Equivariant diffusion solution for inorganic crystal structure determination from powder X-ray diffraction data

Domain: physics
Published: 2026-02-27
Editorial outcome: major_revision

Abstract
Determining the crystal structures of inorganic materials is critical because atomic arrangement governs their physical, chemical, and mechanical properties. Powder X-ray diffraction (PXRD) remains a cornerstone characterization technique, yet solving structures directly from experimental PXRD patterns is challenging and traditionally depends heavily on expert manual interpretation. Even leading databases contain thousands of entries with incomplete or implausible structural models. In this work, we develop an equivariant graph neural network-based diffusion model that directly infers atomic coordinates from PXRD patterns. Starting from random noise, the model iteratively refines coordinates to produce chemically valid structures that match the target pattern. It solves crystal structure in an average of 0.6 second on a single GPU, several orders of magnitude faster than previous methods, and achieves success rates of 82.3% and 81.6% on simulated and experimental datasets, respectively. We use the model to correct 39 energetically unfavorable database entries and to complete high quality structural models for 912 entries lacking atomic positions, including difficult cases with light elements, natural minerals, and chemical disorder. This conditional equivariant generative approach enables robust automated crystal structure solution from diffraction data and supports future developments in closed-loop autonomous materials discovery.

Introduction
X-ray diffraction (XRD) is one of the most widely used structural characterization techniques for inorganic crystalline materials. Since Bragg’s law was first proposed in 1912, X-ray crystallography has become a fundamental tool that has revolutionized many scientific fields, including mineralogy, geology, chemistry, biology, and materials science1. As a core structural characterization technique, X-ray crystallography plays a pivotal role in studying the structure-property relationship of materials. The earliest batch of crystal structures solved by XRD used single-crystal samples2,3, which are not always readily accessible. Soon after, PXRD techniques were developed to analyze polycrystalline samples, significantly reducing sample preparation requirements. Beyond phase identification, PXRD is also used to extract information on texture, lattice parameters, grain size, and crystallinity. These advantages make PXRD one of the most widely used structural characterization techniques for inorganic crystalline materials.

However, determining crystal structure directly from PXRD patterns is challenging. When diffraction patterns are collected from polycrystalline samples, the three-dimensional atomic spatial information is compressed into one-dimensional pattern, resulting in a significant loss of information. In most scenarios, PXRD is used jointly with large diffraction or structure databases, such as the International Centre for Diffraction Data (ICDD) and the Inorganic Crystal Structure Database (ICSD). If the acquired PXRD patterns match any existing patterns within the database, the phase identity is revealed. If not, it becomes necessary to infer the crystal structure directly from the PXRD pattern, a difficult task requiring domain knowledge of crystallography and solid-state chemistry.

A reasonably inferred crystal structure model should meet two requirements: chemical reasonableness and matching the target PXRD patterns. Chemical reasonableness requires that the crystal structures adhere to explicit and implicit rules of chemical bonding and atomic arrangement. Pattern matching involves determining whether the reconstructed pattern matches the target pattern while considering various factors, such as texture, background noise, sample size, and X-ray polarization. Therefore, ab initio crystal structure determination from PXRD patterns has long been a manual task that heavily relies on the expertise of human experts, during which determining atomic coordinates is one of the most crucial and challenging steps. The traditional approach generally involves first determining the lattice parameters and possible space groups, then exploring reasonable combinations of atom coordinates, and finally performing Rietveld refinement. Specifically, determining atomic coordinates usually relies on prior knowledge, prerequisite information, and the expertise of expert crystallographers. As a result, for complex structures or those lacking prior information, even experienced crystallographers may require multiple rounds of tedious trial-and-error attempts to arrive at the correct structure. Besides, most researchers who perform PXRD analysis focus on specific materials of their interests rather than crystallography. In these cases, structural analysis based on PXRD could be challenging, potentially resulting in incomplete, controversial, or even incorrect solutions. For example, the ICSD database, which is the largest database for completely identified inorganic crystal structures, contains about 300,000 entries. However, a previous study suspected that among the unique entries with ordered structures, about one-fifth may be spurious4. The ICDD database, one of the most widely used diffraction databases, contains more than 480,000 sets of inorganic diffraction data5. Nevertheless, it also includes tens of thousands of entries without atomic coordinates in their associated structures (Supplementary Table 1). Given that these two databases already represent cutting-edge resources for structural data on inorganic materials, developing efficient and accurate approaches to solve crystal structures from PXRD is urgently important.

Apart from the traditional method solely relying on the experience and intuition of crystallographers, several alternative approaches have been proposed to tackle the challenge of structure determination from PXRD patterns, including structure optimization algorithms6–9 and database-based prototype search methods10. Density functional theory (DFT) calculations are commonly used to select plausible solutions among candidates. FPASS employed a DFT-supported genetic algorithm for automatic crystal structure solutions based on given PXRD patterns, stoichiometry, and space group6. Evolve & Morph combined a DFT-supported evolutionary algorithm with crystal morphing to optimize structure solutions from initial random guesses8. However, these approaches typically require hundreds to thousands of DFT calculations to solve a single structure, which is computationally expensive. Besides, they require predetermined space group information, which may be unavailable or unreliable11. For database-based methods, Griesemer utilized prototypes as initial guesses to solve about 500 previously unsolved compounds from experimental PXRD patterns in ICDD10. Nevertheless, the effectiveness of this approach is limited by the availability and reliability of known structural prototypes.

Recently, machine learning approaches, particularly generative diffusion models, have provided a paradigm for materials structure generation12–15. Xie et al. built the Crystal Diffusion Variational Autoencoder (CDVAE), which employs score matching to generate realistic crystal structures by learning from the data distribution of known stable materials12. Without any predefined prototypes, CDVAE shows its ability to generate valid and diverse crystal structures that captures the physical inductive bias of material stability. Equivariant diffusion can effectively preserve reflection, translation, rotation, and mirror symmetries12–16, enabling generation of physically reasonable crystal structures. Such equivariant denoising models were employed to generate inorganic crystal structures12,14,15 and molecules13,16, and showed high performance. Besides generating crystal structures without constraints, conditioning diffusion models allow crystal generation with desired chemistry, mechanical, electronic, or magnetic properties, supporting the developments towards generative AI-assisted materials design paradigms. Recently, Lipson et al. proposed a generative diffusion model, PXRDnet, to determine the crystal structures from nanocrystalline powder diffraction patterns17. This approach addresses the challenge of peak broadening effects due to nanosize crystallites, and solved 200 structures of varying symmetry and complexity from simulated nanocrystalline PXRD patterns. Consequently, the conditional equivariant diffusion model holds tremendous potential for determining atomic coordinates under PXRD pattern conditions.

In this work, we introduce XRDSol, an equivariant diffusion model that can propose crystal structure solutions based on given PXRD patterns. Besides the diffraction patterns, XRDSol requires only stoichiometry and unit cell as inputs, and can reasonably infer atomic coordinates without human intervention. By taking advantage of the equivariance inherently implemented in the model, the reflection, translation, rotation, and mirror symmetries in crystal structures are guaranteed in solutions. XRDSol takes, on average, about 0.6s to provide a crystal structure solution based on a given PXRD pattern, making it ~10^4–10^5 times faster than previous works6–8. The success rate of XRDSol is 82.3% on the simulated PXRD test dataset (MP-20), and 81.6% on the experimental PXRD dataset (ICDD-20). We revisited hundreds of previously solved (mostly by human experts) but energetically unfavorable structures, and XRDSol successfully provides more plausible structure solutions for at least 39 entries from ICSD. Moreover, for 912 ICDD entries without documented atomic coordinates, XRDSol is able to derive structure solutions from PXRD patterns, and we verified that these solutions are highly plausible with both DFT calculations and manual inspections. The performance of XRDSol is robust across different chemistry systems, including those involving light-element materials, natural minerals, and structures with substitutional disorder. Such performance demonstrates that successfully trained equivariant diffusion models can implicitly learn crystallographic knowledge, and can be deployed to provide crystal structure solutions based on PXRD patterns, paving the way for automated, efficient, and accurate structural analysis for inorganic materials.

Results
Conditional equivariant diffusion for crystal structure solutions
Solving crystal structures from PXRD patterns typically begins with determining lattice parameters and identifying space group, both of which are relatively well-established tasks18–20 if the pattern has reasonable resolution and especially when the unit cell is not large. The information on the number and type of atoms within the determined unit cell can also be obtained using well-established techniques. However, determining atomic coordinates could be challenging due to the vast number of possibilities in the configurational space. Our XRDSol is specifically designed to address this challenging step of determining atomic coordinates.

The training of XRDSol includes a forward noising process and a reverse denoising process (Fig. 1a and Fig. S1). In the forward noising process, we disrupt the atomic arrangement of a known crystal structure S0 within its unit cell by applying random displacements with gradually increasing noise level over 1000 steps (t=0–1000). To account for the periodic invariance of the crystalline materials, the random displacements were set to follow a wrapped normal distribution in all three lattice dimensions during the forward process. At the noise limit, atoms in structure ST acquire random atomic coordinates. In the reverse denoising process, the model is trained to restore the atomic coordinates from ST to S0 with the guidance of target PXRD patterns. The PXRD patterns are compressed into a feature space using a neural network architecture, and then mapped to nodes feature distribution in an equivariant graph neural network (EGNN), which predicts the atomic coordinates. Since our training set (MP-20) consists of thermodynamically stable structures, the model implicitly learns crystallographic knowledge throughout the denoising process. Consequently, it can provide structures that are both chemically reasonable and match the target PXRD patterns, thus enabling crystal structure determination from PXRD patterns.

After successful training, the model is then capable of using given PXRD patterns to infer the correct crystal structures from random initial guesses through the denoising process. We illustrate this process using Sm(CoB)2 (mp-4073) from the MP-20 dataset as an example (Fig. 1b). 5 atoms (1 Sm, 2 Co, and 2 B atoms) with random atomic coordinates were initialized in a primitive cell (a=b=3.56 Å, c=5.47 Å, α=β=108.97°, γ=90°). We use cosine similarity Rcos to quantify the consistency between the target and reconstructed PXRD patterns (more details in SI). The randomly initialized structure differs significantly from the ground truth structure, reflected by a low Rcos of 0.735. Under the corresponding PXRD pattern condition, the atomic positions are iteratively denoised, and the simulated PXRD pattern gradually aligns with the target pattern. At t=300, the intermediate reconstructed structure still shows unreasonable atomic coordinates and a dissimilar PXRD pattern (Rcos=0.761). By t=600, the Rcos increases sharply to 0.959, indicating closer alignment with the target structure. Finally, at t=1000, the resulting crystal structure has a high Rcos of 0.999, demonstrating excellent reconstruction performance guided by the target PXRD pattern. The reconstructed crystal structure also closely resembles the target structure (Fig. S2). In this way, a successfully trained model can reconstruct a reasonable crystal structure based on a given PXRD pattern.

