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PXRD-Bench: How Close Are We to Determining Crystal Structures from Powder Diffraction?
Dongyeop Woo, Nofit Segal, Nayoung Kim, Minkyu Kim, Rafael Gomez-Bombarelli, Sungsoo Ahn
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Abstract
Abstract:Determining crystal structures from powder X-ray diffraction (PXRD) would enable routine characterization of newly synthesized materials, including those absent from reference databases. Generative models offer a promising approach, but existing evaluations remain fragmented and largely assume single-phase samples with known composition. To fill this gap, we introduce PXRD-Bench, a unified benchmark that extends evaluation to unknown per-element atom counts and recovery of both structures in two-phase mixtures. It provides approximately 4.1 million training structures with simulated PXRD, tests generalization to unseen compositions and structural prototypes, and evaluates recovery from experimental PXRD after excluding structurally matching training entries. To provide a strong reference baseline, we also introduce Bragglite, a PXRD-to-structure model developed through a systematic study of PXRD modeling choices. It achieves a 44.2\% Top-1 match rate on experimental RRUFF patterns with known composition, 24.6\% above the strongest prior baseline. Further analysis yields several key findings: some models fail to benefit from experimental PXRD, while generalization to unseen structural prototypes and recovery from two-phase mixtures remain challenging. Together, our benchmark, systematic studies, and Bragglite clarify the capabilities and limitations of current PXRD-to-structure models.
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- Source: https://arxiv.org/abs/2610.16480v1
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1]Korea Advanced Institute of Science & Technology (kaist.edu) 2]Massachusetts Institute of Technology (mit.edu) [Code] PXRD-Bench , PXRD-Bench: How Close Are We to Determining Crystal Structures from Powder Diffraction? Dongyeop Woo Nofit Segal Nayoung Kim Minkyu Kim Rafael Gomez-Bombarelli Sungsoo Ahn Affiliation: [ Affiliation: [ Email: dongyeop@kaist.ac.kr Email: sungsoo.ahn@kaist.ac.kr Abstract Determining crystal structures from powder X-ray diffraction (PXRD) would enable routine characterization of newly synthesized materials, including those absent from reference databases. Generative models offer a promising approach, but existing evaluations remain fragmented and largely assume single-phase samples with known composition. To fill this gap, we introduce PXRD-Bench, a unified benchmark that extends evaluation to unknown per-element atom counts and recovery of both structures in two-phase mixtures. It provides approximately 4.1 million training structures with simulated PXRD, tests generalization to unseen compositions and structural prototypes, and evaluates recovery from experimental PXRD after excluding structurally matching training entries. To provide a strong reference baseline, we also introduce Bragglite, a PXRD-to-structure model developed through a systematic study of PXRD modeling choices. It achieves a 44.2% Top-1 match rate on experimental RRUFF patterns with known composition, 24.6% above the strongest prior baseline. Further analysis yields several key findings: some models fail to benefit from experimental PXRD, while generalization to unseen structural prototypes and recovery from two-phase mixtures remain challenging. Together, our benchmark, systematic studies, and Bragglite clarify the capabilities and limitations of current PXRD-to-structure models. 1 Introduction Inferring atomic structures from experimental measurements is central to understanding material properties and guiding materials discovery (Oganov et al., 2019). Powder X-ray diffraction (PXRD) is widely used to characterize crystalline materials because it is relatively inexpensive, broadly applicable, and does not require large, high-quality single crystals (Harris et al., 2001; Kaduk et al., 2021). Reliable structure determination from PXRD would enable routine characterization of newly synthesized materials, including in high-throughput experiments and self-driving laboratories. However, averaging diffraction over differently oriented crystallites compresses three-dimensional structural information into a one-dimensional pattern, causing peak overlap and allowing different structures to yield similar patterns (David and Shankland, 2008; Kaduk et al., 2021; Segal et al., 2026b). Noise, peak broadening, intensity distortion, and contributions from multiple crystalline phases further complicate interpretation (Altomare, 2022; Hollarek et al., 2026; Li et al., 2026). Conventional methods combine crystallographic analysis, structure search, and refinement, often requiring substantial expert guidance (David and Shankland, 2008; Kaduk et al., 2021). Figure 1: Overview of benchmark tasks and evaluation settings. Three tasks recover crystal structures from PXRD under known composition, element set, and two-phase conditions. Models trained on simulated PXRD are evaluated on experimental RRUFF and opXRD patterns. Generalization splits test unseen composition–prototype pairs, new compositions, and new structural prototypes. Generative models offer a promising approach by predicting lattice parameters and atomic positions conditioned on PXRD patterns and available chemical information (Riesel et al., 2024; Li et al., 2025; Li et al., 2026; Segal et al., 2026a). Training on known structures paired with simulated PXRD provides abundant data but requires transfer to experimental patterns. Recent work has advanced PXRD encoders and conditional generators (Li et al., 2025), post-generation refinement (Riesel et al., 2024; Li et al., 2025), and representations and augmentation strategies for simulation-to-experiment transfer (Li et al., 2026; Segal et al., 2026a). Despite these advances, the lack of a common benchmark makes it difficult to assess progress and identify remaining challenges. Differences in datasets, chemical inputs, and refinement procedures hinder direct comparison and obscure which design choices drive improvements. Existing evaluations also cover a limited range of settings. Most assume single-phase samples with known composition, leaving recovery with unknown per-element atom counts or overlapping signals from multiple phases underexplored. Furthermore, testing unseen compositions does not establish whether models can recover unfamiliar structural arrangements, since the same structural prototypes may remain represented in training. Contribution. In this paper, we introduce PXRD-Bench, a unified benchmark for PXRD-based structure determination. The focus of our new benchmark is twofold: • We construct a large-scale corpus of 4.2 million crystal structures paired with simulated PXRD patterns. For experimental evaluation, we curate a benchmark subset from opXRD and adopt the existing RRUFF benchmark, enabling systematic assessment of simulation-to-experiment transfer. We additionally construct simulated two-phase mixtures from thermodynamically plausible phase pairs and evaluate two-phase recovery on experimental GALAXI mixtures. • We provide leakage-controlled evaluation protocols that hold out entire compositions, structural prototypes, or composition–prototype pairs. To evaluate models under real-world scenarios, we extend evaluation beyond single-phase samples with known composition to (1) element-set conditioning and (2) two-phase structure recovery. We further introduce Bragglite as a strong baseline for PXRD-based structure determination. Bragglite incorporates recent advances in CSP architectures and achieves state-of-the-art performance in our benchmark. Using this architecture, we systematically study how PXRD representations, augmentation strategies, and training-data scale and distribution affect structure recovery and simulation-to-experiment transfer. The combination of our new benchmark and systematic studies reveals that current PXRD-to-structure models remain limited in exploiting experimental diffraction and recovering unfamiliar structural prototypes or multiple phases. For example, several existing PXRD-conditioned models underperform DiffCSP on experimental RRUFF patterns, even though DiffCSP receives chemical composition but no PXRD. We observe that data augmentation can improve experimental recovery, but its benefits vary across datasets and may come at the cost of accuracy on simulated patterns. Models also struggle substantially more with unseen structural prototypes than with unseen compositions. 2 Related Work PXRD-to-structure models. Prior work typically formulates structure determination from PXRD as crystal generation conditioned on diffraction patterns and chemical information (Riesel et al., 2024; Li et al., 2025; Li et al., 2026; Segal et al., 2026a). Existing methods vary in training data, chemical inputs, and post-generation refinement procedures, complicating direct comparison. Most assume known primitive-cell composition (Riesel et al., 2024; Li et al., 2025; Li et al., 2026), while XRDiff additionally considers element-set conditioning with a known atom count (Segal et al., 2026a). Prior evaluations also largely focus on in-distribution or composition-level generalization, leaving unseen structural prototypes underexplored. We benchmark models under controlled training data and evaluation settings with held-out compositions, structural prototypes, and composition–prototype pairs. Transfer to experimental PXRD. Because PXRD-to-structure models are typically trained on simulated PXRD, recent work has explored representations designed for experimental robustness and experiment-inspired augmentation for transfer to measured patterns (Riesel et al., 2024; Li et al., 2026; Segal et al., 2026a). However, these strategies have been studied under different models, training settings, and evaluation protocols, including different experimental PXRD benchmarks, while validation and test sets are often not separated. We compare representation and augmentation choices under a common protocol, use separate experimental validation and test sets, and exclude training structures that match the experimental references. Multiphase PXRD analysis. Prior learning-based work on multiphase PXRD has focused on phase identification, i.e., determining which known crystalline phases are present in a mixture, or on separating mixed patterns into phase-specific components. PhaseDifformer (Ito et al., 2026) and XDecomposer (Gao et al., 2026) learn to decompose mixed PXRD signals, while GALAXI (Tong et al., 2026) addresses multiphase identification from experimental PXRD. These approaches do not directly recover the atomic structures of all constituent phases. We extend multiphase analysis to two-phase structure recovery by generating a crystal structure for each decomposed component. To evaluate transfer to the measured pattern, we use GALAXI’s multiphase experimental PXRD dataset. 3 Benchmark design We design PXRD-Bench around three complementary directions: practically motivated task extensions, simulation-to-experiment transfer, and generalization beyond the training distribution. Figure 1 summarizes these tasks and evaluation axes. 3.1 Task formulations We consider three PXRD-based structure determination tasks: single-phase recovery with known composition, single-phase recovery with element-set conditioning (Segal et al., 2026a), and two-phase recovery. The latter two settings reflect cases where compositional analysis reveals the elements without determining stoichiometry or where a powder sample contains multiple crystalline phases. We formulate each PXRD inversion task as conditional generation of plausible crystal structures. Let =(,,)C=(A,F,L) denote a primitive-cell crystal structure with N atoms, where =(A1,…,AN)A=(A_1,…,A_N) lists their element types, ∈[0,1)N×3F∈[0,1)^N× 3 gives their fractional coordinates, and ∈ℝ3×3L ^3× 3 is the lattice matrix. For each element z, let cz=|i∈1,…,N:Ai=z|c_z= |\i∈\1,…,N\:A_i=z\ | be its primitive-cell atom count. We call =(cz)zc=(c_z)_z the composition and ℰ=z:cz>0E=\z:c_z>0\ the element set. Thus c specifies the multiplicity of every element, whereas ℰE specifies only which elements are present. We denote the observed PXRD profile by x. Single-phase PXRD tasks. In the known-composition setting, the model receives the single-phase PXRD profile x and composition c. The atom types are thereby fixed up to their ordering, and the model generates the remaining structural degrees of freedom as (,)∼p(⋅∣,)(F,L) p_ θ(· ,c). In the element-set setting (Segal et al., 2026a), the model instead receives ℰE and the primitive-cell atom count N, but not the element counts c. It generates ∼p(⋅∣,ℰ,N)C p_ θ(· ,E,N), requiring it to infer the element multiplicities, fractional coordinates, and lattice. Multiphase PXRD task. For a powder sample containing two crystalline phases, we aim to recover both constituent structures from a mixed PXRD profile. We provide mixture-level rather than phase-specific chemical information: the profile mixx_mix and the element set ℰmix=ℰ1∪ℰ2E_mix=E_1 _2, where ℰiE_i is the element set of phase iC_i. We additionally provide the primitive-cell atom counts N1N_1 and N2N_2, but neither phase’s composition ic_i. The task is to recover 1,2∼p(⋅∣mix,ℰmix,N1,N2)\C_1,C_2\ p_ θ(· _mix,E_mix,N_1,N_2). The model must therefore determine which elements belong to each phase, their multiplicities, and both phases’ fractional coordinates and lattices. 