}(\\text{start}=1,\\text{end}=32,\\text{steps}=8) \\in \\mathbb{R}^{1 \\times 8}\\\\\\\\\n\\nu_i &=\\operatorname{flatten}(p_i f)\\in \\mathbb{R}^{24}\\\\\\\\\n\\nu &=\\frac{1}{n_w}\\sum_{i=1}^{n_w} [\\sin(\\nu_i),\\cos(\\nu_i)]\\in\\mathbb{R}^{48}\n\\end{aligned}\n\\]\n\n> is 1731 the total number of Wyckoff positions across space groups?\n\nYes. We will edit line 255 accordingly.\n\n> SymmCD symmetry representations could be an alternative. Was any ablation performed on token design?\n\nWe performed the ablation, replacing our space group and Wyckoff features with SymmCD’s. On MP20, our features yielded better validity, Wasserstein distance for density, structural divergence, and validation log-likelihoods of Wyckoff positions. SymmCD features yielded a better UN rate, Wasserstein distance for number of elements, and compositional diversity. Results with standard deviations over 3 training runs are below. We believe the result on Wyckoff log-likelihoods motivates our featurization.\n| |Wyckoff log-likelihood|Structure validity|Composition validity|UN rate|W_\\rho|W_{N_{el}}|JSD_{d_{Wyckoff}}|CMD|Structure diversity|Composition diversity|\n|---|---|---|---|---|---|---|---|---|---|---|\n|SGEquiDiff|**-0.3206+/-0.0083**|**99.47+/-0.30**|**85.01+/-0.88**|83.40+/-0.79|**0.5128+/-0.1030**|0.2055+/-0.0067|0.0327+/-0.0078|**0.1968+/-0.0078**|0.9149+/-0.0241|15.58+/-0.06|\n|+SymmCD features|-0.4925+/-0.0030|98.43+/-0.18|81.75+/-0.49|**85.23+/-0.35**|0.7565+/-0.0680|**0.1654+/-0.0232**|0.0229+/-0.0039|0.3470+/-0.0150|0.8960+/-0.0116|**16.06+/-0.08**|\n\n> Is LDiff equivalent to the representation of lattice matrices in DiffCSP++?\n\nYes. We will edit lines 330-331 for clarity.\n\n> \"next best\" model is misleading. Given acknowledged variability in SUN rate, lines 334–335 is not well-supported\n\nWe will rephrase to “achieved the highest evaluated SUN rate”. We are currently running ~800 more DFT calculations per model to address potential variability.\n\n> Beyond efficiency, are there other advantages to the asymmetric unit? The conventional cell may capture more structural information\n\nComputational efficiency is the main advantage to using the ASU. We used the conventional cell and its periodic boundaries for graph construction (lines 1183-1185) to avoid losing structural information and to symmetrize the model with space group equivariance (Eq 3). Our efficiency gains from using the ASU stem from only computing score matching losses on ASU atoms, only storing ASU atoms in the dataset, and only generating atom types/Wyckoff positions of ASU atoms. We tried excluding fractional coordinates from the representation such that atoms in the same orbit have the same embeddings, removing the need to compute embeddings for atoms outside the ASU. However, performance was better with fractional coordinates.\n\nA possible approach to remove the need to compute “non-ASU” atom embeddings while maintaining performance is to replace periodic sinusoidal embedding functions with space group invariant functions [12]. We defer this to future work.\n\n[12] arXiv:2306.05261 (2023)\n\n**Table 1**\n> What does T represent? What is T in the SGEquiDiff row?\n\nT is the inference-time temperature for all discrete distributions, and T=1.0. We will add these to the table.\n\n> There appears to be an inverse relationship between composition validity and UN rate\n\nWe observed the uniqueness rate was nearly 100% for all models. Thus, the relationship occurs because exotic compositions are more likely to be novel but also invalid.\n\n> It is incorrect that SymmCD uses fixed templates\n\nThank you, we will fix that.\n\n> Please report variance across multiple seeds for SGEquiDiff\n\nWe trained SGEquiDiff with 3 random seeds per dataset and will update the tables with average metrics and standard deviations, as below.\n| |Sampling time|Structure valid|Composition valid|UN rate|W_rho|W_{N_{el}}|JSD_G|JSD_{d_{Wyckoff}}|CMD|Structure diversity|Composition diversity|\n|---|---|---|---|---|---|---|---|---|---|---|---|\n|MP20|266+/-40|99.4+/-0.30|85.501+/-0.88|83.4+/-0.79|0.5128+/-0.1030|0.2055+/-0.0067|0.1723+/-0.0041|0.0327+/-0.0078|0.1968+/-0.0078|0.9149+/-0.0241|15.58+/-0.06|\n|MPTS52|678+/-58|97.19+/-0.43|78.21+/-0.18|86.5+/-0.7|1.405+/-0.274|0.4409+/-0.0472|0.3179+/-0.0252|0.1539+/-0.0393|0.4263+/-0.0412|0.8490+/-0.0363|16.21+/-0.3|\n\n> Why is Wasserstein distance for density significantly higher for SGEquiDiff\n\nGood question - we aren’t sure. We checked 1D histograms of unit cell volumes and numbers of atoms per unit cell for generated vs ground truth crystals. We did not see qualitative differences.\n\nSome insight can be drawn from the LDiff ablation, which has a comparable density metric with the best baselines. It's possible that (1) iterative refinement from lattice diffusion improves the density metric (aligned with the result of CDVAE, which doesn't diffuse lattices), (2) the higher density metric is an artifact of discretizing the lattice, and/or (3) choosing the lattice before atom types/Wyckoff positions is harder to learn than the reverse.\n\nRegardless, we don't think the higher density metric is significant in practice since both CDVAE and SGEquiDiff attain better SUN rates (Table 1) and RMSDs (Table 3) than DiffCSP++ despite having higher Wasserstein distances for density.\n> The table caption would benefit from more description\n\nWe will add the following to table 1 (similar for table 2): “We report metrics of validity for 10,000 sampled crystals; uniqueness and novelty for 1,000 valid crystals; diversity and goodness-of-fit for valid, unique, and novel (UN) crystals; and stability for 100 UN crystals. Stability is defined as having a DFT-predicted $E_\\mathrm{hull}$ < 0.1 eV/atom compared to Materials Project v2024.11.14. Error bars are the standard deviation of three training runs. $T$ represents inference-time temperature of discrete distributions in SGEquiDiff.”\n\n---\nWe hope this addresses your questions. If our responses are helpful, we would be grateful if you would consider raising your score."}]

Author rebuttal:

Meta-review: The paper proposes a new equivariant diffusion method for materials using explicit representation of space group symmetries. The paper's motivation is clear, is well written, and the experimental results show good improvements over prior work. Despite some concerns about evaluations, and comparisons to recent works (like FlowLLM), this paper is a valuable contribution to material generative models. Therefore, I recommend acceptance.
Decision: Accept (poster)