Ultrafast and high-quality crystalline materials reconstruction
We evaluated the performance of XRDSol in terms of efficiency and accuracy. The test dataset contains 9046 set-aside unseen crystal structures in the MP-20 dataset. Starting from a randomly initialized structure, XRDSol reconstructs the crystal structure with a simulated PXRD pattern as a condition. The average solution time is only 0.6s per crystal structure with a batch size of 128 on a single NVIDIA V100 GPU (Fig. S3). In contrast, previous approaches that start from randomly initialized structures but are based on evolution algorithms or particle swarm optimization8,21, typically use a runtime of 0.5–1 day and require multiple DFT calculations8 to solve a single structure. Our XRDSol is about 10^4–10^5 times faster than the previous works6–8 without using any database or first principles calculation (Fig. 2a). Furthermore, we evaluated the accuracy of both the reconstructed crystal structures and their reconstructed PXRD patterns in the test dataset. Compared to the ground truth, 91.5% of the reconstructed PXRD patterns exhibit a high similarity to the target patterns, with an Rcos exceeding 0.9, and the average Rcos reaches 0.974 (Fig. 2b).

Structural reasonableness is an important metric to assess the quality of crystal structure solutions from PXRD patterns. To evaluate the quality, we compared the reconstructed crystal structures to the ground truth solutions from the test dataset. A reconstruction is considered successful only if site-to-site matching is achieved and the maximum distance between all paired atomic sites (MaxDist) falls below a specified threshold. For all successfully reconstructed solutions, we also computed the scaled root mean squared distance (sRMS) between the original and reconstructed crystal structures to quantify the accuracy of the reconstruction. All these structure analysis were performed with pymatgen and a typical threshold for MaxDist is chosen as 0.522. XRDSol successfully restored the atomic coordinates of 82.3% of structures in the MP-20 test dataset (Fig. 2c). Furthermore, the reconstructed crystal structures from XRDSol generally exhibit low sRMS values. The average and median of the sRMS are 0.024 and 0.006, respectively (Fig. 2c). Furthermore, we also calculated the root mean square deviation (RMSD) of atom positions to reflect the reconstruction quality of the crystal structures (Fig. S4). 72.15% of the structures achieved an RMSD less than 0.1 Å and 78.13% achieved an RMSD less than 0.3 Å. Among the structures with RMSD<0.1 Å, the mean RMSD is 0.0215 Å, the median RMSD is 0.0129 Å, the first quartile is 0.0024 Å, and the third quartile is 0.0325 Å. These results confirm that most of the reconstructed crystal structures align well with the original ones. Such performance indicates that XRDSol is capable of reconstructing high-quality structures based on the given PXRD patterns. We showcase six materials from different chemical systems (KNaYCl6, LiSbS2, CaEuNbFeO5, CuNi, CsYbBr3, and InGa(AgTe)2) as examples (Fig. 2d). These examples contain 2–5 different elements in the unit cell, covering halides, sulfide, oxide, intermetallic, and tellurite. XRDSol successfully inferred the correct crystal structure for all of them, highlighting its robust and excellent performance across various chemical systems. Furthermore, quantitative evaluation reveal that XRDSol’s strong dependence on structural symmetry, with higher success rates for high-symmetry crystals (e.g., 97.7% success rate for cubic) compared to low-symmetry ones (e.g., 48.5% success rate for triclinic). This performance trend across all seven crystal systems and individual space groups is detailed in Figs. S5 and S6.

Revisiting energetically unfavorable structures in the ICSD database
Thermodynamic stability is a crucial property for inorganic crystalline materials, and convex hull analysis is commonly used for thermodynamic assessment4. Although first-principles calculations confirm that most experimentally reported structures have energy close to the convex hull, Sun et al. suspected that about 20% of the structures in the ICSD database exhibit poor thermodynamic stability with implausibly high energy above the convex hull (Ehull)4. Manual inspection suggests that some of these structures, previously solved and reported by human experts, are probably incorrect or incomplete, due to unreasonable or partially missing atomic positions. We revisited these high-energy structures sourced from ICSD, all of which contain no more than 20 atoms in the primitive cell and their Ehull are higher than 0.1 eV/atom (Supplementary Table 2). The workflow of revisiting high-energy structures from ICSD is shown in Fig. S7. Calculated PXRD patterns were used as inputs to see if XRDSol can propose more reasonable or complete structures that are energetically favorable. At least 39 structures proposed by XRDSol have a reasonable pattern match and Ehull lower than 0.1 eV/atom, suggesting that these structures are highly plausible (Supplementary Tables 3, 4, Figs. S8‒S16). We manually inspected these structure solutions with a few representative examples analyzed below.

HoGeAg is an intermetallic compound with Ho ions forming a distorted Kagome lattice, leading to the frustrated spin ice state23,24. An earlier report based on neutron diffraction claimed that silver occupies two distinct Wyckoff positions, while 3 germanium atoms are symmetrically equivalent24. However, the Ehull of this originally reported structure is 0.17 eV/atom, extremely high for an intermetallic compound. XRDSol proposed a different structure from the original one, in which Ho and Ag atoms arrange in alternating layers and occupy 3f and 3g Wyckoff positions, respectively. Ge occupies two distinct Wyckoff positions, 1a site within the Ho plane, and 2d sites within the Ag plane. Our proposed structure agrees with a recent literature report23 (RMSD=0.04 Å) (Fig. S17), and it is predicted as a stable compound based on first principles calculations (Ehull=0 eV/atom), suggesting the solution is highly plausible.

SrPbF6, a hexafluoride plumbate, was first synthesized and characterized by Hoppe in 195825. It was originally proposed to crystallize in space group P4/mmc. However, the original report noted the presence of several unindexed peaks in the PXRD pattern, which were attributed to another unknown phase. First principles calculations reveal that the original structure has a high Ehull of 0.16 eV/atom. It was not until half a century later, in 2018, that this hexafluoride plumbate was experimentally revisited with its crystal structure corrected26. This revisited structure perfectly matches the PXRD profile obtained from the experiment with no unindexed peaks, suggesting its correctness. The crystal structure proposed by XRDSol is identical to the later corrected one with RMSD of 0.08 Å (Fig. S18), and is predicted to be a stable compound (Ehull=0 eV/atom). The corrected structure belongs to space group P4/mcm, in which Sr is eight-coordinated instead of ten-coordinated by F. This example demonstrates the capability of XRDSol to propose structures with more chemically reasonable coordination environments.

Half-Heusler is a group of ternary intermetallic compounds, among which many are promising thermoelectric materials. Three equal molar elements form three interpenetrating face-centered cubic sublattices, and only one element is eightfold coordinated. There are two types of half-Heusler materials, with anion (MgAgAs-type) and cation (MgCuSb-type) occupying the eight-coordinated site, respectively. Correct assignment of lattice sites is crucial to understand their structure-property relationships27. Among the ICSD entries we investigated, XRDSol proposed different lattice site assignments for 10 half-Heusler compounds, including NbRhSn, TiNiSn, ZrPdSn, HfPtSn, HfRhSb, NiTiSb, HoPdSb, TmSbPd, MgCuSn, and MgAgAs. The Ehull of the original structures range from 0.170 eV/atom to 0.704 eV/atom. In contrast, structures proposed by XRDSol exhibit much lower Ehull, ranging from 0 eV/atom to 0.079 eV/atom, suggesting solutions from XRDSol are more plausible.

CsIO3, a covalent perovskite material28, was first reported in 1928 with a cubic perovskite structure in the Pm3m space group, where Cs occupies the six-coordinate B-site29. The calculated Ehull of this original structure is extremely high at 1.97 eV/atom. In contrast, our solution from XRDSol exhibits a much reasonably lower Ehull of 0.02 eV/atom. The larger Cs cations are moved to twelve-coordinate A-sites. Our structure solution is very close to the recently synthesized and experimentally characterized structure from a single crystal with RMSD of 0.28 Å (Fig. S19)30, validating its plausibility.

NbP was first characterized by PXRD in 1954 and reported as centrosymmetric with I4/amd space group. It contains a very uncommon four-fold square planar coordination for Nb and P. DFT calculation reveals an extremely high Ehull of 0.6 eV/atom, further raising doubts about the validity of this structure. In contrast, XRDSol suggests a non-centrosymmetric structure with a different space group I4md, where Nb is in a trigonal prismatic coordination environment with six P atoms. DFT calculation confirms this proposed structure is stable (Ehull=0 eV/atom), and it agrees with a few other literature reports31–33. In particular, our solution matches that reported by Zhao et al. with RMSD of 0.07 Å (Fig. S20)33.

Locating hydrogen atomic coordinates is an intricate task due to their weak scattering property, heavily dependent on the crystallographer’s expertise. Typically, people choose to combine other characterization techniques, such as neutron scattering, with PXRD to determine the positions of hydrogen atoms, as it is very challenging to determine hydrogen atom positions solely based on the PXRD pattern. For hydrogen-containing materials, XRDSol can determine the positions of other heavy atoms while suggesting the most reasonable and likely positions for hydrogen atoms based on its embedded crystallographic knowledge. Here, we demonstrate such capability of XRDSol using three examples: Fe(OH)2, Tm(OH)3, and SrNH, which are three compounds that had unassigned hydrogen atoms structures in early reports (Fig. 3b)34–36. These incomplete structures violate charge neutrality, resulting in significantly high Ehull of 0.28, 0.46, and 0.43 eV/atom, respectively. XRDSol can infer reasonable hydrogen positions from its implicitly learned crystallographic knowledge. For Fe(OH)2, it places H at 2d Wyckoff position (1/3, 2/3, 0.453), in good agreement with neutron diffraction results37, with a much lower Ehull of 0.005 eV/atom. Similarly, in Tm(OH)3, XRDSol restored hydrogen at 6h Wyckoff sites (0.721, 0.863, 3/4), reducing Ehull to 0.06 eV/atom. While we failed to find experimental reports on H positions in Tm(OH)3, our solved structure is isomorphous to other lanthanide trihydroxide, such as Er(OH)3, Ho(OH)3, and Dy(OH)3 38. For SrNH, early PXRD reports showed Sr and N forming a rocksalt-type lattice and H coordinates were missing. Structure proposed by XRDSol agrees with the rocksalt framework. Complex rotational disorder of the imide ion may lead to multiple partially occupied H sites39, which XRDSol is incapable to handle. This issue will be further elaborated in the next section. Instead, it places H along the <111> direction, consistent with literature reports suggesting this is the most energetically favorable site with minimized electrostatic energy39. The SrNH with a complete crystal structure has a reduced Ehull of 0.04 eV/atom.