3.2 Single-phase PXRD benchmarks For the single-phase benchmarks, we evaluate simulation-to-experiment transfer and generalization to unseen compositions and structural prototypes. We support these evaluations with a large simulated PXRD–structure corpus, controlled held-out splits, and experimental PXRD datasets. (a) Element occurrence. (b) Element-pair frequencies. (c) Source provenance. (d) Crystal systems. (e) Primitive-cell size. Figure 2: Coverage of PXRD-Bench’s simulated structure/PXRD corpus. (a) Structures containing each element (%); (b) element-pair frequencies normalized by total pair occurrences; (c) deduplicated source shares; (d) crystal systems; and (e) primitive-cell atom counts. Simulated PXRD dataset. We construct a corpus of about 4.2 million structures from MP (Jain et al., 2013; Horton et al., 2025), Alexandria (Cavignac and others, 2026), OQMD (Kirklin et al., 2015), and AFLOW/AFLUX (Curtarolo et al., 2012; Rose et al., 2017), each paired with a simulated PXRD profile. Preprocessing includes energy-based filtering at 100 meV/atom where energy-above-hull values are available, structural validation, primitive-cell conversion, and cross-source deduplication. The resulting corpus spans all crystal systems and a broad range of elements, with crystal-system, elemental, and element-pair distributions summarized in figure 2. Full preprocessing details and additional statistics are provided in Sec. A.1. Held-out split design. We construct one in-distribution (IID) and three out-of-distribution (OOD) splits, all structure-disjoint from training. Composition denotes the reduced composition, i.e., the simplest integer ratio of constituent atom counts, and a structural prototype is defined by the space group and species-anonymized Wyckoff-orbit pattern. We target approximately 20K samples each for validation and testing in every split. • Pair-OOD: Specific composition–prototype pairs are held out, while both the composition and prototype remain represented in training. This tests generalization to unseen combinations of familiar compositions and prototypes. • Composition-OOD: Entire compositions are held out from training, testing generalization to unseen chemical compositions. • Prototype-OOD: Entire structural prototypes are held out across all compositions, testing generalization to unseen structural prototypes. Experimental PXRD datasets. To assess simulation-to-real transfer, we use 148 measured RRUFF patterns curated by Crystalyze (Riesel et al., 2024; Lafuente et al., 2015), reserving 48 for validation and 100 for testing. We use the validation set to select PXRD representation and augmentation settings. Additionally, we curate 123 measured profiles paired with structures from opXRD (Hollarek et al., 2026). Among samples satisfying our benchmark constraints, we use 40 for validation and 83 for testing. To prevent data leakage, we exclude training structures that structurally match any reference structure in either experimental dataset. Structural matching without refinement. Prior work varies in post-generation refinement and structural matching criteria, complicating direct model comparisons. We therefore evaluate unrefined predictions using pymatgen’s StructureMatcher (Ong et al., 2013) with tolerances (ltol,stol,angle_tol)=(0.2,0.3,5∘)( ltol, stol, angle\_tol)=(0.2,0.3,5 ), stricter than the more permissive (0.3,0.5,10∘)(0.3,0.5,10 ) commonly used in the literature (Jiao et al., 2023). These stricter criteria help avoid accepting distorted structures or structures with different space groups (Martirossyan et al., 2025; Wei et al., 2024). Under this protocol, we report top-k match rates for k∈1,20k∈\1,20\, where a target is considered recovered if at least one of the k candidates matches the reference structure. Metric details are provided in Sec. A.3. 3.3 Multiphase PXRD benchmark For the multiphase benchmark, we construct a simulated two-phase dataset for training and controlled evaluation and additionally evaluate on experimental mixtures. Multiphase PXRD dataset. We construct a separate two-phase benchmark based on MP-20. Within binary metallic chemical systems, we use the corresponding phase diagrams to select thermodynamically plausible phase pairs and generate mixtures at multiple relative phase fractions. We retain equilibrium or near-hull assemblages within 50 meV/atom of the convex hull. For each mixture, we combine the constituent PXRD profiles by weighted superposition using Rietveld scale factors, followed by maximum-intensity normalization. We apply this procedure separately within the original MP-20 train, validation, and test splits, yielding 100,798 training, 12,418 validation, and 11,848 test mixtures. Full construction details are provided in Sec. A.2. Experimental multiphase PXRD dataset. To assess simulation-to-experiment transfer, we use 21 test samples from GALAXI’s impurity-phase experiments (Tong et al., 2026). We select this subset because its primary–secondary phase mixtures reflect realistic multiphase outcomes of synthesis. Multiphase recovery evaluation. We use the same structural matching criterion as in the single-phase benchmark. For each phase, the model generates k candidates, and we report dominant-, minor-, and all-phase top-k match rates for k∈1,20k∈\1,20\. A sample is counted as an all-phase top-k match only if both constituent phases are recovered within their respective k candidates. 4 Bragglite: A baseline for PXRD-to-structure generation We introduce Bragglite, a PXRD-to-structure reference baseline resulting from our systematic study of PXRD representation, encoding, and augmentation choices. Our model builds on Crystalite (Hadži Veljković et al., 2026), a diffusion Transformer with pair-geometry attention biases, and adds a Transformer PXRD encoder to condition on diffraction. We describe the PXRD representation, conditioning, and augmentation used in Bragglite; full architecture details are provided in Appx. C. 4.1 PXRD representation and conditioning Following prior peak-based approaches (Li et al., 2026; Segal et al., 2026a), we represent each PXRD pattern as detected peaks =pii=1Kx=\p_i\_i=1^K, with each peak described by its position, intensity, width, and prominence. Peak position representation. PXRD peak positions provide information about the lattice parameters. They are conventionally expressed in terms of the diffraction angle 2θ2θ, which depends on both crystal structure and X-ray wavelength λ. We also consider the wavelength-independent interplanar spacing d=λ/(2sinθ)d=λ/(2 θ) and scattering-vector magnitude q=4πsinθ/λ=2π/dq=4π θ/λ=2π/d. We compare 2θ2θ, d, and q, and use q by default based on our ablation. For 2θ2θ models, we convert each pattern to a canonical wavelength before extracting peaks to remove variation due to measurement wavelength. Peak features. Peak intensities depend on the atomic species and their arrangement within the unit cell. Because the absolute intensity scale varies with experimental conditions, we normalize each profile by its maximum intensity and use relative peak intensities. We additionally use peak prominence and width. Prominence measures how strongly a peak stands out from its local background and can help when measured patterns contain background scattering or noise. Peak width captures broadening and can indicate uncertainty in peak localization when peaks are poorly resolved. Peak encoding. To provide a richer representation of peak position, we encode q using sinusoidal Fourier features with 16 logarithmically spaced frequencies. We embed the base peak features and Fourier features separately and sum the resulting embeddings to obtain the peak embedding. PXRD conditioning. We retain up to 128 highest-intensity peaks and encode them with a Transformer into a profile-level PXRD embedding. For in-context conditioning, we project this embedding to the Crystalite hidden dimension and replicate it across eight context tokens with learnable positional embeddings. The context, atom, and lattice tokens are jointly processed by the Crystalite Transformer. 4.2 PXRD augmentation Figure 3: Schematic illustration of PXRD profile augmentations. Bragg reflections derived from a crystal structure are perturbed by weak-peak removal, position jitter, peak-wise intensity scaling, a common angular shift, pseudo-Voigt broadening, background addition, and Gaussian noise. Blue denotes perturbed signals and gray denotes reference signals. To improve transfer from simulated to experimental PXRD, we augment simulated PXRD during training to capture common measurement distortions. Starting from simulated Bragg reflections, we first apply reflection-level augmentations and then construct and augment a dense diffraction profile. Figure 3 illustrates the full pipeline. Exact augmentation definitions, sampling distributions, and hyperparameters are provided in Sec. C.3. After augmentation, we redetect peaks from the dense profile and use their positions, intensities, widths, and prominences as model input. Reflection-level augmentations. We randomly remove weak reflections, independently jitter their positions, and rescale their intensities to model missing peaks, local position errors, and variations in relative peak strength. To model systematic angular calibration errors, we additionally apply a shared zero shift to all reflections. Profile-level augmentations. We convert the perturbed Bragg reflections into a dense diffraction profile using pseudo-Voigt peak shapes (Thompson et al., 1987). The peak width varies with diffraction angle, with additional randomization to capture variation in peak broadening. We then add a smooth background and Gaussian noise to model background scattering and measurement noise. 5 Experiments We first present the single-phase benchmark results (Sec. 5.1), then analyze simulation-to-real transfer in more detail (Sec. 5.2), and finally evaluate two-phase structure recovery on simulated and experimental mixtures (Sec. 5.3). 5.1 Single-phase benchmark results Table 1: Main benchmark results. Top-K match rates (%) on simulated and experimental PXRD datasets. Results on experimental datasets are reported as mean ± standard deviation over five sampling seeds. Bold and underlined values indicate the best and second-best means, respectively, within each dataset and chemical-input setting. Simulated PXRD RRUFF opXRD Dataset Model Top-1 Top-20 Top-1 Top-20 Top-1 Top-20 MP-20 Chemical input: exact stoichiometry PXRDGen 54.75 77.99 30.80 ± 3.270 57.00 ± 1.410 30.00 ± 1.960 36.43 ± 0.980 RealPXRD-Solver 56.35 80.61 27.80 ± 4.760 51.20 ± 1.100 27.14 ± 0.800 36.07 ± 0.800 XRDiff 55.10 78.11 17.80 ± 0.840 39.80 ± 0.840 21.07 ± 3.870 31.43 ± 2.040 Crystalyze 29.32 48.03 05.00 ± 1.580 14.80 ± 1.480 11.07 ± 1.490 17.86 ± 1.790 DiffCSP (no PXRD) 52.17 77.28 31.60 ± 2.700 57.00 ± 2.120 23.21 ± 1.790 36.43 ± 0.980 Crystalite (no PXRD) 61.42 78.64 39.20 ± 2.390 52.60 ± 1.140 26.43 ± 0.800 41.43 ± 1.960 Bragglite (Ours) 70.85 82.59 52.40 ± 0.890 62.60 ± 0.890 32.14 ± 2.820 42.50 ± 0.800 Chemical input: element set + N XRDiff 34.29 46.13 15.00 ± 1.580 22.40 ± 0.550 09.64 ± 0.980 14.64 ± 0.800 Bragglite (Ours) 47.48 71.51 38.20 ± 2.490 52.80 ± 1.920 23.57 ± 1.490 28.21 ± 1.490 PXRD-Bench Chemical input: exact stoichiometry PXRDGen 62.50 88.90 00.00 ± 0.000 03.20 ± 1.480 00.96 ± 0.540 07.71 ± 1.830 RealPXRD-Solver 60.45 89.50 19.60 ± 2.190 48.00 ± 1.220 17.59 ± 1.830 27.47 ± 1.790 XRDiff 48.58 83.42 08.20 ± 3.350 37.60 ± 3.210 09.16 ± 3.140 24.10 ± 0.850 Crystalyze 13.92 36.26 00.60 ± 0.550 02.60 ± 1.340 00.72 ± 0.660 03.13 ± 1.080 Crystalite (no PXRD) 30.54 75.49 16.20 ± 4.090 57.80 ± 2.680 11.08 ± 1.570 31.57 ± 3.550 Bragglite (Ours) 76.48 93.81 44.20 ± 2.490 69.20 ± 2.050 23.61 ± 1.080 38.80 ± 1.320 Chemical input: element set + N XRDiff 22.45 49.78 02.20 ± 1.480 13.80 ± 1.480 06.51 ± 1.830 11.57 ± 1.080 Bragglite (Ours) 56.53 81.59 36.00 ± 2.120 58.80 ± 1.300 15.18 ± 0.660 24.58 ± 2.020 We evaluate single-phase structure recovery under the known-composition and element-set settings defined in Sec. 3.1. We benchmark models trained on PXRD-Bench and additionally report MP-20 (Jain et al., 2013; Jiao et al., 2023) results for reference. We compare Bragglite with Crystalyze (Riesel et al., 2024), PXRDGen (Li et al., 2025), RealPXRD-Solver (Li et al., 2026), and XRDiff (Segal et al., 2026a) under controlled training data and evaluation settings, and also include DiffCSP (Jiao et al., 2023) and Crystalite (Hadži Veljković et al., 2026) as composition-only baselines without PXRD input. Table 1 reports match rates on the simulated in-distribution test sets and on experimental RRUFF and opXRD patterns. For opXRD, MP-20-trained models are evaluated only on ≤20≤ 20-atom structures, while PXRD-Bench-trained models use the full set. Detailed baseline configurations and experimental settings appear in Appx. D. Bragglite outperforms baselines across all settings. In the known composition setting, Bragglite achieves higher match rates than existing baselines on both simulated and experimental PXRD across the two training datasets. It also outperforms XRDiff under element-set conditioning. PXRD-conditioned models can underperform CSP baselines. On MP-20 and RRUFF, Crystalite achieves higher Top-1 match rates than existing PXRD-conditioned baselines. This suggests that a strong composition prior alone often recovers the reference structure when polymorph ambiguity is limited. In contrast, Crystalite performs substantially worse on the polymorph-rich PXRD-Bench, where PXRD is needed to distinguish structures sharing the same composition. Existing baselines struggle with experimental PXRD. To isolate the contribution of PXRD conditioning from backbone differences, we train a composition-only DiffCSP model with the same backbone and model size as the existing baselines. On RRUFF, this CSP-only model matches or outperforms its PXRD-conditioned counterparts, suggesting that PXRD representations optimized on simulated data do not transfer robustly to experimental measurements. Table 2: Out-of-distribution generalization. Top-K match rates (%) on the OOD simulated PXRD splits of PXRD-Bench, grouped by chemical input. Bold indicates the best result within each setting. Pair OOD Prototype OOD Composition OOD Model Top-1 Top-20 Top-1 Top-20 Top-1 Top-20 Chemical input: exact stoichiometry PXRDGen 62.40 89.98 7.55 33.97 63.80 89.61 RealPXRD-Solver 58.94 88.88 3.31 17.52 62.90 90.66 XRDiff 47.48 83.87 2.36 18.51 50.13 84.16 Crystalyze 16.06 39.53 0.58 3.19 14.00 36.91 Crystalite (no PXRD) 16.81 69.75 1.39 11.12 27.30 73.32 Bragglite (Ours) 75.26 94.18 12.37 35.89 77.27 94.55 Chemical input: element set + N XRDiff 21.89 49.57 0.59 4.49 24.02 51.39 Bragglite (Ours) 54.46 80.01 4.08 13.86 57.27 82.13 Generalization beyond training distribution. We evaluate models on held-out splits with unseen composition–prototype pairings, chemical compositions, and structural prototypes. As shown in table 2, performance remains strong on unseen pairings and compositions but drops substantially on unseen structural prototypes. This indicates that generalization to unseen pairings and compositions is relatively tractable when corresponding structural prototypes are represented during training, whereas generalization to unseen structural prototypes remains challenging. To assess whether recovery of unseen structural prototypes is driven solely by the learned crystal structure prior, we compare Bragglite with its composition-only counterpart, Crystalite. Despite low absolute match rates, Bragglite substantially outperforms the CSP baseline, indicating that PXRD provides structure-specific information beyond the composition prior even for unseen structural prototypes. Figure 4: Data scaling. Performance vs. training data fraction. Data scaling experiments. We train Bragglite on progressively larger subsets of PXRD-Bench to study the effect of simulated training scale. As shown in figure 4, Top-1 match rates improve across all simulated splits and on experimental RRUFF as data scale increases. The gains diminish at larger scales but remain consistent across simulated and experimental evaluations. 5.2 Understanding sim-to-real transfer We study how PXRD representation and augmentation affect simulation-to-real transfer while fixing the model architecture and training configuration. Unless otherwise stated, we train Bragglite on MP-20 for 200K optimizer steps and vary only the PXRD representation or augmentation strategy. We focus here on the effects of PXRD augmentation and training distribution, while additional ablations of PXRD representations are provided in Sec. B.1. Ablation on PXRD augmentation. We progressively add profile-augmentation components to assess their contributions. As shown in figure 5, training without augmentation achieves high Top-1 accuracy on simulated PXRD but transfers poorly to experimental RRUFF. Adding augmentations consistently improves RRUFF performance while generally reducing simulated accuracy, indicating a trade-off between simulation performance and experimental robustness. Furthermore, individual components vary substantially in effectiveness across experimental datasets, indicating that augmentations effective for one dataset may not transfer equally well to others. Developing sim-to-real augmentations that generalize across diverse experimental artifacts and conditions therefore remains important. Figure 5: Analysis of PXRD augmentation and simulation-to-experiment transfer. (a) Additive ablation of the PXRD augmentation components. (b) Relationship between simulated MP-20 and experimental validation performance on RRUFF and opXRD. Filled markers denote the final training checkpoint; solid lines connect results across training checkpoints. Correlation between simulated and experimental performance. We examine whether simulated and experimental performance follow similar trends across augmentation strategies. We compare no augmentation, peak-only augmentation, and the full pipeline with reduced or proposed augmentation magnitudes. During 200K-step training, we evaluate Top-1 validation match rates on MP-20, RRUFF, and opXRD every 50K steps and fit linear trends across four checkpoints. As shown in figure 5, no augmentation and peak-only augmentation yield little experimental improvement as simulated performance increases, whereas both full-augmentation settings show positive trends, with the proposed magnitudes yielding larger experimental gains. Effect of training distribution. Despite its smaller size, MP-20 transfers better to RRUFF than PXRD-Bench. RRUFF is primarily composed of minerals and is enriched in elements such as Si and O, which are more frequently represented in MP-20. As shown in table 3, oversampling training structures containing these mineral-associated elements further improves RRUFF performance, suggesting that distribution-aware fine-tuning on PXRD-Bench toward the expected target domain may further improve performance on downstream experimental PXRD. RRUFF Training setup Top-1 Top-20 MP-20 48.33 ± 1.740 58.33 ± 2.080 PXRD-Bench 43.75 ± 2.950 70.42 ± 1.740 + oversampling 54.58 ± 3.420 74.17 ± 1.140 Table 3: Effect of Si/O oversampling. Validation match rates (%) on RRUFF. MP-20 GALAXI Model Top-1 Top-20 Top-1 Top-20 MatterGen 2.13 30.67 0.00 42.86 XRDiff 6.76 14.64 0.00 9.52 Bragglite 13.26 25.34 19.05 47.62 Table 4: Multiphase recovery. All-phase Top-K match rates (%) on MP-20 and GALAXI. 5.3 Two-phase recovery We evaluate two-phase structure recovery on the simulated MP-20 benchmark and experimental PXRD mixtures from GALAXI (Tong et al., 2026). Each mixed profile is first decomposed into phase-specific profiles using XDecomposer (Gao et al., 2026), after which a structure generator is applied independently to each profile. Starting from the element-set-conditioned Bragglite, we fine-tune it so that each phase-specific generation uses the shared mixture-level element set. We use MatterGen (Zeni et al., 2025) and XRDiff as per-phase candidate generators, but treat them as reference methods rather than strictly controlled baselines: MatterGen is not leakage-controlled against our evaluation sets, while XRDiff receives phase-specific element sets. As shown in table 4, Bragglite outperforms the reference methods in most settings. Its substantially higher Top-1 match rate than the PXRD-free MatterGen reference further supports the value of diffraction conditioning. 6 Conclusion We introduce PXRD-Bench, a unified benchmark for PXRD-based structure determination spanning simulation-to-experiment transfer, out-of-distribution generalization, element-set conditioning, and two-phase recovery. We also develop Bragglite, a strong PXRD-conditioned baseline, and use it to systematically study PXRD representations and augmentation strategies. Our experiments show that Bragglite substantially improves experimental structure recovery over prior methods, while revealing that robust use of experimental PXRD, generalization to unseen structural prototypes, and recovery of multiple phases remain challenging. These results establish PXRD-Bench as a controlled framework for measuring progress toward practical PXRD-based structure determination and identify key limitations that future methods must address. AI use statement We used OpenAI’s ChatGPT and Codex to support implementation across the research workflow, including data processing, model development, training, evaluation, and interpretation of experimental results. These tools were also used to assist with plotting and analysis code, figure and table preparation, and drafting, revising, and formatting the manuscript. Their use was guided by the authors’ research ideas, experimental goals, and intended analyses. Generative AI did not independently determine the research questions, core methodology, or final scientific conclusions. The authors reviewed the resulting analyses, figures, tables, and manuscript text and take full responsibility for the final results, claims, and presentation of this work. Reproducibility statement We provide the details needed to reproduce the benchmark construction and experiments throughout the main text and appendix. 