Solving crystal structures without atomic coordinates information
Experimental PXRD patterns differ from simulated PXRD profiles, especially in peak intensities, due to various factors, including texture, polarization, background noise, and temperature effects. Therefore, it is necessary to benchmark the performance of XRDSol on experimental diffraction data. We assembled the ICDD-20 dataset, which contains 1000 sets of experimental PXRD data and their ground truth crystal structures (up to 20 atoms per primitive cell). XRDSol achieved a 81.6% success rate on this dataset, demonstrating robustness against intensity variations in experimental PXRD profiles. Rcos is expectedly lower compared to the performance on the MP-20 test dataset. The average Rcos is 0.67, with 71.1% of the samples exceeding 0.6 (Fig. 4a). The distribution of Rcos on ICDD-20 dataset provides a gauge of the fitting quality by using simulated PXRD patterns to fit experimental PXRD patterns. Since structural reasonableness is more important than achieving better R factors40, arriving at the correct structure is the ultimate goal of structure solutions.

We then deployed XRDSol to solve crystal structures from ICDD entries missing atomic coordinates. Obviously, no ground truth is available for these entries. XRDSol inferred missing atomic coordinates based on its training experience (Fig. 4b). The input for solving these entries includes known target PXRD patterns, lattice parameters, number and type of atoms. Among 912 entries we investigated, the average Rcos reaches 0.708, with 79.6% exceeding 0.6, indicating high pattern reconstruction quality. Note that the distribution of Rcos is quite similar to that on ICDD-20 dataset, indicating that such a distribution is typical for experimental PXRD patterns.

Thermodynamic assessment is another important metric to evaluate the plausibility of structure solutions. We performed first principles calculations to evaluate Ehull of these solved structures (Fig. 4c). All 912 structures exhibit Ehull below 0.1 eV/atom, which is a typical threshold for determining structure plausibility, with 81.3% exhibiting Ehull below 0.05 eV/atom. To the best of our knowledge, this is the largest dataset containing the most structures solved from experimental PXRD patterns using an automated algorithm-based solver (Fig. S21)6–8,10. The reconstructed structures and their PXRD patterns of 912 entries are available in the supporting dataset. The performance demonstrates the capability of XRDSol to infer crystal structures from experimental PXRD patterns.

Light elements such as hydrogen and lithium exhibit weak X-ray scattering due to their low atomic numbers, making it challenging to determine their lattice sites directly only based on PXRD patterns. (NH4)3CoF6 is an ammonium cobalt trifluoride, and was documented in 1970 without atomic coordinates41. XRDSol solved its structure as cubic perovskite, with cobalt at the B-site and ammonium at the A-site. Nitrogen is tetrahedrally coordinated with hydrogen, forming the correct ammonium structure. The structure has a low Ehull of 0.03 eV/atom and a good diffraction pattern match with the original one (Rcos=0.846) (Fig. S22). Moreover, it aligns well with a previous report with RMSD of 0.28 Å (Fig. S23)42. Li2SrGeO4 is a candidate host material for luminescent phosphors. It was synthesized in 1986 and characterized as a tetragonal phase with I-42m space group, but its atomic coordinates were not identified. The structure XRDSol proposed is a Sr-analog of LiCaGeO4 phase. It is thermodynamically stable (Ehull=0 eV/atom), and its diffraction pattern matches well with the original pattern (Rcos=0.767) (Fig. S24). Given the chemical similarity between Ca and Sr, the proposed structure is highly plausible and consistent with literature10,43. In particular, our solution agrees with the recently reported structure by Griesemer et al. with RMSD of 0.08 Å (Fig. S25)10.

The presence of impurities represents a significant factor that can adversely impact the success rate of structure determination from PXRD patterns. At least 15 structures of natural minerals proposed by XRDSol have the Ehull lower than 0.1 eV/atom, suggesting that these structures are highly plausible (Supplementary Table 5). NaSrPO4 is a natural olgite mineral discovered in 1980. Early XRD experiments revealed that it belongs to the hexagonal lattice system and was originally assumed to have the P3 space group44. Lattice parameters were extracted from diffraction patterns but without atomic coordinates. XRDSol provided a solution with a different space group P3m1. Despite that the reconstructed pattern matches most of the major peaks, strong intensity deviation was observed, leading to a relatively low Rcos of 0.313 (Fig. S26). This is probably due to the strong texture of the sample and potentially associated impurity phases45. DFT calculation confirms that the structure is highly plausible with a low Ehull 0.01 eV/atom. We confirmed that our proposed structure agrees with a recent report on olgite crystal structure, except that one lattice site might be co-occupied by Na and Sr45. Natural ramsdellite MnO2 was documented by ICDD Grant-in-Aid in 1989. However, the atomic coordinates for this entry were missing, possibly due to the absence of indexed peaks46,47. The structure solved by XRDSol agrees with literature reports46,47, and it has a low Ehull of 0.009 eV/atom. In particular, our solution agrees with the recently reported structure by Post et al. with RMSD value of 0.1 Å (Fig. S27)46. We believe the proposed structure is highly plausible. Compared to the simulated pattern of proposed structure, a few low-intensity peaks were missing in the experimental XRD pattern (Fig. S28), which is probably the cause of preferred orientation. In contrast, XRDSol shows robust performance in these natural mineral samples.

Chemical disorder is a common phenomenon in inorganic crystalline materials, where different types of atoms are randomly distributed within a crystal lattice. Currently, XRDSol cannot propose crystal structures with chemical disorder, since all atoms have pre-assigned element types, similar to the limitation of DFT calculations. Nevertheless, we challenged XRDSol with two PXRD patterns from disordered materials: Sr0.5Ca0.5TaO2N48 and Eu0.5Dy0.5TiO3 49. Surprisingly, we all reached solutions that are “ordered approximations” of the ground truth structures that contain chemical disorder. Sr0.5Ca0.5TaO2N is an oxynitride perovskite material, in which complex anion order behavior may exist. The A-site should be co-occupied by Sr and Ca. In contrast, in the structure proposed by XRDSol, two A-sites in a supercell are occupied by Sr and Ca, respectively. For the anion-sites, XRDSol proposed an ordered trans-type anion configuration, in which nitrogen occupies axial sites while oxygen occupies the equatorial sites. This is in excellent agreement with neutron diffraction results50. The ordered structure that we solved reaches Rcos of 0.879 (Fig. S29). Similarly, for perovskite Eu0.5Dy0.5TiO3 with Eu/Dy chemical disordering, XRDSol successfully proposed a fully ordered structure model, which shows a good match between the reconstructed and the original PXRD patterns (Fig. S30). Atoms at all Wyckoff positions are correctly assigned except for the presence of chemical disordering49 (Fig. S31). Such performance demonstrates that for materials with chemical disorder, XRDSol is capable of proposing reasonably ordered structure models, which is crucial for further structural refinement tasks.

Methods
Conditional equivariant diffusion
Graph representation of crystal structure. For each given crystal structure, its primitive reduced cell was chosen as its unique representation of the infinite 3D periodic crystal structure. The 3D periodic crystal structures were represented as a graph G=(L,A,P,E). L is a 3×3 matrix that represents a set of lattice vectors of the crystal structure, which is equivalent to the conventional lattice parameters representation (a, b, c, α, β, and γ). A is the atom type matrix A=[a1, a2, …, an]T ∈ Rn×da, P=[p1, p2, …, pn]T ∈ Rn×3, where pi=(xi, yi, zi) ∈ R3 denotes the 3D position of ai, and E ∈ Rn×n is the adjacency matrix and denotes the edge features.

Conditional equivariant diffusion model. The diffusion model is a generative model inspired by non-equilibrium thermodynamics. It includes a forward noising process and a reverse denoising process (Fig. S1). During the forward process, random normal distribution noise is gradually added to the input data x0 via T steps, thereby establishing a Markov chain x0 ! x1 ! … ! xt ! … ! xT. This noising process can be described by the transition distribution as q(xt | xt-1) = N(xt | αt|t-1 xt-1, σ2t|t-1 I), where the parameter αt controls the amount of x0 retained and σt controls the amount of noise added. For XRDSol, we perform diffusion on the atomic coordinates of the given structure S, which defines a Markov chain S0 ! S1 ! … ! St ! … ! ST12,14. Due to the periodicity of crystal structures, we added wrapped normal distribution noise NW on the atomic coordinates to account for the intrinsic periodicity. We define our forward process via transition distribution as q(St | St-1) = NW(St | αt|t-1 St-1, σ2t|t-1 I). During the reverse denoising process, the EGNN ϕ predicts the noise to construct the desired samples. EGNN is a type of Graph Neural Network that satisfies the equivariance constraint51. In this work, the graph is initialized by a fully connected graph G based on atomic positions and lattice information. Each node is endowed with coordinates as well as features. The edge construction follows a fully connected graph G, where every atom in the unit cell is connected to every other atom, forming a fully connected graph. The edge features are defined as fractional coordinate differences between connected atoms. In this setting, EGNN consists of 3 EGNN blocks, each of which composed of equivariant convolutional layers. The noisy structure St is passed through the EGNN to extract structure features. And the structure features are concatenated with the PXRD features that are derived from the target PXRD pattern by a neural network. Next, the concatenated features are fed into a neural network to predict the atomic coordinates of St-1. This process is repeated for 1000 steps and gradually restores the atomic coordinates from ST to S0 with the guidance of target PXRD patterns. We define the reversed denoising process via transition distribution as p(St-1 | St) = NW(St-1 | ϕ(St), σ2t|t-1 I). To solve crystal structures conditioned on given PXRD patterns C, we input the PXRD conditional on g with atomic coordinates into the neural network to guide the correct placement of atoms. The final denoising process was defined as p(St-1 | St, C) = NW(St-1 | ϕ(St, C), σ2t|t-1 I). The pseudo-code of XRDSol is provided to illustrate noising and denoising steps (Supplementary Table 6).

Experimental details
Datasets. (1) MP-20 dataset (simulated PXRD patterns with ground truth structures): We employed the MP-20 dataset (45231 structures) to train and evaluate our XRDSol model. The dataset consists of materials sourced from ICSD database. Their energy above the hull and the formation energy are constrained to be less than 0.08 and 2 eV/atom, respectively. These thermodynamic stability constraints ensure the experimental validity of the materials under consideration. (2) ICDD-20 dataset (experimental PXRD patterns with ground truth structures): We assembled the ICDD-20 dataset (1000 structures) with at most 20 atoms in the primitive cell and collected their corresponding experimental PXRD patterns. As an experimental dataset, ICDD-20 serves as a crucial benchmark for validating the performance of XRDSol under real-world conditions. (3) ICSD-20-HE dataset (high-energy structures with simulated PXRD patterns): We have created the ICSD-20-HE dataset (3122 structures) to revisit energetically unfavorable structures in the ICSD. Their Ehull are constrained to be higher than 0.1 eV/atom, respectively. This dataset is used to explore the model’s ability to handle the probably incorrect or incomplete structures. (4) Unsolved experimental XRD pattern dataset: We use the trained XRDSol model to solve the entries from the ICDD database, of which the atomic coordinates are missing but lattice parameters are known. We start with the entries whose diffraction quality is listed as “star”, “good”, or “indexed”, indicating that the diffraction pattern represents a single-phase crystal with minimal impurities. The lattice and number of formula units per unit cell are already known. The number of atoms should be less than 20.

Together, these four datasets enable a comprehensive evaluation of the model across both simulated and experimental XRD patterns, with or without ground truth structural data. The target PXRD patterns, lattice parameters, number and type of atoms are already known and used as the inputs of XRDSol.

Training XRDSol model. We provide detailed configuration parameters for the training and sampling process within the conditional XRDSol model. The conditional XRDSol model was trained for 1000 steps with a batch size of 256 across four V100 GPUs. The Adam optimizer was employed. The initial learning rate was set to 0.001 and was subjected to decay using the ReduceLROnPlateau scheduler, with a decay factor of 0.6, patience of 30, and a minimum learning rate of 10−4. The loss function includes LXRD and Lcoordinates, which is defined as L=λ1 × LXRD + λ2 × Lcoordinates, where λ1 = 1000 and λ2 = 1. The LXRD is given a much higher weight, emphasizing the dominant role of XRD data in guiding the crystal structure solution. For the denoising process, we discretized the reverse diffusion process over the continuous time interval [0, 1] into T=1000 steps. For each timestep, we sample (Xt-1) given (Xt) using the EGNN.

Evaluating the crystal structure solutions. We used the Ehull, Rcos, and sRMS to evaluate the quality of resolved structures. (1) Energy assessment. We used DFT to compute the formation energy of all our crystal structure solutions. All DFT calculations were performed using the Vienna Ab-Initio Simulation Package (VASP) using the PBE exchange-correlation functional, and potentials supplied by VASP with the projector augmented-wave method. The parameters of DFT calculations, such as the plane-wave energy cutoff and k-points density, were consistent with the parameters used for the Materials Project52. The energy correction schemes for oxides and transition metals were implemented as the Materials Project setup. To measure the energetic stability of a candidate structure, we compute the convex hull of formation energies in the relevant phase space and compute the difference between the candidate structure’s formation energy and the convex hull energy Ehull at the target composition. (2) XRD match. The simulated diffraction data were calculated by XRD Calculator module in pymatgen22. We evaluated the difference between the target and solved XRD diffraction data using Rcos. Rcos is based on cosine similarity, which measures the similarity of two vectors8. In this work, the two vectors are the original XRD pattern and reconstructed XRD patterns, respectively. The cosine similarity is calculated as the cosine of the angle between the vectors. The value ranges from −1 to 1, where −1 means completely dissimilar, 1 means completely similar, and 0 means no correlation. (3) Structural match. We used the get_rms_dist method in pymatgen package to calculate the sRMS22, which is normalized by the average free length per atom. Furthermore, we used StructureMatcher in pymatgen22 to evaluate the success rate by calculating the MaxDist between the structures. The fit function in StructureMatcher compares the normalized MaxDist against the specified site tolerance (stol). If the MaxDist is less than stol (stol=0.5), the fit function returns True, indicating a successful match. We also calculated the RMSD of atom positions between the original and reconstructed crystal structures to quantitatively measure the structure similarity.