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The frozen snapshots of MP [Jain et al., 2013, Horton et al., 2025], OQMD [Kirklin et al., 2015], Alexandria [Cavignac and others, 2026], and AFLOW/AFLUX [Curtarolo et al., 2012, Rose et al., 2017] contain approximately 9.78 million input records in total. This aggregate should not be interpreted as a count of unique structures: it includes repeated AFLOW API records, and the source-specific counts refer to different processing stages. We apply energy-based selection to MP, OQMD, and Alexandria, but no energy filter to AFLOW because the AFLOW snapshot does not store energy-above-hull values for assessing thermodynamic stability. This source-specific selection retains 5,634,620 candidate records. Table 5 summarizes the snapshots and selection criteria. Structural validation. We next validate the selected structures and convert them to primitive cells, retaining only three-dimensionally periodic structures with fully occupied, ordered sites. We exclude structures with invalid cell volumes (non-finite or ≤10−6≤ 10^-6 Å3), no atomic sites, non-finite lattice parameters or atomic coordinates, or elements with atomic numbers greater than 94. At this stage, we also discard structures with more than 208 atoms in the primitive cell. Composition-bucket filtering and deduplication. Before deduplication, we group candidate records by reduced composition and discard every bucket containing more than 128 records in its entirety. We then deduplicate the remaining structures across all sources using sequential, representative-centered, species-aware structural matching with (ltol,stol,angle_tol)=(0.2,0.3,5∘)(ltol,stol,angle\_tol)=(0.2,0.3,5 ). Representatives are selected using a fixed source priority of MP, OQMD, Alexandria, then AFLOW. This priority is fixed for reproducibility and does not rank source quality. We also retain the provenance of matching records from other sources. Atom-count restriction and chemical curation. For the benchmark used in our experiments, we further restrict the corpus to structures with at most 52 atoms in the primitive cell and apply chemical curation to focus the dataset on extended inorganic crystalline materials. We screen structures using periodic bond connectivity to identify discrete molecular units and extended networks. We exclude molecular crystals composed of finite, neutral molecular units, including both organic and inorganic molecules. We additionally exclude structures with evidence of organic components, including organic ions, polymeric organic structures, metal–organic coordination networks, COF-like networks, and organic guests in inorganic hosts. This broader screening does not require definitive classification as a salt, MOF, or COF, nor confirmed porosity, and conservatively excludes ambiguous candidates with evidence of organic components. This curation is an operational exclusion policy rather than a guarantee that the remaining corpus is entirely free of molecular or organic structures. After all filtering and split-consistency checks described below, the resulting benchmark contains 4,262,691 structures, including an effective training set of 4,107,627 structures. Alexandria and AFLOW account for the largest shares of representative structures in the final corpus (figure 2(c)). For each retained structure, we simulate a PXRD profile using Cu Kα radiation over 2θ∈[5∘,90∘]2θ∈[5 ,90 ]. Table 5: Frozen source snapshots, selection criteria, and record counts before and after dataset curation. Candidate counts are measured after structural validation and before composition-bucket filtering, cross-source deduplication, and the 52-atom restriction. Final counts are measured after the 52-atom restriction, chemical curation, OOD-consistency filtering, and benchmark-overlap exclusion, totaling 4,262,691 structures, including 4,107,627 training structures. Each retained structure is attributed to the source of its representative record, selected during deduplication using the fixed priority MP, OQMD, Alexandria, then AFLOW. The final counts therefore reflect representative provenance rather than independent source contributions or source quality. EhullE_hull denotes energy above hull; OQMD uses its signed stability field, including negative values. All energy values are in eV/atom. Source Snapshot Selection criteria Record counts Candidates Final corpus Materials Project 2026-04-13 0≤Ehull≤0.100≤ E_hull≤ 0.10 100,384 77,493 OQMD v1.5 (2021-10) Signed stability ≤0.10≤ 0.10; negative values retained 219,945 166,102 Alexandria PBE 3D, 2025-07-02 0≤Ehull≤0.100≤ E_hull≤ 0.10 2,331,796 2,139,337 AFLOW AFLUX, 2026-08-16 No energy filter 2,981,679 1,879,759 Total 5,633,804 4,262,691 A.1.2 Dataset splits and overlap audits We use a sequential holdout procedure to partition the corpus into a training set and validation/test sets for four evaluation settings: IID, prototype-OOD, composition-OOD, and composition–prototype-pair-OOD (pair-OOD). We construct the three OOD holdouts first, followed by the IID holdouts. We define a structural prototype by its space group, primitive-cell site count, and species-anonymized per-species multiset of Wyckoff orbits, using a symmetry tolerance of symprec=0.1symprec=0.1 Å. To prevent structurally similar examples from crossing split boundaries, we additionally group structures using species-aware structural matching with (ltol,stol,angle_tol)=(0.3,0.5,10∘)(ltol,stol,angle\_tol)=(0.3,0.5,10 ). Structural similarity groups are defined as connected components of the resulting matching graph and are kept intact during splitting. First, we construct the prototype-OOD set. A candidate prototype is eligible for holdout only if every associated reduced composition remains represented in the training pool by another prototype. We hold out all structures belonging to each selected prototype, targeting approximately 40,000 structures in total. Second, we construct the composition-OOD set, excluding reduced compositions associated with the prototype-OOD set from consideration. We hold out all structures with a candidate composition only if every associated prototype remains represented in the training pool by another composition, again targeting approximately 40,000 structures. Third, we construct the pair-OOD set from structures not assigned to either preceding OOD set. We hold out complete composition–prototype pairs only if the composition and prototype each remain represented in the training pool through other pairs. A pair is eligible only if it can be held out without splitting any structural similarity group. We divide each OOD set into validation and test subsets, targeting approximately 20,000 structures per subset. The defining holdout units—prototypes, reduced compositions, or composition–prototype pairs—are kept intact, so that no unit is shared between the corresponding validation and test subsets. After constructing the OOD splits, we randomly hold out structural similarity groups from the remaining pool to form IID validation and test sets of approximately 20,000 structures each. We refer to these random structural-group holdouts as IID splits because they do not explicitly enforce novelty in composition or structural prototype. Consequently, rare compositions or prototypes may occur only in validation or test under random group assignment. Subsequent filtering can remove the remaining training examples needed to satisfy an OOD holdout condition. We therefore re-evaluate the OOD constraints after the IID holdouts and again after the atom-count, chemical-curation, and benchmark-overlap exclusions described above and below. Any validation or test structures that no longer satisfy their defining OOD condition with respect to the final effective training set are removed. To prevent training overlap with the external benchmarks, we exclude training structures that match reference structures in RRUFF-148 or opXRD-123. Matching requires identical element species and uses primitive-cell conversion and lattice scaling, without supercell or subset matching. For this exclusion, we use more permissive structural matching tolerances than those used for the match-rate metric: (ltol,stol,angle_tol)=(0.3,0.5,10∘)(ltol,stol,angle\_tol)=(0.3,0.5,10 ). Audits of the finalized splits confirm that no structural similarity group is shared across splits and that no training structure matches an RRUFF-148 or opXRD-123 reference after exclusion. These checks separately address internal split leakage and training–benchmark structural overlap. Figure 7 reports the finalized split sizes. A.1.3 Additional dataset coverage statistics The main text summarizes source provenance, crystal-system frequencies, element occurrence, and element-pair frequencies (Secs. 3 and 2). Here, we complement that overview with distributions of primitive-cell size, composition arity (the number of distinct elements), prototype frequency, and simulated Bragg line counts (figure 7). All statistics reported below are computed over the final curated benchmark corpus of 4,262,691 structures. Together, these statistics characterize cell size, chemical diversity, structural-family coverage, and diffraction-profile complexity. Primitive cells have a median of 6 atoms, with the 95th percentile at 22 atoms (figure 7(a)). Ternary compositions dominate the corpus at 69.5%, followed by quaternary compositions at 20.4% and binary compositions at 8.0%. Compositions with five and six element types account for 1.7% and 0.3%, respectively, while unary compositions and those with seven or eight element types each account for less than 0.1% (figure 7(b)). The prototype distribution is strongly long-tailed. Among 46,658 structural prototypes, 41.4% occur only once (figure 7(c)). Thus, despite the large size of the corpus, coverage across structural families remains highly nonuniform. The simulated profiles also span a wide range of diffraction complexity (figure 7(d)). Within 2θ∈[5∘,90∘]2θ∈[5 ,90 ], the number of simulated Bragg lines follows a long-tailed distribution that extends well beyond the 128- and 512-line budgets indicated in the figure. These counts refer to the peak positions returned by the diffraction calculator, which can merge multiple reflections at the same position into a single line. They therefore do not count individual reflections and need not correspond to the number of experimentally resolved peaks. The broad distribution motivates examining how much diffraction information is retained by fixed-size peak representations. Figure 6: Finalized split sizes of the large-scale simulated corpus. The plot reports the training-set size and validation/test sizes for IID, pair-OOD, prototype-OOD, and composition-OOD evaluations on a logarithmic horizontal axis. No structural similarity group is shared across splits, and training–benchmark structural overlap with RRUFF-148 and opXRD-123 is zero after exclusion. (a) Primitive-cell size. (b) Composition arity. (c) Prototype frequencies. (d) Simulated Bragg line counts. Figure 7: Structural and diffraction-profile distributions of the large-scale simulated corpus. (a) Primitive-cell atom counts. (b) Number of distinct elements per structure. (c) Structures per prototype, ordered by decreasing frequency on logarithmic axes. (d) Simulated Bragg line counts for Cu Kα radiation over 2θ∈[5∘,90∘]2θ∈[5 ,90 ]; multiple reflections at the same position can be merged into one line. Vertical lines mark 128- and 512-peak budgets. The displayed histogram ends at 800 lines, while the annotated maximum is 7,107. A.2 Construction of the Multiphase Dataset We construct the multiphase dataset independently within the original MP-20 train, validation, and test splits, so that no constituent structure crosses the original split boundaries. We first identify binary metallic chemical systems and collect split-local unary and binary structures belonging to each system. Using a frozen Materials Project GGA/GGA+U snapshot, we build the corresponding binary phase diagrams and enumerate pairs of phases with distinct compositions. For each phase pair, we consider nine phase fractions. Let w1w_1 and w2w_2 denote the weight fractions of the two constituent phases, with w1+w2=1,w1∈0.1,0.2,…,0.9,w2=1−w1.w_1+w_2=1, w_1∈\0.1,0.2,…,0.9\, w_2=1-w_1. (1) For non-tie mixtures, we define the phase with the larger weight fraction as the dominant phase and the other as the minor phase. Mixtures with w1=w2=0.5w_1=w_2=0.5 have no dominant–minor assignment. Each combination of a phase pair and weight fraction defines a candidate mixture. We retain candidate mixtures that either correspond to equilibrium two-phase assemblages between adjacent stable phases on the binary convex hull, or whose mixture energy lies within 50 meV/atom of the convex hull at the corresponding overall composition. In the latter case, each constituent phase is also required to satisfy the corresponding stability criterion. We simulate the constituent PXRD profiles over 2θ∈[5∘,90∘]2θ∈[5 ,90 ] using Cu Kα radiation. To convert the phase weight fractions into diffraction scale factors, let MiM_i and ViV_i denote the unit-cell mass and volume of phase i∈1,2i∈\1,2\. Using the Hill–Howard relation between phase weight fraction and Rietveld scale factor [Hill and Howard, 1987], we define s~i=wiMiVi,si=s~is~1+s~2. s_i= w_iM_iV_i, s_i= s_i s_1+ s_2. (2) Given constituent profiles x1(2θ)x_1(2θ) and x2(2θ)x_2(2θ), we construct the mixed profile by linear superposition, x~mix(2θ)=s1x1(2θ)+s2x2(2θ), x_mix(2θ)=s_1x_1(2θ)+s_2x_2(2θ), (3) followed by maximum-intensity normalization, xmix(2θ)=x~mix(2θ)max2θ′x~mix(2θ′).x_mix(2θ)= x_mix(2θ) _2θ x_mix(2θ ). (4) Each record stores the mixed and constituent profiles, both crystal structures, phase weight fractions, Rietveld scale factors, compositions, and primitive-cell atom counts. The final dataset contains 100,798 training, 12,418 validation, and 11,848 test mixtures (125,064 total). For the reported recovery experiment, we use the 10,657 non-tie test mixtures. A.3 Evaluation Metrics Top-K match rate. We evaluate structure recovery using the Top-K match rate. For a dataset D containing |||D| target structures, let CnC_n denote the reference structure for example n, and let C^n(j) C_n^(j) denote the j-th generated candidate. We define the per-example Top-K success indicator as mn(K)=[∃j∈1,…,K:C^n(j)≃Cn],m_n^(K)=I [∃ j∈\1,…,K\: C_n^(j) C_n ], (5) where ≃ denotes structural equivalence under pymatgen’s StructureMatcher. The Top-K match rate is then Top-K=1||∑n=1||mn(K).Top-K= 1|D| _n=1^|D|m_n^(K). (6) Thus, an example is counted as successfully recovered if at least one of its first K generated candidates matches the reference structure. For all single-phase evaluations, we use StructureMatcher with (ltol,stol,angle_tol)=(0.2,0.3,5∘)( ltol, stol, angle\_tol)=(0.2,0.3,5 ). We report Top-1 and Top-20 match rates throughout the benchmark. Multiphase recovery metrics. For the two-phase benchmark, each example n contains two reference structures, Cn,1C_n,1 and Cn,2C_n,2. Let C^n,i(j) C_n,i^(j) denote the j-th generated candidate for phase i∈1,2i∈\1,2\. We define the per-phase Top-K success indicator as mn,i(K)=[∃j∈1,…,K:C^n,i(j)≃Cn,i].m_n,i^(K)=I [∃ j∈\1,…,K\: C_n,i^(j) C_n,i ]. (7) Let dnd_n and rnr_n denote the dominant and minor phases of example n, respectively. The dominant-phase and minor-phase Top-K match rates are Top-Kdom=1|multi|∑n=1|multi|mn,dn(K),Top-K_dom= 1|D_multi| _n=1^|D_multi|m_n,d_n^(K), (8) and Top-Kmin=1|multi|∑n=1|multi|mn,rn(K).Top-K_min= 1|D_multi| _n=1^|D_multi|m_n,r_n^(K). (9) Finally, the all-phase Top-K match rate requires both constituent structures to be recovered within their respective candidate sets: Top-Kall=1|multi|∑n=1|multi|mn,dn(K)mn,rn(K).Top-K_all= 1|D_multi| _n=1^|D_multi|m_n,d_n^(K)m_n,r_n^(K). (10) The all-phase metric is therefore a joint, sample-level success rate rather than an average of the dominant- and minor-phase match rates. We report dominant-, minor-, and all-phase Top-1 and Top-20 match rates. A.4 Existing datasets used in additional experiments This subsection documents the existing datasets used in our additional experiments. We describe MP-20, which serves as a compact reference training corpus for comparisons with prior PXRD-to-structure models, and MP-20-polymorph-split, which we use for supplementary evaluation under a composition-disjoint split. MP-20. MP-20 is a widely used benchmark for crystal structure prediction derived from the Materials Project [Jain et al., 2013, Jiao et al., 2023]. It contains inorganic crystal structures with at most 20 atoms in the primitive cell and provides standard train, validation, and test splits. We use MP-20 as a compact reference training corpus for comparisons with prior PXRD-to-structure models and for the representation and augmentation studies reported in the main text and appendix. For experimental PXRD evaluation, we remove from the training set all structures that structurally match reference structures in RRUFF or opXRD. MP-20-polymorph-split. MP-20-polymorph-split is a composition-grouped repartition of MP-20 introduced by Martirossyan et al. [2025]. Whereas the standard MP-20 split can place structures with the same reduced composition in different subsets, MP-20-polymorph-split assigns all structures sharing a reduced composition to the same split while preserving the overall composition-arity distribution. It therefore removes composition overlap between training and evaluation while retaining the underlying MP-20 structure collection. Appendix B Additional experimental results B.1 PXRD representation study We study the design choices for PXRD input representation and encoding in three stages. First, we compare alternative representations of the physical peak position using 2θ2θ, d-spacing, and q-spacing, with intensity as the only additional peak feature. Second, starting from the best peak-position representation, we examine the effect of adding peak width and prominence. Third, we augment the resulting peak features with Fourier encodings of the physical peak position and compare different frequency ranges. Throughout the study, we keep the model architecture and size fixed, train each model on MP-20 for 200K steps, and use our full PXRD profile augmentation. We compare the representations using Top-1 match rates on the RRUFF and opXRD validation sets. Results are summarized in Table 6. Table 6: Top-1 validation match rates (%) for ablations of the PXRD input representation and peak-position encoding. The first three rows compare peak-position representations using intensity only. Subsequent rows cumulatively augment the q,Iq,I representation with peak prominence and width, followed by Fourier embeddings of the peak position. Input feature MP-20 RRUFF opXRD Peak-position representation 2θ,I2θ,I 60.94 36.67 ± 1.860 36.67 ± 6.180 d,Id,I 59.96 39.17 ± 2.280 35.83 ± 2.280 q,Iq,I 64.84 41.25 ± 1.740 37.50 ± 2.950 Additional peak features ++ prominence, width 66.21 43.33 ± 2.280 41.67 ± 4.170 Peak-position Fourier embedding ++ log16 69.99 47.92 ± 1.250 45.56 ± 3.940 ++ log64 69.73 44.86 ± 1.460 40.83 ± 2.500 Peak-position representation. We first compare 2θ2θ, d-spacing, and q-spacing as representations of the physical peak position, using peak intensity I as the only additional feature. Among the three representations, (q,I)(q,I) gives the strongest validation performance overall. Unlike 2θ2θ, both d- and q-spacing are independent of the measurement wavelength, avoiding the need to represent the same scattering geometry differently across measurement wavelengths. Based on the validation results, we use q as the peak-position representation for the remainder of the study. Peak-shape features. Starting from the (q,I)(q,I) representation, we additionally provide the width w and prominence p of each detected peak. These descriptors characterize local peak shape and distinguish peaks that may have similar positions and intensities but different profiles. Adding width and prominence improves validation performance on both RRUFF and opXRD, indicating that peak-shape information provides useful conditioning beyond peak positions and intensities alone. We therefore use (q,I,w,p)(q,I,w,p) as the base peak representation for the following experiments. Fourier encoding of physical peak positions. We further augment the representation with Fourier features of the physical peak position q. This encoding operates on the value of q itself rather than on the index of a peak in the input sequence, and is therefore distinct from positional encoding for token order. We use 16 logarithmically spaced frequencies, fk=Fmaxk/15,k=0,…,15,f_k=F_ ^k/15, k=0,…,15, (11) where Fmax=16F_ =16 and Fmax=64F_ =64 define the log16 and log64 settings, respectively. Thus, both settings use 16 frequencies spanning 11 to FmaxF_ on a logarithmic scale. Further details of the Fourier encoding and peak embedding are provided in Sec. C.2. As shown in Table 6, adding Fourier features with the log16 setting improves validation performance on both RRUFF and opXRD and gives the best overall results. Increasing the maximum frequency to 64 reduces performance on both datasets. We therefore use the log16 Fourier encoding in the final model. B.2 Architecture ablation We ablate three architectural components of Bragglite: in-context conditioning, the PXRD encoder architecture, and the Geometry Enhancement Module (GEM). To isolate architectural effects from simulation-to-real robustness, we train all models on MP-20 for 300K steps without PXRD augmentation and evaluate them on the clean simulated MP-20 validation set. Table 7: Architecture component study on the MP-20 validation set. Top-K match rates (%) are reported as mean ± sample standard deviation over three training seeds. All models are evaluated at the final 300K-step EMA checkpoint. Top-1 Top-20 Top-1 (p) Bragglite 78.86 ± 0.400 90.62 ± 0.400 – −- In-context fusion 78.64 ± 0.340 90.33 ± 0.280 −0.22-0.22 −- GEM 76.35 ± 0.590 89.90 ± 0.050 −2.51-2.51 Table 8: PXRD encoder study on the MP-20 validation set. Values are Top-K match rates (%). PXRD encoder Top-1 Top-20 Transformer encoder 75.37 89.04 MLP encoder 72.70 87.96 In-context fusion and GEM. Table 7 evaluates the contribution of in-context PXRD conditioning and GEM. Removing in-context fusion produces a small reduction in both Top-1 and Top-20 match rates, whereas removing GEM leads to a larger drop, particularly in Top-1 performance. These results indicate that the geometry-aware attention bias provided by GEM is important for structure recovery, while in-context fusion provides a smaller additional improvement. PXRD encoder architecture. Table 8 compares Transformer and MLP encoders for the PXRD peak sequence. The Transformer encoder improves both Top-1 and Top-20 match rates over the MLP encoder, supporting the use of a sequence-modeling architecture for constructing the PXRD representation used to condition the structure generator. B.3 In-depth analysis of benchmark results Figure 8: Overlap of Top-1 successes between Bragglite and prior baselines on the RRUFF and opXRD test sets. Each bar partitions the test samples into cases where both models succeed, only the baseline succeeds, only Bragglite succeeds, or neither succeeds. Retained denotes the fraction of baseline successes that are also recovered by Bragglite. Overlap of successful predictions. To better understand whether the improvement of Bragglite comes from solving different test cases or consistently recovering those solved by prior methods, we analyze the overlap of Top-1 successes on the RRUFF and opXRD test sets in Figure 8. For each baseline, we partition test samples according to whether both models succeed, only the baseline succeeds, only Bragglite succeeds, or neither succeeds. Across most baselines, Bragglite recovers the large majority of samples solved by the baseline while additionally solving a substantially larger set of previously unsuccessful cases. This trend is consistent across both experimental datasets, suggesting that the improved aggregate match rate primarily reflects additional structure recoveries rather than a trade-off in which previously solved samples are lost. The overlap is lower for XRDiff under element-set conditioning, indicating greater complementarity between the two models in this more weakly conditioned setting. B.4 Detailed two-phase recovery results We further report phase-wise recovery results for the two-phase benchmark, separating dominant-phase, minor-phase, and joint all-phase recovery. Table 9 complements the all-phase results in Sec. 5.3 by showing how often each constituent phase is recovered individually. Table 9: Detailed two-phase recovery results. Dominant-phase, minor-phase, and all-phase Top-K match rates (%) on the MP-20 and GALAXI two-phase benchmarks. All-phase recovery requires both constituent phases to be recovered within their respective candidate sets. Dominant Minor All-phase Dataset Model Top-1 Top-20 Top-1 Top-20 Top-1 Top-20 MP-20 MatterGen 14.28 55.69 15.48 55.41 2.13 30.67 XRDiff 25.28 38.28 23.14 36.48 6.76 14.64 Bragglite 33.99 48.78 30.59 45.02 13.26 25.34 GALAXI MatterGen 0.00 76.19 4.76 71.43 0.00 42.86 XRDiff 28.57 38.10 9.52 14.29 0.00 9.52 Bragglite 57.14 76.19 33.33 66.67 19.05 47.62 B.5 Benchmark results on MP-20-polymorph-split Table 10: Top-K match rates (%) for models trained on MP-20-polymorph-split, following the MP-20 configuration in Table 1. Simulated PXRD RRUFF opXRD Dataset Model Top-1 Top-20 Top-1 Top-20 Top-1 Top-20 MP-20- polymorph- split Chemical input: exact stoichiometry PXRDGen 55.43 78.92 30.00 ± 3.080 59.00 ± 1.410 24.29 ± 1.600 36.79 ± 2.040 RealPXRD-Solver 57.42 79.54 26.60 ± 2.610 47.80 ± 2.680 24.64 ± 1.490 33.21 ± 1.600 XRDiff 60.26 77.54 23.00 ± 2.350 45.60 ± 1.520 18.93 ± 2.710 29.29 ± 1.600 Crystalyze 31.07 49.46 02.00 ± 1.410 11.00 ± 1.220 11.79 ± 0.980 17.50 ± 0.800 DiffCSP (no PXRD) 51.85 75.71 29.00 ± 3.160 58.60 ± 1.520 20.71 ± 1.600 36.07 ± 0.800 Crystalite (no PXRD) 62.28 79.45 41.20 ± 3.560 60.40 ± 1.820 25.36 ± 1.490 35.71 ± 1.260 Bragglite 68.43 81.48 49.00 ± 1.220 65.40 ± 1.340 27.86 ± 2.040 38.57 ± 2.040 Chemical input: element set + N XRDiff 35.92 50.79 15.60 ± 1.520 26.60 ± 1.670 10.36 ± 1.490 17.14 ± 0.980 Bragglite 44.20 69.76 30.60 ± 1.520 55.20 ± 1.920 17.86 ± 1.260 23.93 ± 1.600 Table 10 follows the model lineup, chemical-input settings, and evaluation columns of the MP-20 block in Table 1, using MP-20-polymorph-split as the training dataset and its held-out test split for simulated PXRD evaluation. Appendix C Bragglite model details This section provides additional implementation details for Bragglite. We first summarize the Crystalite backbone, then describe the PXRD representation, encoder, in-context conditioning, and augmentation pipeline. Finally, we describe how Bragglite is combined with XDecomposer for two-phase structure recovery. Bragglite builds on Crystalite [Hadži Veljković et al., 2026] and preserves its diffusion state, EDM formulation, sampler, and structural prediction heads. To condition the denoiser on diffraction, we introduce a PXRD encoder that maps the observed peak list to a short sequence of context tokens. These tokens are concatenated with the structural tokens and jointly processed by the Crystalite Transformer, allowing the observed diffraction pattern to influence structural predictions throughout denoising. C.1 Crystalite backbone Crystalite [Hadži Veljković et al., 2026] is a geometry-aware diffusion Transformer for crystal generation. It represents a unit cell as a continuous state consisting of atom-type features, fractional coordinates, and a six-dimensional lattice latent. Each atomic site is represented by a single token that combines its atom-type and fractional-coordinate embeddings, while the lattice is represented by a separate global token. The resulting sequence is processed by a noise-conditioned Transformer under the EDM denoising formulation. To explicitly incorporate periodic crystal geometry, Crystalite uses the Geometry Enhancement Module (GEM). GEM computes minimum-image pair geometry from the current fractional coordinates and lattice and injects the resulting features into atom–atom attention as additive geometric biases. This provides the Transformer with explicit lattice-dependent periodic geometry during denoising. For atom types, Crystalite uses Subatomic Tokenization, which represents each element using chemically structured descriptors derived from its periodic-table position and electronic configuration. These descriptors are projected onto a PCA basis and normalized to form continuous atom-type features. During de novo generation, atom-type features, coordinates, and lattice variables are jointly denoised, and the predicted atom-type features are mapped back to discrete elements by nearest-prototype decoding. In the CSP setting, atom types are fixed and only the coordinates and lattice are denoised. These two operating modes align naturally with our benchmark settings. Under exact-stoichiometry conditioning, atom types are fixed throughout generation. Under element-set conditioning, atom-type features are additionally denoised and decoded while restricting the output elements to the supplied element set. C.2 PXRD encoding and in-context conditioning Peak extraction. For both simulated and experimental PXRD, we begin with a dense intensity profile =(x1,…,xM)∈ℝ≥0M,M=850,x=(x_1,…,x_M) _≥ 0^M, M=850, (12) sampled at 0.1∘0.1 intervals over 2θ∈[5∘,90∘)2θ∈[5 ,90 ). We detect peaks from x using scipy.signal.find_peaks [Virtanen et al., 2020] with prominence=0.0 and no additional minimum-height or minimum-distance constraint. Peak widths are computed using scipy.signal.peak_widths at relative height 0.50.5 and converted from bins to the units of the corresponding peak-position representation. If more than 128 peaks are detected, we retain the 128 peaks with the highest intensities and sort them by increasing 2θ2θ. Let K≤128K≤ 128 denote the number of retained peaks. We represent the resulting peak sequence as x=(1,…,K),k=(qk,Ik,wk,πk)∈ℝ4,P_x=(p_1,…,p_K), _k= (q_k,I_k,w_k, _k ) ^4, (13) where qkq_k, IkI_k, wkw_k, and πk _k denote the scattering-vector magnitude, normalized intensity, peak width, and prominence, respectively. Let I~k I_k denote the unnormalized intensity of peak k. We normalize peak intensities by the maximum value of the full profile, Ik=I~kI~max,I~max=maxjxj.I_k= I_k I_ , I_ = _jx_j. (14) Fourier encoding for peak positions. To provide a richer representation of the physical peak position, we additionally encode qkq_k using sinusoidal Fourier features. We first convert qkq_k to the dimensionless coordinate ξk=qkλref4π,λref=1.54184Å. _k= q_k _ref4π, _ref=1.54184 \ . (15) We then use 16 logarithmically spaced frequencies fm=Fmaxm/15,m=0,…,15.f_m=F_ ^m/15, m=0,…,15. (16) For each peak, the corresponding 32-dimensional Fourier feature is ϕ(qk)=[sin(2πfmξk),cos(2πfmξk)]m=015∈ℝ32. φ(q_k)= [ (2π f_m _k), (2π f_m _k) ]_m=0^15 ^32. (17) This encoding provides multiple frequency scales for representing variations in physical peak position while retaining qkq_k itself in the base peak descriptor. Peak encoding. The four base peak features are jointly embedded with a two-layer MLP, while the normalized Fourier feature is projected through a separate linear layer. The two branches are summed to obtain the peak embedding, k=MLPpeak(k)+FNormalize(ϕ(qk))∈ℝdx,dx=256,h_k=MLP_peak(p_k)+W_FNormalize ( φ(q_k) ) ^d_x, d_x=256, (18) where MLPpeak=Lineardx→dx∘SiLU∘Linear4→dx,MLP_peak=Linear_d_x→ d_x _4→ d_x, (19) and F∈ℝdx×32W_F ^d_x× 32 denotes the bias-free Fourier projection. Profile encoding. We prepend a learned [CLS] embedding CLS∈ℝdxc_CLS ^d_x to the peak embeddings and process the resulting sequence using a four-layer, eight-head Transformer: x=PeakTransformer([CLS;1;…;K])∈ℝ(K+1)×dx.U_x=PeakTransformer ([c_CLS;h_1;…;h_K] ) ^(K+1)× d_x. (20) The output corresponding to the [CLS] token provides a global representation of the observed PXRD profile. We project this representation to the hidden dimension of the Crystalite backbone, x=Linearproj(x,CLS)∈ℝd.z_x=Linear_proj (U_x,CLS ) ^d. (21) In-context conditioning. We form R=8R=8 context tokens by repeating the global PXRD representation and adding a distinct learned positional embedding to each copy: x=[x+1;…;x+R]∈ℝR×d,R=8,Z_x= [z_x+e_1;…;z_x+e_R ] ^R× d, R=8, (22) where r∈ℝde_r ^d is a learned positional embedding for the r-th context token. We prepend these context tokens to the original Crystalite structural token sequence, (0)=[x;str(0)]∈ℝ(R+N+1)×d,T^(0)=[Z_x;T_str^(0)] ^(R+N+1)× d, (23) where str(0)∈ℝ(N+1)×dT_str^(0) ^(N+1)× d is the original Crystalite structural sequence, consisting of the N atomic-site tokens and one global lattice token. The PXRD context depends only on the observed profile and is therefore reused across diffusion steps, whereas str(0)T_str^(0) is recomputed from the current noisy crystal. Within each denoiser evaluation, the context and structural tokens are jointly processed by the Crystalite Transformer, allowing the atom and lattice tokens to attend to the diffraction representation throughout denoising. C.3 PXRD augmentation Overview. For simulated training data, we augment PXRD profiles online to mimic distortions commonly observed in experimental measurements. Starting from Bragg reflections computed for each crystal structure under Cu Kα radiation (λ=1.54184λ=1.54184 Å), we first apply reflection-level perturbations to peak presence, position, and intensity. We then render the perturbed reflections into a dense profile using randomized pseudo-Voigt peak shapes and apply profile-level background and noise augmentation. The resulting profile is converted to the peak representation using the extraction procedure described in Sec. C.2. All random augmentation parameters are resampled whenever a training example is loaded. Weak-peak removal. Weak-reflection removal has been used in prior PXRD augmentation schemes, including RealPXRD-Solver [Li et al., 2026], to model weak peaks that may be missing or unresolved in experimental patterns. We randomly remove a fraction of the lowest-intensity Bragg reflections during training. Let the simulated pattern contain M reflections. We sample rdrop∼(0,rdropmax),ndrop=⌊rdropM⌋,r_drop (0,r_drop ), n_drop= r_dropM , (24) and remove the ndropn_drop weakest reflections while retaining at least one reflection. In our experiments, we use rdropmax=0.2r_drop =0.2. Peak-position jitter. Following prior experiment-inspired PXRD augmentation [Li et al., 2026], we independently perturb the position of each retained reflection to model local peak-position errors: 2θi′=2θi+δi,δi∼(−Δjitter,Δjitter).2 _i =2 _i+ _i, _i (- _jitter, _jitter). (25) Unlike the shared zero shift described below, these perturbations are sampled independently for individual reflections. We use Δjitter=0.2∘ _jitter=0.2 . Peak-intensity scaling. Prior PXRD augmentation schemes perturb relative peak intensities to improve robustness to experimental intensity distortions, including effects associated with preferred orientation [Li et al., 2026]. We independently scale each retained reflection as Ii′=siIi,si∼(smin,smax).I_i =s_iI_i, s_i (s_ ,s_ ). (26) The perturbed intensities are then jointly rescaled so that their maximum matches that of the original reflection list. We use (smin,smax)=(0.6,1.4)(s_ ,s_ )=(0.6,1.4). This independent intensity scaling provides a simple approximation to experimental variations in relative peak intensities, including those that may arise from preferred orientation, but does not constitute a crystallographic preferred-orientation model because it does not depend on reflection orientation. Zero shift. Global angular shifts have been used in physics-informed XRD augmentation to model systematic measurement offsets [Oviedo et al., 2019]. We apply a single shift shared by all reflections in a profile, z∼(−Δzero,Δzero),2θi′=2θi′+z.z (- _zero, _zero), 2 _i =2 _i +z. (27) This perturbation models coherent offsets such as zero-point calibration or sample-displacement errors, in contrast to the independent peak-position jitter above. We use Δzero=0.3∘ _zero=0.3 . Pseudo-Voigt broadening. Prior PXRD models have incorporated peak broadening and profile-level augmentation to improve robustness to experimental diffraction patterns [Segal et al., 2026a]. We render the perturbed Bragg reflections into a dense profile using pseudo-Voigt peak shapes. The base full width at half maximum of reflection i follows the Caglioti form Hi(0)=Utan2θi+Vtanθi+W.H_i^(0)= U ^2 _i+V _i+W. (28) We use (U,V,W)=(0.1,0.01,0.1).