Solving details. For each PXRD pattern, we ran XRDSol 25 times, generating 25 crystal structures independently. First, we rank the 25 potential solutions using Rcos (top-25). The solution with the highest Rcos is chosen as the final crystal structure solution of the corresponding given XRD pattern. We observed a systematically improving performance for the XRDSol as the number of total independent runs increased (Figs. S32 and 33).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents a data-driven approach to the structure prediction problem in extended solids. Neural networks trained on known crystal structures are a promising tool for structure prediction of polycrystalline solids without expensive ab initio or semi-empirical calculations. Yu et al describe XRDSol, an equivariant diffusion model that allows rapid crystal structure prediction taking only elemental composition and an observed PXRD pattern as inputs. Authors test their model on both observed and predicted PXRD patterns corresponding to known crystal structures with >80% success rate of reproducing the solved structure. Further, authors predict structures for crystals that have been called into question and show that XRDSol predictions fall in line with more recent experiments.\nGenerally, XRDSol is shown to effectively reproduce manually solved crystal structures, however the presented work does not demonstrate the effectiveness of XRDSol at predicting structures for crystals that are difficult or impossible to solve. The manuscript also does not provide an appropriate comparison of the presented work to relevant works in machine-learning based crystal structure prediction that do aim to predict much more complicated structures. For these reasons, we recommend that the current manuscript should only be accepted following major revisions that both expand the scope of the approach and better delineate its limitations.\nBecause manual solving allows each step to be checked and justified, there is an additional level of rigor to manually solved structures that should not be readily sacrificed when new utility is not gained by a structure prediction algorithm. We believe that the utility of this approach should lie in predicting structures of unsolvable or unusually difficult to solve phases. Thus, it remains unclear how the authors envision their approach being used, as the manuscript, by focusing inherently on solved structures, provides limited improvements to structure solution methods that are actively being made more readily implementable by improvements in crystallographic data analysis software (such that a student of crystallography should be equipped to tackle these types of structures).\nTo clarify this point, it would be helpful if the authors explicitly stated the constraints of the training set employed, such as limits on the symmetry and number of atoms in the unit (for example, it does not seem like this tool is appropriate for molecular crystals or proteins), and ensure that the language used throughout the manuscript is consistent with the specific challenges being tackled (c.f. comments on the bottom of page 4, ln 86-87, describing DFT based approaches with the statement that “the information on the number and type of atoms within the determined unit cell is also usually known” describing their approach on page 6, ln 121-123). Can the authors also clarify what inputs are required? How does the stated need for knowing the “periodic lattice” differ from knowing the space group?\nHow did the authors choose the examples focused on in the manuscript? They are mainly well-described as ionic materials that are chemically limited, with some notable exceptions (the HoGeAg is particularly intriguing). For example, does predicted magnetic order (or other symmetry breaking phenomena) influence the efficacy of this structure solution approach? More generally, the examples shown raise questions regarding bias towards solving specific types of structures. We suggest the application of XRDSol to a more chemically diverse set of materials. Particularly, we believe that predicting structures of high energy or low symmetry materials would provide a more robust test of XRDSol and we suggest that authors verify that XRDSol does not have an implicit bias towards any space groups, substructures, structure types.\nAn additional challenge to XRDSol that would demonstrate the effectiveness of the model is the ability to explicitly predict structures that present particularly thorny crystallographic challenges, such as (in)commensurate order, patterns that arise from mixtures of components, or patterns collected on poor quality samples with limited resolution. The authors’ testing on Sr0.5Ca0.5TiO2N and Eu0.5Dy0.5TiO3 provides a starting point for modelling substitutional disorder (see comment on substitutional and positional disorder), however the model seems limited in that it does not explicitly predict random placement of Sr/Ca and Eu/Dy at otherwise chemically equivalent positions, which may be indicated by multiple closely related solutions that differ simply by atom identity at equivalent sites. Rather, XRDSol predicts a larger unit cell with the empirical formulas SrCaTiO4N2 and EuDyTi2O6. Can the authors comment on this distinction? Extending XRDSol to consider long range order and demonstrating that solutions are accurate for a larger variety of materials would warrant its use for structure prediction over manual solving methods and allow this work to stand among other machine learning based structure prediction algorithms.\nWe further highlight the importance of appropriate comparison to existing works. Authors claim to have “demonstrated that conditional equivariant generative model(s) can tackle the ab initio structure determination problem and provide high-quality structure models for inorganic crystalline materials” (ln 28-30). While XRDSol does provide structures with minimal input, neural networks are not ab initio methods. Rather, XRDSol is a data-driven model that “implicitly learns crystallographic knowledge” (ln 387-388). We believe that comparison to ab-initio and manual structure determination methods is inappropriate considering the existence of other machine-learning based structure prediction methods.\nWe encourage the authors to continue to develop XRDSol. We are particularly encouraged by the highly accessible computational cost of XRDSol in comparison to other structure prediction approaches. We also believe that machine learning structure prediction is a promising area of development given the ability of a machine learning model to go beyond what can be done by manual human analysis. We provide the following comments as additional notes for the reference of the authors who we hope to see present a more robust demonstration of their structure prediction algorithm in future works. --- Authors refer to how in a PXRD experiment “the three-dimensional atomic spatial information is compressed into one-dimensional spectra” (ln 50-51). The result of a PXRD experiment is a pattern not a spectrum. Similarly, the use of Rcos as a metric is not standard, and this variable should be defined in the manuscript.\n--- Authors claim that XRDSol can provide a crystal structure in an average time of 0.6 seconds (ln 103). We suggest a comment on the distribution of run times for XRDSol or a figure demonstrating this distribution. A demonstration of consistent fast structure generation provides confidence in the computational cost of XRDSol that is a point of strength for this work. --- Authors claim that XRDSol performs well in systems with chemical disorder (ln 113). While this claim is based on the predicted structures of Sr0.5Ca0.5TiO2N and Eu0.5Dy0.5TiO3, we suggest clarifying that the model performs in crystals with substitutional disorder as opposed to positional disorder within the unit cell. Additionally, we believe that the future development of XRDSol should include a more robust treatment of disordered structures. The standard representation of disorder is not multiple unit cells each with one possible microstate of the disordered crystal, but rather a single unit cell that has a specified variation at a certain position that corresponds to the relevant disorder. A model that generates multiple unit cells containing different microstates of a disordered position does not accurately depict the symmetry of the crystal and does not address either the dynamics or the site-to-site variation of the material\n--- Authors attribute uncharacteristically low Rcos values for NaSrPO4 to low phase impurity and a highly textured sample (ln 355-356). Further, authors attribute the absence of predicted peaks in the MnO2 observed pattern to Rietveld refinement failure (ln 364-365). We suggest providing a reference that justifies these statements in the relevant system or a chemically similar system.", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_3": "The authors\n- developed a conditioned diffusion model based on an equivariant graph neural net\n- trained it on MP20 (~45k structures with simulated data, 9k held out for testing) and ICDD20 (~1000 structures with experimental data).\n- conditioned it on simulated x-ray powder diffraction data\n- the model is able to predict atomic structures given a powder diffraction pattern.\nTo summarize my assessment of this manuscript:\n- the paper started very slowly, reading a little bit the way I would expect a computer scientist explaining materials science to a materials scientist. However, later it got much better with quite detailed descriptions of structures from different materials classes. I strongly recommend to get rid of most of the introduction and get right to the point.\n- This work is quite similar to a related paper on arxiv: arXiv:2406-10796. It is nature policy that papers in preprint servers cannot be used to assert precedent, so the existence of the arxiv paper should not affect whether the current manuscript is accepted. However, it would seem appropriate to cite that paper (and possibly place the current work in context of that work).\n- The problem the current authors tackled was, given a known unit cell and composition, what are the fractional coordinates of the atoms in the unit cell. This is justified on the basis that the indexing of the diffraction pattern to get the unit cell information is straightforward. This is not completely unreasonable, but in many interesting cases it is actually not super straightforward to get the unit cell for powder patterns, especially when the unit cell is larger (not considered in this work). This is a limitation of the broader applicability of the current work.\n- This is a significantly simplifying assumption because it results in the Bragg peaks being in the right positions and the model simply changes the intensities of the peaks when it atoms around. It gives a subjective impression that the model is working better than it is because the Bragg peaks are all in the right locations, but close inspection of the figures shows that even \"successful\" solutions have the intensities that don't match. It is not so clear that a crystallographer would be satisfied with these structure solutions. This is expanded on a bit below.\n- These authors did a decent job of assessing the outcomes of their studies. In common with most studies in this domain they faced the challenge of determining some way of assessing whether a structure was successfully solved or not. They used standard methods (Rcos of the initial and final diffraction pattern) and similarity tools. I wasn't able to find the MaxDist measure so not sure how it works (I looked in the reference that was cited and it doesn't seem to appear in that paper, or I couldn't find it anyway). This should be addressed.\nIn general, the structure solution field needs better measures of when structures are successfully solved. The problem with this is that there is disagreement even among crystallographers about the exact definition.\nFor example, an Rcos of 0.899 sounds pretty good (almost 90% good!), but a crystallographer looking at the powder patterns in figure S7 of the supplementary would be pretty surprised with how bad it is. Clearly wrong. Even the Rcos of 0.974 in figure S2 has a bunch of problems. These could possibly be fixed by a final Rietveld refinement step, which was not done by these authors, but this would normally be part of a minimal requirement from a crystallographer.\n- I did like that the authors computed the formation energies using DFT and compared to the complex hull. It was very reassuring that the model was able to find on hte whole low energy solutions, and indeed some solutions that had lower formation energy than the reported ones. This gives good confidence that the diffusion model is learning structures that are energetically stable (it was trained on stable structures).\n- They tested against experimental data with solved structures from ICDD and the model performed well in most cases. This is good.\n- They also used it to propose structure solutions from some experimental patterns that didn't have structure solutions proposed in the ICDD and the models performed well. It wasn't completely clear the inputs in this case. Was the stoichiometry known and input? And the number of formula units in the unit cell? This needs to be placed more prominently.\nSome comments on the validity and areas for improvement are:\n- if the model architecture was built from or inspired by or reused architecture from the literature I encourage the authors to mention this in the methods section where the model is described. I would be a bit surprised if it were not built off an existing one or ones because this structure is quite similar to CDVAE for example (ref 12 in the current manuscript). It is ok to admit where the inspiration came from and what was reused. It gives correct credit to prior workers. This needs to be fixed before publication.