(U,V,W)=(0.1,0.01,0.1). (29) To vary the overall broadening across profiles, we sample a profile-wide width scale sw∼(swmin,swmax),Hi=swHi(0).s_w (s_w ,s_w ), H_i=s_wH_i^(0). (30) We use (swmin,swmax)=(0.4,2.2)(s_w ,s_w )=(0.4,2.2). We additionally sample one pseudo-Voigt mixing coefficient per profile, η∼(0,ηmax),η (0, _ ), (31) with ηmax=0.8 _ =0.8, and render each reflection as Pi(t)=Ii′[(1−η)exp(−4ln2(t−2θi′Hi)2)+η11+4(t−2θi′Hi)2], splitP_i(t)=I_i [&(1-η) (-4 2 ( t-2 _i H_i )^2 )\\ &+η 11+4 ( t-2 _i H_i )^2 ], split (32) where t denotes the 2θ2θ coordinate. The broadened profile is P(t)=∑iPi(t)P(t)= _iP_i(t). Background augmentation. Smooth background variation is a common experimental artifact and has been included in physics-informed XRD augmentation pipelines [Oviedo et al., 2019]. We normalize the profile coordinate to u∈[−1,1]u∈[-1,1] and add a quadratic background B(u)=Amax(0,b0+b1u+b2u2).B(u)=A (0,b_0+b_1u+b_2u^2 ). (33) We sample A A ∼(0,Amax), (0,A_ ), b0 b_0 ∼(0,1), (0,1), (34) b1 b_1 ∼(−0.5,0.5), (-0.5,0.5), b2 b_2 ∼(0,0.5), (0,0.5), and use Amax=0.35A_ =0.35. Gaussian noise. Additive noise is also commonly used to approximate measurement noise in experiment-inspired PXRD augmentation [Oviedo et al., 2019, Li et al., 2026]. We sample one noise scale per profile, σn∼(0,σnmax),ϵj∼(0,σn2), _n (0, _n ), _j (0, _n^2), (35) and use σnmax=0.04 _n =0.04. The final augmented profile is P~j=normalizemax[max(0,Pj+Bj+ϵj)]. P_j=normalize_ [ (0,P_j+B_j+ _j ) ]. (36) The resulting profile is processed using the same peak-extraction and encoding pipeline described in Sec. C.2. C.4 Multiphase recovery pipeline We construct the two-phase recovery pipeline by fine-tuning both XDecomposer [Gao et al., 2026] and the element-set-conditioned Bragglite on the simulated multiphase training set. Ordered phase decomposition. Starting from the publicly available XDecomposer checkpoint, we fine-tune the model to produce an ordered decomposition of each mixed PXRD profile, with the first output corresponding to the dominant phase and the second to the minor phase. Given a mixed profile mixx_mix, the fine-tuned decomposer predicts (1,2)=XDecomposer(mix),(x_1,x_2)=XDecomposer(x_mix), (37) where 1x_1 and 2x_2 are the predicted dominant- and minor-phase profiles, respectively. This ordering allows each decomposed profile to be paired with the oracle primitive-cell atom count of the corresponding phase, denoted by N1N_1 and N2N_2. Phase-wise structure recovery. We additionally fine-tune the element-set-conditioned Bragglite on the multiphase training set so that it predicts each constituent structure from a phase-specific PXRD profile while receiving only mixture-level chemical information. Let ℰmix=ℰ1∪ℰ2E_mix=E_1 _2 (38) denote the element set of the full mixture. For each phase i∈1,2i∈\1,2\, the structure generator receives the decomposed profile ix_i, the shared mixture-level element set ℰmixE_mix, and the oracle atom count NiN_i: Ci∼pθ(⋅∣i,ℰmix,Ni).C_i p_θ (· _i,E_mix,N_i ). (39) The two phases are generated independently using the same fine-tuned model. At evaluation time, we therefore first decompose the mixed profile into ordered dominant- and minor-phase profiles, pair them with N1N_1 and N2N_2, respectively, and then apply Bragglite independently to recover the two structures. Importantly, the structure generator is not given phase-specific element sets or compositions; it must infer which elements from ℰmixE_mix belong to each phase, together with their multiplicities, atomic coordinates, and lattice. Reference methods. For XRDiff, we use the same ordered XDecomposer outputs as for Bragglite and apply the structure generator independently to the predicted dominant- and minor-phase profiles. XRDiff additionally receives the phase-specific element set ℰiE_i and oracle atom count NiN_i for each phase, since we do not fine-tune XRDiff to condition on the mixture-level element set. MatterGen instead receives the shared mixture-level element set ℰmixE_mix and oracle atom count NiN_i for each phase, but does not condition on the decomposed PXRD profiles. For each phase and each reference method, we generate 20 candidate structures and evaluate dominant-, minor-, and all-phase recovery using the protocol in Sec. A.3. Appendix D Experimental details In this section, we provide detailed experimental settings for the results reported in the main text, including our model architecture, training and evaluation configurations, and the training and evaluation settings of the benchmark baselines. D.1 Model configurations We train separate Bragglite models for the known-composition and element-set settings. All models use Crystalite’s PCA-16 subatomic type representation and Geometry Enhancement Module (GEM). PXRD observations are represented by up to 128 intensity-selected peaks, with peak extraction and preprocessing detailed in Sec. C.2. PXRD encoder. We use a four-layer Transformer with hidden dimension 256 and eight attention heads as our PXRD encoder. Each peak is represented by (q,I,Δq,prominence)(q,I, q,prominence), where q=2π/dq=2π/d and Δq q is obtained by converting the detected angular peak width to q space using the Jacobian at the peak center. We additionally encode q using 16 logarithmically spaced Fourier frequencies with Fmax=16F_ =16, yielding a 32-dimensional Fourier encoding. The encoder uses [CLS] pooling and a padding mask for variable-length peak sequences, with no dropout. The resulting PXRD embedding is projected to the Crystalite backbone width and injected into the token sequence for in-context conditioning. Backbone architecture. We use the same Bragglite backbone configuration for models trained on PXRD-Bench and MP-20, with hidden dimension 512, 14 Transformer layers, and 16 attention heads. Each Transformer block uses a feed-forward expansion ratio of 4 with GELU activations, pre-LayerNorm without affine parameters, and no attention or feed-forward dropout. The backbone also applies an affine-free LayerNorm before the output heads. Time conditioning is incorporated through adaLN-Zero, while PXRD conditioning is provided in context as described above. Coordinate embedding. Fractional atomic coordinates are first wrapped to [0,1)[0,1) and encoded independently along each coordinate axis using integer Fourier frequencies k=1,…,32k=1,…,32: ϕ()=[sin(2πkfj),cos(2πkfj)]j=1,2,3;k=1,…,32.φ(f)= [ (2π kf_j), (2π kf_j) ]_j=1,2,3;\,k=1,…,32. (40) This produces a 192-dimensional feature vector, which is mapped to the backbone hidden dimension by 192→512→512192→ 512→ 512 with a SiLU activation between the two linear layers. This coordinate encoding is separate from the Fourier encoding used for PXRD peak positions. Noise conditioning. Following the EDM parameterization, we first transform the noise level as cnoise=14logσ.c_noise= 14 σ. (41) We encode cnoisec_noise using 128 fixed sinusoidal frequencies ωj=10000−j/128 _j=10000^-j/128 for j=0,…,127j=0,…,127, producing a 256-dimensional embedding from sin(2πcnoiseωj) (2π c_noise _j) and cos(2πcnoiseωj) (2π c_noise _j). In our single-σ setting, the same 256-dimensional embedding is supplied to both noise-input slots and concatenated into a 512-dimensional vector. A two-layer MLP, 512→512→512512→ 512→ 512, with a SiLU activation maps this vector to the noise-conditioning embedding used for adaLN-Zero modulation in each Transformer block. Geometry Enhancement Module. We use a single GEM shared across all Transformer layers. GEM computes head-specific attention biases once from the current noisy coordinates, lattice, and noise level, and reuses them across all 14 backbone layers. The distance bias, edge-bias MLP, and noise gate are therefore shared across layers. Atom ordering and positional encoding. For the element-set model, we follow the atom-ordering scheme introduced in MCFlow [Seong et al., 2026], which organizes atoms according to their symmetry orbits. We first identify symmetry orbits using SpacegroupAnalyzer with symprec=0.1Å symprec=0.1\, , grouping symmetry-related atoms of the same element into individual orbits. We then order the orbit blocks first by increasing Pauling electronegativity of the element and then by Wyckoff letter, while keeping atoms within each orbit contiguous. During training, we apply a hierarchical orbit permutation to avoid dependence on an arbitrary atom enumeration. At the inter-orbit level, orbit blocks with the same element and the same Wyckoff label, including multiplicity, are randomly permuted without merging distinct orbits. At the intra-orbit level, atoms within each orbit block are independently permuted. The same permutation is applied jointly to atom types and fractional coordinates, while padding positions remain unchanged. Thus, the procedure changes only the ordering of atomic tokens and leaves the underlying crystal structure and PXRD pattern invariant. These random permutations are applied during training and disabled during validation and testing. Together, this ordering and permutation scheme provides a consistent element- and symmetry-aware organization while avoiding dependence on any single equivalent atom enumeration. The element-set model further applies sinusoidal atom-slot positional encodings after ordering and permutation. These encodings specify atomic token positions and are distinct from the Fourier encoding of fractional atomic coordinates. Output heads. The backbone predicts atom-type, coordinate, and lattice outputs using separate bias-enabled linear heads. The atom-type and coordinate heads operate on each atomic token and map 512→16512→ 16 and 512→3512→ 3, respectively, while the lattice head maps the lattice token from 512→6512→ 6 to predict the lower-triangular lattice latent. No additional activation is applied after these heads. Their raw outputs are converted to denoised estimates using EDM preconditioning, Dθ(x,σ)=cskip(σ)x+cout(σ)Fθ(⋅),D_θ(x,σ)=c_skip(σ)x+c_out(σ)F_θ(·), (42) where cskip(σ)=σdata2σ2+σdata2,cout(σ)=σσdataσ2+σdata2.c_skip(σ)= _data^2σ^2+ _data^2, c_out(σ)= σ\, _data σ^2+ _data^2. (43) Here, σdata _data denotes the data-scale parameter used in EDM preconditioning. Thus, the model heads predict continuous denoising outputs rather than categorical atom-type logits or coordinate velocities. Table 11 summarizes the model configurations. Table 11: Bragglite model configurations. All configurations are shared between the known-composition and element-set models unless specified otherwise. Parameter counts include the PXRD encoder and conditioning modules. Component Known composition Element set +N+N Backbone Hidden dimension 512 Transformer layers 14 Attention heads 16 FFN expansion ratio 4 FFN activation GELU Normalization Pre-LN LayerNorm affine / ϵε No / 10−610^-6 Attention / FFN dropout 0 / 0 Final normalization Affine-free LN Time conditioning adaLN-Zero PXRD conditioning In-context Coordinate encoding Fourier (kmax=32k_ =32) Noise embedding Input 14logσ 14 σ Sinusoidal frequencies / dimension 128 / 256 MLP →512512\!→\!512\!→\!512 Activation SiLU Geometry Enhancement Module Sharing across layers Shared Distance / edge bias Yes Noise gate Yes PXRD encoder Hidden dimension 256 Transformer layers 4 Attention heads 8 Pooling [CLS] Dropout 0 Peak position encoding Fourier (Fmax=16F_ =16, d=32d=32) Output heads Coordinate →3512\!→\!3 Lattice →6512\!→\!6 Head activation None Setting-specific configuration Atom-type head Unused →16512\!→\!16 Atom ordering None MCFlow Atom-slot positional encoding None Sinusoidal Parameters 71.01M 71.32M D.2 Training objective and optimization Training objective. All Bragglite models are trained with the EDM denoising objective used by the Crystalite backbone. We combine the atom-type, fractional-coordinate, and lattice losses as ℒ=λHℒH+λFℒF+λLℒL,L= _HL_H+ _FL_F+ _LL_L, (44) where ℒHL_H, ℒFL_F, and ℒLL_L denote the atom-type, fractional-coordinate, and lattice denoising losses, respectively. For the known-composition model, atom types remain fixed throughout diffusion, so the atom-type loss is inactive and we use (λH,λF,λL)=(0,50,5)( _H, _F, _L)=(0,50,5). For the element-set model, atom types are jointly denoised with coordinates and lattice parameters, and we use (64,150,5)(64,150,5). The two settings also use different EDM noise distributions. For known composition, we sample logσ∼(−1.2,1.22), σ (-1.2,1.2^2), (45) with (σmin,σmax)=(0.002,80)( _ , _ )=(0.002,80) for coordinates and lattice variables; atom types remain fixed and are not noised. For element-set conditioning, we instead use logσ∼(−1.2,1.52), σ (-1.2,1.5^2), (46) with (σmin,σmax)=(0.001,40)( _ , _ )=(0.001,40) for atom types, coordinates, and lattice variables. Table 12 summarizes these objective settings. Table 12: Diffusion-objective settings. Loss weights and EDM noise-distribution parameters for the two chemical-input settings. Known composition Element set +N+N Loss weights (λH,λF,λL)( _H, _F, _L) (0,50,5)(0,50,5) (64,150,5)(64,150,5) EDM (Pmean,Pstd)(P_mean,P_std) (−1.2,1.2)(-1.2,1.2) (−1.2,1.5)(-1.2,1.5) EDM (σmin,σmax)( _ , _ ) (0.002,80)(0.002,80) (0.001,40)(0.001,40) Optimization. We train all models with AdamW [Loshchilov and Hutter, 2019] using (β1,β2)=(0.9,0.999)( _1, _2)=(0.9,0.999) and zero weight decay. We clip the global gradient norm at 1.0, use bfloat16 precision, and maintain an exponential moving average (EMA) of the model parameters with decay 0.99990.9999. Learning-rate settings depend on the chemical-input setting. The known-composition model uses a learning rate of 2×10−42× 10^-4 with 1,000 linear warmup steps followed by a constant learning rate, whereas the element-set model uses a learning rate of 1×10−41× 10^-4 with 500 warmup steps followed by cosine decay. Table 13 summarizes these conditioning-specific settings. Training budgets depend on the training corpus. Models trained on MP-20, which contains structures with at most 20 atoms, are trained for 800K optimizer steps with a batch size of 512. Models trained on PXRD-Bench, which allows up to 52 atoms, are trained for 4M optimizer steps with a batch size of 256. These budgets correspond to approximately 15K and 250 epochs, respectively. All runs use a single NVIDIA RTX 3090 GPU without gradient accumulation and a training seed of 123. Table 14 summarizes these corpus-specific settings. Table 13: Conditioning-specific optimization settings. Known composition Element set +N+N Learning rate 2×10−42× 10^-4 1×10−41× 10^-4 LR schedule Constant Cosine decay Warmup steps 1,000 500 Table 14: Corpus-specific training settings. MP-20 PXRD-Bench Maximum atom count NmaxN_ 20 52 Training steps 800K 4M Approx. epochs ∼ 15K ∼ 250 Batch size 512 256 GPUs / gradient accumulation 1 / 1 1 / 1 Training seed 123 123 Ablation training settings. Unless otherwise specified, the ablation experiments use the same MP-20 architecture, training objective, and optimization settings described above. Each ablation model is trained from scratch for 200K optimizer steps, corresponding to approximately 3,846 epochs, and comparisons use the final 200K-step EMA checkpoint rather than validation-based checkpoint selection. For the augmentation ablation, we train three independent models with training seeds 123,124,125\123,124,125\. Each trained model is evaluated using five sampling seeds 777,778,779,780,781\777,778,779,780,781\. We first average over the five sampling seeds for each trained model and then report the mean and standard deviation across the three training seeds. For the peak-representation and encoding studies, we instead train a single model with training seed 123123 and evaluate it using the same five sampling seeds. For the representation and encoding studies, we keep the full augmentation setting fixed and vary only the component under study. For the augmentation ablation, we vary the specified augmentation components while keeping the remaining training configuration fixed. Table 15 summarizes the settings that differ from the main training protocol. Table 15: Ablation training settings. Settings that differ from the main training protocol. Setting Value Training steps 200K (∼ 3,846 epochs) Checkpoint Final 200K-step EMA checkpoint Training seed (representation) 123123 Training seeds (augmentation) 123,124,125\123,124,125\ Sampling seeds 777,778,779,780,781\777,778,779,780,781\ per training run Multiphase fine-tuning. We initialize the phase-wise structure generator from the pretrained element-set Bragglite checkpoint and fine-tune it on the simulated multiphase training set. We use AdamW with a learning rate of 1×10−51× 10^-5, 500 linear warmup steps followed by cosine decay, and a batch size of 128 phase examples. We fine-tune the model for 50K optimizer steps using a training seed of 123. XDecomposer is initialized from its publicly released checkpoint and fine-tuned separately on the simulated multiphase training set. We use AdamW with an initial learning rate of 3×10−63× 10^-6, reducing the learning rate by a factor of two when the validation metric plateaus. We use a batch size of 16 mixtures and fine-tune for 100K optimizer steps, corresponding to approximately 21 epochs, with a training seed of 123. For both models, we use the final fine-tuned checkpoint for evaluation. D.3 Sampling and checkpoint selection Sampling. At inference time, we sample from the EMA model using the stochastic Heun sampler from EDM [Karras et al., 2022]. For each input PXRD pattern, we run 150 sampling steps and generate 20 candidate structures. The same sampling procedure is used for models trained on MP-20 and PXRD-Bench. We follow the sampling configuration of Crystalite [Hadži Veljković et al., 2026], including the Karras noise schedule, stochastic churn, and anti-annealing settings. Table 16 summarizes the sampler configuration. Table 16: Sampling configurations. Settings are shared across MP-20 and PXRD-Bench and follow the Crystalite sampling configuration. Setting Known composition Element set +N+N Solver / steps Heun / 150 Noise schedule Karras, ρ=7ρ=7 SchurnS_churn 60 Churn range [Smin,Smax][S_ ,S_ ] [0,999][0,999] SnoiseS_noise 1.003 σdata _data 0.3 Anti-annealing (H,F,L)(H,F,L) (0,10,10)(0,10,10) (4,4,4)(4,4,4) Checkpoint selection. We select checkpoints using a fixed random subset of 512 structures from the simulated PXRD validation set. Every 10K training steps, we evaluate one generated candidate per validation structure and compute the top-1 structure match rate. We report the EMA weights from the checkpoint with the highest validation top-1 match rate rather than automatically using the final training checkpoint. D.4 Baseline configurations We use the publicly released implementation of each baseline rather than reimplementing the models. Across training corpora, all baselines are trained using Adam with (β1,β2)=(0.9,0.999)( _1, _2)=(0.9,0.999), ϵ=10−8ε=10^-8, and no weight decay. We use a training seed of 42, a single NVIDIA GeForce RTX 3090 GPU, no gradient accumulation, and elementwise gradient clipping at a maximum absolute value of 0.5. For all baselines, we maintain exponential moving average (EMA) parameters and use EMA checkpoints for evaluation. PXRD-Bench training. For baselines trained on PXRD-Bench, we use a batch size of 256 and initialize the learning rate to 10−310^-3. We use ReduceLROnPlateau, monitoring the Top-1 structure match rate on a fixed subset of 512 structures from the simulated validation set. The learning rate is reduced by a factor of two if this metric does not improve for five epochs, and training is terminated when the learning rate would fall below 10−510^-5. We select the EMA checkpoint with the highest monitored validation Top-1 match rate for evaluation. MP-20 training. For baselines trained on MP-20 or MP-20-polymorph-split, we use the same optimization protocol but a longer learning-rate patience and a fixed maximum training budget. Starting from a learning rate of 10−310^-3, we use ReduceLROnPlateau with a patience of 30 epochs and a minimum learning rate of 10−510^-5, again monitoring the Top-1 structure match rate on a fixed subset of 512 structures from the corresponding simulated validation set. Training is run for at most 300K optimizer steps, and we use the EMA checkpoint with the highest monitored validation Top-1 match rate for evaluation. DiffCSP. DiffCSP [Jiao et al., 2023] serves as a composition-only CSP baseline and does not receive PXRD input. We condition the model on the exact primitive-cell composition and use a six-layer CSPNet with hidden dimension 512 and a time-embedding dimension of 256. The diffusion process uses 1,000 steps. XRDiff. For XRDiff [Segal et al., 2026a], the PXRD profile is represented on 850 bins over 2θ∈[5∘,90∘]2θ∈[5 ,90 ]. We retain up to the 20 strongest detected peaks and represent each peak using its intensity, position, prominence, and width, yielding an 80-dimensional fixed-size PXRD feature. The structure generator uses a six-layer CSPNet with hidden dimension 512 and FiLM conditioning on the PXRD representation. We do not apply additional stochastic PXRD augmentation during training. We train both the known-composition and element-set +N+N variants; for the latter, the atom types are additionally predicted and included in the training objective. RealPXRD-Solver. RealPXRD-Solver [Li et al., 2026] uses a 750-bin profile over 2θ∈[5∘,80∘]2θ∈[5 ,80 ], from which a Bragg-stick representation is constructed. We use its two-layer Transformer PXRD encoder together with a six-layer CSPNet of hidden dimension 512. For simulated-to-experimental transfer, we follow the PXRD augmentation procedure described in the original method. PXRDGen. PXRDGen [Li et al., 2025] receives a 7,500-bin Bragg-derived PXRD profile over 2θ∈[5∘,80∘]2θ∈[5 ,80 ]. We first contrastively pretrain its PXRD CNN encoder on the PXRD-Bench training corpus and subsequently train the diffusion model while continuing to update the CNN jointly. We use a contrastive temperature of 1.0 and do not use an externally pretrained PXRD encoder. Crystalyze. Crystalyze [Riesel et al., 2024] operates on an 8,500-bin PXRD profile using its PXRD CNN encoder, a latent dimension of 256, and a GemNet-T structure decoder. During training, we sample one of the four provided Caglioti broadening settings and add the publicly released RRUFF-derived Gamma background-noise augmentation. We do not apply an additional peak-intensity modulation. Baseline sampling. For each baseline, we use the default sampling procedure provided by its public implementation. Crystalyze uses 5,000-step annealed Langevin dynamics. DiffCSP, XRDiff, and PXRDGen use 1,000 predictor–corrector sampling steps, corresponding to 2,000 neural network evaluations. RealPXRD-Solver uses a 200-step Euler integrator. D.5 Compute resources All Bragglite models are trained on a single NVIDIA GeForce RTX 3090 GPU with 24 GB of memory. Table 17 reports the training configuration and computational cost of the main models, including batch size, number of optimization steps, wall-clock training time, and peak GPU memory usage. Table 17: Training cost. Wall-clock training time and peak GPU memory usage for the main Bragglite configurations. All models are trained on a single NVIDIA GeForce RTX 3090 GPU. Training data / input Batch size Steps Time (h) Peak memory (GiB) PXRD-Bench / Known composition 256 4M 200.18 7.97 PXRD-Bench / Element set 256 4M 201.29 7.98 MP-20 / Known composition 512 800K 37.10 7.15 MP-20 / Element set 512 800K 37.89 7.16 Appendix E Qualitative examples We provide visualizations of representative predictions from the evaluated models in figure 9. Figure 9: Qualitative comparison of structures generated by PXRDGen, XRDiff, and Bragglite on experimental PXRD patterns, using Skutterudite and Stolzite examples from the RRUFF test split. For each method, the generated crystal structure and its simulated PXRD pattern are shown alongside the reference structure and experimental pattern. The reported r denotes the correlation between the simulated and experimental PXRD profiles. Appendix F Scope and limitations Candidate generation vs. end-to-end determination. Our benchmark focuses on the candidate-generation stage of PXRD-based structure determination rather than complete end-to-end crystallographic analysis. We evaluate generated structures before downstream refinement and use Top-K structural match rate as a direct and reproducible measure of structure recovery. This enables controlled comparison of generative models without confounding differences in refinement or other post-processing procedures. However, structural match rate is not equivalent to end-to-end structure-determination success: unmatched candidates may still be refinable to the correct structure, while matched candidates do not necessarily guarantee successful refinement against the measured pattern. Developing standardized automated refinement and validation procedures with minimal expert intervention would enable more complete end-to-end evaluation and is an important direction for future work. Oracle atom counts. A limitation of the current benchmark is that the primitive-cell atom count is provided as oracle information. As in many diffusion-based crystal structure prediction and generation models, the number of atomic sites is specified before structure generation. Thus, our element-set setting requires the model to infer elemental multiplicities but not the total atom count, while the two-phase setting provides the atom count of each constituent phase. A more general formulation would infer this quantity from the chemical and diffraction information, for example by learning an atom-count distribution or predicting the count jointly with the crystal structure. Limited experimental PXRD data. A further limitation is the availability of experimental PXRD data paired with reliable structural labels. Publicly available measured PXRD patterns are limited, particularly when a corresponding crystal structure is required as a trustworthy reference for evaluation or training. This constrains both the scale and coverage of experimental benchmarks and the ability to study adaptation directly on measured diffraction data. Building larger experimental PXRD datasets with reliable and traceable structure annotations would enable broader evaluation across materials and measurement conditions, as well as systematic study of experimental fine-tuning and adaptation.