\n- is it possible to show one or more schematics of the model architecture in a figure? this can be very helpful\n- In the description of the model the L vector is described but I had to think hard to figure out it was the lattice vectors. Makes sense in hindsight (L) but please add a few words so that all the letters are fully defined in the place where they appear.\n- The work would benefit from better baselining. It was baselined against some non ML approaches that use evolutionary algorithms, For example, and the trained model was much faster (not a huge surprise). But can it be benchmarked against other ML models and simple models such as nearest neighbor models or something like that?", "reviewer_4": "Strength:\n- The writing is good and easy to follow.\n- This work is both interesting and novel. Unlike previous methods focused on accurate 3D reconstruction, this study explores the use of a diffusion model to approximate spatial structures.\n- The adoption of a graph model to represent the spatial structure of XRD is intuitive and naturally lends itself to being extended and learned by the diffusion model.\n- Compared with numeric methods (e.g., the DFT), the diffusion-based method is efficient with a significant speedup of at least 10,000x. The estimation accuracy under both the simulation and observation environment is also acceptable for further study of the crystal.\nWeakness&Questions:\n(1) Some algorithm details are not clear.\n(a) How is the graph initialized? Is any prior knowledge required? If not, could incorporating prior knowledge, such as partial spatial structures, improve prediction accuracy?\n(b) How is the number of diffusion steps determined? Is it consistent across all crystals, or does it vary? Would increasing the number of steps consistently enhance performance?\n(c) What is the detailed structure of the model used for the denoising process?\n(2) The diffusion process involves inherent randomness. What is the statistical significance of the reported results, considering this randomness?\n(3) Figure 2(b) shows examples with relatively low estimation accuracy. What distinguishes these \"bad\" examples from others? Do they share specific characteristics, such as belonging to a similar group? While the paper effectively highlights the success of using the diffusion model for accurate XRD predictions in most cases, it would be helpful to provide a more detailed analysis of these failed examples.\n(4) Does the number of cells affect estimation accuracy? Do different space groups exhibit similar accuracy trends?\n(5) Is there any way to use DFT or something else to further refine the structure?\n(6) It would be beneficial to include more details about the diffusion model to enhance clarity for the materials science community.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Responses to Reviewers\n(Black italic: Reviewer’s comments; Black type: Our response; Blue type: Key revisions)\nResponses to Reviewer #1:\nComments #0: This manuscript presents a data-driven approach to the structure prediction problem\nin extended solids. Neural networks trained on known crystal structures are a promising tool for\nstructure prediction of polycrystalline solids without expensive ab initio or semi-empirical\ncalculations. Yu et al describe XRDSol, an equivariant diffusion model that allows rapid crystal\nstructure prediction taking only elemental composition and an observed PXRD pattern as inputs.\nAuthors test their model on both observed and predicted PXRD patterns corresponding to known\ncrystal structures with >80% success rate of reproducing the solved structure. Further, authors\npredict structures for crystals that have been called into question and show that XRDSol predictions\nfall in line with more recent experiments. Generally, XRDSol is shown to effectively reproduce\nmanually solved crystal structures, however the presented work does not demonstrate the\neffectiveness of XRDSol at predicting structures for crystals that are difficult or impossible to solve.\nThe manuscript also does not provide an appropriate comparison of the presented work to relevant\nworks in machine-learning based crystal structure prediction that do aim to predict much more\ncomplicated structures. For these reasons, we recommend that the current manuscript should only\nbe accepted following major revisions that both expand the scope of the approach and better\ndelineate its limitations.\nResponse #0: We truly appreciate your thorough evaluation and valuable comments of our work.\nWe have addressed your concerns point by point below.\nComments #1: Because manual solving allows each step to be checked and justified, there is an\nadditional level of rigor to manually solved structures that should not be readily sacrificed when\nnew utility is not gained by a structure prediction algorithm. We believe that the utility of this\napproach should lie in predicting structures of unsolvable or unusually difficult to solve phases.\nThus, it remains unclear how the authors envision their approach being used, as the manuscript,\nby focusing inherently on solved structures, provides limited improvements to structure solution\nmethods that are actively being made more readily implementable by improvements in\ncrystallographic data analysis software (such that a student of crystallography should be equipped\nto tackle these types of structures).\nalgorithm should not compromise the rigor and accuracy of the crystal structure solution results. This\nis why we implemented multi-level validation tests to verify the reliability of our approach. We agree\nthat manual structure solving provides a level of rigor since the crystallographer can check each step.\nAt this stage, XRDSol is designed to complement rather than replace manual approaches.\nParticularly, our approach does not target PXRD patterns that are difficult or even impossible for\nhuman experts to solve. Since XRDSol has demonstrated a decent (>80%) success rate and rapid\nsolution time, we envision three practical application scenarios for XRDSol:\n(1) Integration into autonomous materials discovery platforms\nAutonomous materials discovery has become a major focus in materials science recently. These\nplatforms typically employ fully automated, high-throughput synthesis-characterization-\nexperimentation workflows. In such closed-loop systems, rapid and automated structural\ncharacterization of large numbers of synthesized samples forms an essential component. Our\nalgorithm is particularly well-suited for these applications due to its minimal human intervention\nrequirements, rapid processing capability, high accuracy, and standardized outputs that enable\nseamless system integration. These characteristics make our approach ideal for maintaining the\nuninterrupted flow required in autonomous materials discovery workflows.\n(2) Maintenance and quality control of large-scale structural databases\nAs illustrated in our manuscript, even state-of-the-art inorganic materials structural databases (e.g.,\nICSD, ICDD) contain entries with incomplete or incorrect structural information. XRDSol provides\na systematic solution for addressing these limitations, by enabling the initial screening of suspicious\nentries, identifying potential errors in existing records, supplementing missing structural parameters,\nand offering cross-validation for ambiguous cases. This combination of accuracy and efficiency\nmakes our method particularly valuable for large-scale database maintenance and quality control.\n(3) Assistant for manual PXRD analysis\nWhile human experts generally possess more domain-specific knowledge and experience than\nXRDSol, our tool can serve as a valuable assistant for manual PXRD analysis. Specifically, in\nmanual PXRD analysis, XRDSol can serve as an efficient first-step tool to provide preliminary\nstructural models that experts can either verify directly or use as optimized starting points for\nRietveld refinement. This complementary approach enables researchers to concentrate their\nspecialized expertise on the critical tasks of validation and refinement while maintaining rigorous\n2\nstandards of accuracy.\nIn addition, although beyond the current capability of XRDSol, we also hope to build algorithms\nthat can handle more difficult cases (or what you called “unsolvable or unusually difficult phases”).\nHowever, there are two practical difficulties: First, when human experts deal with these cases, they\noften need to consider more factors, such as thermal vibration, possible twinning, preferred\norientation, etc., and our current model is not yet compatible with these complex multiple factors.\nSecond, and more importantly, for these complex cases, even human experts can make controversial\njudgments. Since ground truth is usually not available for these cases, we lack effective means to\nverify the reliability of automated algorithms. Therefore, our current algorithm does not target these\nextremely difficult cases.\nAction #1: We appreciate your valuable comments regarding the potential applications of XRDSol.\nWe have explicitly elaborated on three key application scenarios in the revised manuscript as\nfollows:\n(Discussion in the revised Manuscript)\n“We envision three practical application scenarios for XRDSol: integration into autonomous\nmaterials discovery platforms, quality control of large-scale structural databases for inorganic\nmaterials, and AI-assistant for manual PXRD analysis.”\nComments #2: To clarify this point, it would be helpful if the authors explicitly stated the constraints\nof the training set employed, such as limits on the symmetry and number of atoms in the unit (for\nexample, it does not seem like this tool is appropriate for molecular crystals or proteins), and ensure\nthat the language used throughout the manuscript is consistent with the specific challenges being\ntackled (c.f. comments on the bottom of page 4, ln 86-87, describing DFT based approaches with\nthe statement that “the information on the number and type of atoms within the determined unit cell\nis also usually known” describing their approach on page 6, ln 121-123). Can the authors also\nclarify what inputs are required? How does the stated need for knowing the “periodic lattice” differ\nfrom knowing the space group?\nResponse #2: Thanks for the comments.\n(1) Constraints of training set:\nOur model XRDSol was trained on the MP-20 dataset, which is a relatively commonly used benchmark\ndataset for machine learning approaches in the inorganic material community (Tian Xie\n3\net, al. Crystal Diffusion Variational Autoencoder for Periodic Material Generation, ICLR, 2022).\nThe MP-20 dataset consists of inorganic crystal structures with up to 20 atoms per unit cell, most\nof which are thermodynamically stable based on first principle calculations. This dataset can be\nopenly accessed from the Materials Project database.\nFor symmetry, the training structures include different space groups with different symmetry. We\ndid not screen or filter any data in this dataset. As described by Tian Xie et, al. (ICLR, 2022), MP-\n20 is a realistic dataset that includes most experimentally known inorganic materials. Therefore, it\ninherently exhibits an imbalance among different space groups, as some space groups are more\ncommon or rare in experimentally observed inorganic crystal structures. The generated structures\ncan also be naturally influenced by the underlying distribution of different symmetries of the\ntraining data.\nIn addition, XRDSol was designed for inorganic crystal structures. It is not appropriate for\nmolecular crystals or proteins.\nAction #2: To better reflect the scope of our work, we have clarified that XRDSol is specifically\ndesigned for inorganic crystals by updating the manuscript title as follows:\n(Title in the revised Manuscript)\n“Equivariant Diffusion Solution for Inorganic Crystal Structure Determination from Powder X-ray\nDiffraction Data”\n(2) Consistency in language:\nWe understand the concern about consistent language in the manuscript. Below is a clarification of\nour intended meaning:\n• Line 86-87: The statement “Besides, they require predetermined space group information,\nwhich may be unavailable or unreliable” means that space group information is usually known,\nbut there are cases where it is unavailable or incorrect. This claim is supported by our statistical\nanalysis of the ICDD database: among the ICDD entries without reported crystal structures,\n25.8 % entries (5348 out of a total 20741) lack space group information. Additionally, previous\nstudies have also reported cases of incorrect space group assignments (Angewandte Chemie\nInternational Edition, 2021, 60, 17452-17454; Zeitschrift für anorganische und allgemeine\nChemie, 2018, 644, 1721-1726).\n• Line 121-123: The statement “the information on the number and type of atoms within the\n4\ndetermined unit cell is also usually known” refers to the atom type, number of atoms, and\nlattice parameters (a, b, c, α, /3, and γ). The lattice parameters are usually available. This\nclaim is also supported by our statistical analysis of the ICDD database: among the ICDD\nentries without reported crystal structures, only 0.0002% entries (4 out of a total 20741) lack\nlattice information.\nThe key difference is that the first statement refers to the uncertainty in the space group, while the\nsecond statement talks about lattice parameters. These ideas do not contradict each other.\n(3) Clarification of inputs:\nThe inputs for solving the crystal structures used in this work include the target PXRD patterns,\nlattice parameters, number and type of atoms. The lattice parameters refer to the unit cell\nparameters (a, b, c, α, /3, and γ). The lattice parameters define the periodic translation symmetry of\nthe crystal, which differ from the space group. In contrast, space groups provide other symmetry\ninformation, such as reflection, rotation, and mirror symmetries. Our XRDSol uses lattice\nparameters and does not rely on prior knowledge of the space group.\nComments #3: How did the authors choose the examples focused on in the manuscript? They are\nmainly well-described as ionic materials that are chemically limited, with some notable exceptions\n(the HoGeAg is particularly intriguing). For example, does predicted magnetic order (or other\nsymmetry breaking phenomena) influence the efficacy of this structure solution approach? More\ngenerally, the examples shown raise questions regarding bias towards solving specific types of\nstructures. We suggest the application of XRDSol to a more chemically diverse set of materials.\nParticularly, we believe that predicting structures of high energy or low symmetry materials would\nprovide a more robust test of XRDSol and we suggest that authors verify that XRDSol does not have\nan implicit bias towards any space groups, substructures, structure types.\nResponse #3: Thanks for the comments. Regarding examples that we focused on (Fig. 3 and Fig. 4),\nour selection was guided by a principle: we focused on cases where we could be most confident that\nXRDSol had achieved a correct solution. By “correct” we refer to either cases where XRDSol provided\na more reasonable structural solution compared to existing entries in the ICSD, or cases where the\nICSD or ICDD entry originally lacked complete or partial atomic position information that our method\nwas able to successfully determine.\nexercised extreme caution when correcting or completing these entries. A few approaches we\nadopted include manual inspection, literature review, and first-principles calculations.\n(1) Prediction of magnetic order and symmetry breaking:\nXRDSol is not trained to solve magnetic order at the current stage. Since our model is trained on\nstructures in MP-20 datasets, few of which contain magnetic order information, it is not capable of\npredicting magnetic order.\nAs a future research direction, one possible approach to predict magnetic order is to combine the\ncandidate crystal structure solutions proposed by XRDSol with different magnetic ordering\ncalculation based on density functional theory (DFT). As demonstrated in a previous study (npj\nComputational Materials, 2019, 5, 64), it is possible to construct a high throughput workflow to\nevaluate the possible ground state magnetic ordering.\nNote that correctly evaluating the ground state magnetic ordering with DFT calculation is a non-\ntrivial task, as the authors in the reference remarked: “The workflow successfully predicts the\nexperimental ground state in over 60% of the materials benchmarked.” In addition, it is\ncomputationally expensive, as it requires multiple DFT calculations with different proposed\nmagnetic ordering for each candidate crystal structure. Therefore, we have not implemented\nmagnetic ordering prediction in our model yet.\n(2) Chemical and structural diversity of chosen examples.\nThe chosen examples in our manuscript mainly come from existing entries in ICSD and ICDD\ndatabases.\nFrom the ICSD database, we corrected 39 incorrect crystal structures (6 in Fig. 3a while 33 in SI)\nand completed 3 structures lacking H atoms (Fig. 3b). These 39 materials exhibit chemical and\nstructural diversity. Chemically, we have included 20 metallic compounds, 19 ionic or covalent\ncompounds. Structurally, we have included half-Heusler, perovskite etc. To ensure\nrepresentativeness, we highlighted 6 examples across the three types of materials, including 3\nmetallic compounds (HoAgGe, NbSnRh, and HoSbPd), 1 ionic compound (SrPbF ), and 2 covalent\ncompounds (NbP and CsIO ).\nWe focused on entries that lack full atomic positions but no chemical or structural bias was introduced\nin the selection process. As for the examples chosen in Fig. 4, we aimed to highlight\n6\ndifferent types of materials, such as materials containing light elements (2 examples in Fig. 4d:\nstructures containing H and Li), natural minerals (2 examples in Fig. 4e: both oxides as oxides\ndominate natural minerals), and substitutional disorder lattice sites (1 example in Fig. 4f).\nChemically, we have included oxides, fluorides, and oxynitrides.\n(3) Solving crystal structures with high-energy\nThank you for your suggestion. To the best of our understanding, high-energy inorganic crystalline\nmaterials may refer to materials that are less thermodynamically stable, such as materials synthesized\nunder non-ambient conditions. Our model XRDSol was trained using the MP-20 dataset, which is\nsourced from the ICSD database. This dataset only includes crystal structures with relatively good\nthermodynamic stability (energy above hull below 0.08 eV/atom). These limits ensure that the\nstructures in the dataset are stable under common experimental conditions. Because of this, XRDSol\nwas only trained on low-energy structures. Since most high-energy materials are outside the MP-20\ntraining dataset, XRDSol is currently not specifically designed to handle them effectively.\nAnother obstacle is: to test the performance of XRDSol on high-energy materials, we lack such a\nreliable dataset that contains high-energy materials with available ground truth crystal structure\ninformation. One way we could think of is to assemble another dataset from the Materials Project that\nspecifically focuses on materials with poor thermodynamic stability (high energy above hull).\nHowever, based on the analysis from a previous study (Science Advances, 2016, e1600225),\napproximately 20% highest-energy metastable compounds from the ICSD are considered as spurious\nentries. Based on their manual inspection, many of these high-energy entries could either correspond\nto defected structures, which form unphysical bulk crystals when repeated by DFT periodic boundary\nconditions, or correspond to unobserved hypothetical, computationally-constructed structures.\nTherefore, even if we could perform such a validation test, it may still not correctly reflect the\nperformance of XRDSol on real-world high-energy materials.\nNevertheless, we agree with the reviewer that testing the ability (or limit) of XRDSol on high-\nenergy materials is interesting and also important for the future development of our algorithm.\nIn the future, one potential solution is to collect more high-energy crystal structures and train a\ngenerative model specifically for these cases.\n(4) Solving crystal structures with low symmetry\nThank you for your suggestion. To address concerns about potential bias on certain symmetry or space groups, we re-examined\nXRDSol’s performance on the MP-20 benchmark test dataset. This dataset contains 9046 diverse\nstructures, including structures with low symmetry and structures from various space groups.\n(a) Structures from different crystal systems:\nWe first evaluated XRDSol’s performance across different crystal systems, which exhibit different\ndegrees of symmetry. For example, the cubic system has the highest symmetry with multiple high-\norder rotation axes and the triclinic system has the lowest symmetry. As shown in Fig. S3, i.e., Fig.\nR1-1, the success rate varies across the seven crystal systems with different symmetries (cubic\n97.72%, hexagonal 90.33%, trigonal 77.86%, tetragonal 88.02%, orthorhombic 76.32%, monoclinic\n68.91%, and triclinic 48.48%). Among these, the cubic crystal system, which has the highest\nsymmetry, shows the best performance. In contrast, the triclinic system exhibits the poorest accuracy.\nThe model’s performance decreases as the symmetry decreases. It indicates that structures with\nhigher symmetry are easier to solve accurately, while lower-symmetry structures pose greater\nchallenges.\n)%\n(\ne\nta\nr\ns\ns\ne\nc\nc\nu\nS\nFig. R1-1 Success rate across the 7 crystal systems, including triclinic, monoclinic, orthorhombic,\ntetragonal, trigonal, hexagonal, and cubic.\n(b) Structures from various space groups:\nWe further analyzed model accuracy across different space groups. In this work, we excluded those\nspace groups with 10 or fewer samples in the MP-20 test dataset, as they would lack statistical\nsignificance. As shown in Fig. S4, i.e., Fig. R1-2, the accuracy varies across different space groups.\nGenerally, space groups with higher symmetry tend to exhibit higher estimation accuracy.\nSpecifically, 26 space groups achieved an accuracy of 100%, and 41 space groups had accuracy\ngreater than 90%. However, there are 6 space groups with accuracy lower than 50%,\n8\nnamely space groups 2, 19, 46, 51, 55, and 156. This suggests that symmetry plays a significant\nrole in the model’s ability to predict crystal structures accurately.\n2 5 7 9 11 31 51 32 62 33 63 93 34 64 15 55 95 36 56 76 96 17 47 28 78 79 701 411 611 911 221 721 631 931 141 741 941 351 551 061 261 461 761 381 781 191 491 002 502 712 122 422 722\n)%\n(\ne\nta\nr\ns\ns\ne\nc\nc\nu\nS\n1 4 6 8 01 21 41 91 52 13 53 83 24 44 74 25 85 26 46 66 86 07 27 97 68 88 99 901 511 811 121 321 921 731 041 641 841 051 451 651 161 361 661 671 681 981 391 891 402 612 022 322 522 922\nSpace group\nFig. R1-2 Success rate across the 107 space groups.\nAn intuitive explanation of the better accuracy on high symmetry structures is that, although\nsymmetry information is not provided as input for our model, a high symmetry structure\nintrinsically contains fewer unique Wyckoff sites and lower degrees of freedom, which implicitly\nassists our model in identifying the correct crystal structure solutions.\nAction #3: We thank the reviewer for raising this important point about potential bias in different\nstructure types. We have now included a detailed analysis of structure symmetry in the revised\nSupporting Information. Moreover, we have included future direction for low-symmetry in the\nrevised manuscript.\n(Paragraph 1 of Page 5 in the revised Supporting Information)\n“To investigate the potential bias on certain symmetry or space groups, we evaluated XRDSol’s\nperformance on the MP-20 benchmark test dataset. This dataset contains 9046 diverse structures,\nincluding structures with low-symmetry and structures from various space groups. We first evaluated\nXRDSol’s performance across different crystal systems, which exhibit different degrees of symmetry.\nFor example, the cubic system has the highest symmetry with multiple high-order rotation axes and\nthe triclinic system has the lowest symmetry. As shown in Fig. S3, the success rate varies across the\nseven crystal systems with different symmetries (cubic 97.72%, hexagonal 90.33%, trigonal 77.86%,\ntetragonal 88.02%, orthorhombic 76.32%, monoclinic 68.91%, and triclinic 48.48%). Among these,\nthe cubic crystal system, which has the highest symmetry, shows the best performance. In contrast,\nthe triclinic system exhibits the poorest accuracy. The model’s\n9\nperformance decreases as the symmetry decreases. It indicates that structures with higher symmetry\nare easier to solve accurately, while lower-symmetry structures pose greater challenges. We further\nanalyzed model accuracy across different space groups. In this work, we exclude those space groups\nwith 10 or fewer samples in the MP-20 test dataset, as they would lack statistical significance. As\nshown in Fig. S4, the accuracy varies across different space groups. Generally, space groups with\nhigher symmetry tend to exhibit higher estimation accuracy. Specifically, 26 space groups achieved\nan accuracy of 100%, and 41 space groups had accuracy greater than 90%. However, there are 6\nspace groups with accuracy lower than 50%, namely space groups 2, 19, 46, 51, 55, and 156. This\nsuggests that symmetry plays a significant role in the model’s ability to predict crystal structures\naccurately.”\n(Paragraph 2 of Page 19 in the revised Manuscript)\n“In the future, we will focus on overcoming the current limitations in handling large-unit-cell\ncrystals and low-symmetry systems.”\nComments #4: An additional challenge to XRDSol that would demonstrate the effectiveness of the\nmodel is the ability to explicitly predict structures that present particularly thorny crystallographic\nchallenges, such as (in)commensurate order, patterns that arise from mixtures of components, or\npatterns collected on poor quality samples with limited resolution. The authors’ testing on\nSr0.5Ca0.5TiO2N and Eu0.5Dy0.5TiO3 provides a starting point for modelling substitutional\ndisorder (see comment on substitutional and positional disorder), however the model seems limited in\nthat it does not explicitly predict random placement of Sr/Ca and Eu/Dy at otherwise chemically\nequivalent positions, which may be indicated by multiple closely related solutions that differ simply\nby atom identity at equivalent sites. Rather, XRDSol predicts a larger unit cell with the empirical\nformulas SrCaTiO4N2 and EuDyTi2O6. Can the authors comment on this distinction? Extending\nXRDSol to consider long range order and demonstrating that solutions are accurate for a larger\nvariety of materials would warrant its use for structure prediction over manual solving methods and\nallow this work to stand among other machine learning based structure prediction algorithms.\nResponse #4: Thanks for the insightful comments. As you pointed out, we agree with the importance\nof addressing complex crystallographic challenges such as (in)commensurate order, patterns that arise\nfrom mixtures of components, or patterns collected on poor-quality samples with limited resolution.\nWe agree that improving the XRDSol’s ability to solve these challenging cases would make it more\nuseful. However, we also recognize that these issues represent significant\n10\nchallenges. As a team dedicated to developing automated algorithmic frameworks for materials\nstructural characterization, a common dilemma for us lies in balancing a model’s fitting capacity\nwith the risk of overfitting. Human experts leverage domain-specific expertise and accumulated\nempirical knowledge to make nuanced judgments when handling such tasks. Integrating these\ndomain-specific knowledge into trainable machine learning architectures and ensuring reliable\nmodel outputs remain an inherently complex endeavor.\nFor the handling of substitutional disorder, our current XRDSol model cannot directly propose a\nstructure solution containing sites with partial occupancy, as the reviewer correctly pointed out.\nThe primary cause of this limitation comes from the constraints we imposed on the model.\nSpecifically, XRDSol’s training dataset (MP-20) exclusively comprises fully ordered structures, while\natomic species and occupancies remain fixed during the denoising process. Consequently, XRDSol\ncannot directly propose crystal structures containing partial occupancy sites.\nDespite this limitation, we showcase these two examples in the manuscript for two key reasons.\nFirst, even though our model was never exposed to partially disordered structures during training,\nit can still propose a structural model closest to the ground truth under certain constraints,\ndemonstrating non-trivial extrapolation capability beyond its training domain. Second, we\ndemonstrate that XRDSol’s extrapolation ability enables it to generate physically reasonable\nstarting models for disordered structures, which is an essential first step for the following\nrefinement. Therefore, it can effectively assist (rather than fully replace) researchers in interpreting\nPXRD patterns of disordered structures.\nIn the future, we plan to lift the constraints on atomic species in the structural model, allowing\nXRDSol to propose structures with substitutional or positional disorder. Practically, this would\nrequire the model to be retrained on datasets containing partially disordered structures, which is\nprobably beyond the scope of the current work. Nevertheless, we acknowledge that enabling the\nmodel to handle structures with disordering represents a critical direction for future improvement.\nWe will add these features to XRDSol to handle a wider variety of materials in the future. We\nappreciate your feedback and will continue to improve XRDSol to address these challenges.\nComments #5: We further highlight the importance of appropriate comparison to existing works.\nAuthors claim to have “demonstrated that conditional equivariant generative model(s) can tackle\nthe ab initio structure determination problem and provide high-quality structure models for\ninorganic crystalline materials” (ln 28-30). While XRDSol does provide structures with minimal\ninput, neural networks are not ab initio methods. Rather, XRDSol is a data-driven model that\n11\n“implicitly learns crystallographic knowledge” (ln 387-388). We believe that comparison to ab-\ninitio and manual structure determination methods is inappropriate considering the existence of\nother machine-learning based structure prediction methods.\nResponse #5: Thank you for the comments. We agree that comparing our data-driven model to other\nmachine-learning (ML) based structure prediction methods is more appropriate. We compared our\nXRDSol to two recent ML models for crystal structure solutions: CrystalNet (npj Computational\nMaterials, 2024, 10, 209) and Crystalyze&StructSnap (Journal of the American Chemical Society,\n2024, 146, 44, 30340–30348). CrystalNet uses PXRD patterns and chemical composition\ninformation as inputs. It achieves a success rate of 93.4% on simulated PXRD data of 737 test set,\nbut only for cubic (500 samples) and trigonal (237 samples) crystal systems. Its inference time is\nunder a minute per structure. Crystalyze&StructSnap is another ML model based on the CDVAE\nframework. It was tested on 1280 samples from the MP-20 test set (out of 9046 total samples) and\nachieved a success rate of 66.6%. While the paper of Crystalyze&StructSnap does not provide exact\ninference times and source code, it is inferred to be on the order of seconds to minutes based on the\nCDVAE framework. In comparison, our model achieves a success rate of 82.3% on the full MP-20\ntest dataset (9046 samples) with an inference time of 0.6 seconds per structure. This represents a\nscale advantage over the two approaches: CrystalNet (737 samples) and Crystalyze&StructSnap\n(1280 samples). While the results of these models vary depending on the datasets, our XRDSol\nmodel shows at least competitive performance in terms of success rate and speed. Moreover, our\nXRDSol has no crystal system restrictions and has solved the largest number of experimental PXRD\npatterns to date (912 structures, Fig. S11). We appreciate your suggestion to refine our comparison.\nWe have added comparisons to these two ML models in the revised manuscript.\nComments #6: We encourage the authors to continue to develop XRDSol. We are particularly\nencouraged by the highly accessible computational cost of XRDSol in comparison to other structure\nprediction approaches. We also believe that machine learning structure prediction is a promising\narea of development given the ability of a machine learning model to go beyond what can be done\nby manual human analysis. We provide the following comments as additional notes for the reference\nof the authors who we hope to see present a more robust demonstration of their structure prediction\nalgorithm in future works.\nResponse #6: Thank you for your encouraging feedback and for noting the computational cost of\nXRDSol. We appreciate your helpful suggestions for improving XRDSol. For example, XRDSol\n12\ncan be designed to solve high-energy materials and improve the performance of low-symmetry\nstructures. Furthermore, your comments also helped us identify why some cases are harder to solve.\nWe noticed that the success rate decreases when the unit cell has more atoms or when the symmetry\nis lower. Your suggestion to look into how different space groups influence results is also very\nuseful. This will guide us to incorporate more crystallography domain knowledge (e.g., space\ngroups) into XRDSol. Thank you again for your suggestions. Your feedback will help us improve\nthe XRDSol in the future.\nComments #7: Authors refer to how in a PXRD experiment “the three-dimensional atomic spatial\ninformation is compressed into one-dimensional spectra” (ln 50-51). The result of a PXRD\nexperiment is a pattern not a spectrum. Similarly, the use of Rcos as a metric is not standard, and\nthis variable should be defined in the manuscript.\nResponse #7: Thank you for your valuable comments. We realize that calling the result of a PXRD\nexperiment a “spectrum” is incorrect. It should be called a “pattern”. We have corrected this in the\nrevised manuscript. About the term we understand it is not commonly used and needs to be\nRcos,\nexplained better. In the Method section, we have now added a detailed description of The R is\nRcos. cos\na measure of similarity between the original XRD patterns and reconstructed XRD patterns of\nrelated solved crystal structures.\nAction #4: The descriptions of R have been provided in the revised manuscript as follows:\ncos\n(Paragraph 1 of Page 23 in the revised Manuscript)\n“R is based on cosine similarity, which measures the similarity of two vectors (npj Computational\ncos\nMaterials, 2023, 9, 135). In this work, the two vectors are the original XRD pattern and reconstructed\nXRD patterns of related solved crystal structures, respectively. The cosine similarity is calculated as\nthe cosine of the angle between the vectors. The value ranges from -1 to 1, where - 1 means completely\ndissimilar, 1 means completely similar, and 0 means no correlation.”\nComments #8: Authors claim that XRDSol can provide a crystal structure in an average time of 0.6\nseconds (ln 103). We suggest a comment on the distribution of run times for XRDSol or a figure\ndemonstrating this distribution. A demonstration of consistent fast structure generation provides\nconfidence in the computational cost of XRDSol that is a point of strength for this work.\nResponse #8: Thank you for your critical and valuable comments. We appreciate your suggestion\n13\nto provide more details on the distribution of run times for XRDSol to strengthen the confidence in the\ncomputational cost of our model. To illustrate the computational efficiency of XRDSol, we conducted\na statistical analysis of the distribution of run time across 71 batches using the MP-20 test dataset (9046\nsamples, batch size = 128). As shown in Fig. S2, i.e., Fig. R1-3, XRDSol takes an average of 0.57 seconds\n(about 0.6 seconds) to solve each crystal structure from XRD patterns when running on a single NVIDIA\nV100 GPU. The runtime ranges from 0.48 to 0.67 seconds, with a standard deviation of 0.04 seconds.\nThis detailed distribution of run times underscores the efficiency of XRDSol as a key strength of our\nwork. Thank you again for your insightful suggestion.\n)s\n(\ne\nm\nit\nn\nu\nR\n1 2 3 4 5 6 7 8 9 01 11 21 31 41 51 61 71 81 91 02 12 22 32 42 52 62 72 82 92 03 13 23 33 43 53 63 73 83 93 04 14 24 34 44 54 64 74 84 94 05 15 25 35 45 55 65 75 85 95 06 16 26 36 46 56 66 76 86 96 07 17\nBatch\nFig. R1-3 Distribution of run times in 71 batches (MP-20 test dataset (9046 samples), batch size =\n128) for XRDSol.\nComments #9: Authors claim that XRDSol performs well in systems with chemical disorder (ln 113).\nWhile this claim is based on the predicted structures of Sr0.5Ca0.5TiO2N and Eu0.5Dy0.5TiO3, we\nsuggest clarifying that the model performs in crystals with substitutional disorder as opposed to\npositional disorder within the unit cell. Additionally, we believe that the future development of XRDSol\nshould include a more robust treatment of disordered structures. The standard representation of\ndisorder is not multiple unit cells each with one possible microstate of the disordered crystal, but rather\na single unit cell that has a specified variation at a certain position that corresponds to the relevant\ndisorder. A model that generates multiple unit cells containing different microstates of a disordered\nposition does not accurately depict the symmetry of the crystal and does not address either the\ndynamics or the site-to-site variation of the material.\nResponse #9: Thank you for your helpful comments about the chemical disorder in XRDSol. We\nacknowledge our XRDSol currently addresses substitutional disorder rather than positional disorder\nwithin the unit cell. To make it clearer, we have revised line 113 to explicitly state that XRDSol\nperforms well in systems with substitutional disorder. We appreciate the Reviewer’s\n14\nsuggestions about the disorder representation, which is essential for the future development of\nXRDSol. Solving disorder structures from XRD patterns using a generative model remains a\nchallenge and we consider this an important research goal for our future work. Further details on\nthe topic have been provided in Response #4.\nComments #10: Authors attribute uncharacteristically low Rcos values for NaSrPO4 to low phase\nimpurity and a highly textured sample (ln 355-356). Further, authors attribute the absence of\npredicted peaks in the MnO2 observed pattern to Rietveld refinement failure (ln 364-365). We\nsuggest providing a reference that justifies these statements in the relevant system or a chemically\nsimilar system.\nResponse #10: Thank you for your comments. In our original manuscript, we cited references for\nNaSrPO and MnO crystal structures. Specifically, for NaSrPO , we refer to Reference 42\n4 2 4\n(Canadian Mineralogist, 2005, 43, 1521-1526), and for MnO , we refer to Reference 43 (American\nThese references were placed in the section discussing crystal structures rather than specifically in\nthe XRD analysis. To link these references to the XRD analysis, we have added citations above in\nthe section discussing the XRD analysis of the revised manuscript.\nComments #0: I co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training in peer review and\nto provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse #0: We thank the Reviewer for the co-reviewed comments.\nComments #0: The authors\n- developed a conditioned diffusion model based on an equivariant graph neural net\n- trained it on MP20 (~45k structures with simulated data, 9k held out for testing) and ICDD20\n(~1000 structures with experimental data).\n- conditioned it on simulated x-ray powder diffraction data\n- the model is able to predict atomic structures given a powder diffraction pattern.\nTo summarize my assessment of this manuscript:\n- the paper started very slowly, reading a little bit the way I would expect a computer scientist\nexplaining materials science to a materials scientist. However, later it got much better with quite\ndetailed descriptions of structures from different materials classes. I strongly recommend to get rid\nof most of the introduction and get right to the point.\nResponse #0: We thank the Reviewer for the comments and assessment of our work. Following the\nsuggestion to be more concise, we have removed certain sections from the introduction in the\nrevised manuscript. This revision allows us to present our key points more directly.\nComments #1: This work is quite similar to a related paper on arxiv: arXiv:2406-10796. It is nature\npolicy that papers in preprint servers cannot be used to assert precedent, so the existence of the arxiv\npaper should not affect whether the current manuscript is accepted. However, it would seem\nappropriate to cite that paper (and possibly place the current work in context of that work).\nResponse #1: Thank you for the comments regarding the similarity between our work and the\nrelated paper on arXiv (arXiv:2406-10796). We agree that the paper on arXiv is relevant to our work.\nWe have now cited the paper (arXiv:2406-10796) in the Introduction section of our revised\nmanuscript. Furthermore, we have taken this opportunity to place our work in the context of the\npaper on arXiv.\nMoreover, we have observed that while the PXRDnet model assesses solution correctness based solely on\nXRD fitting R , our experience suggests that this approach can sometimes produce solutions with similar\nwp\nPXRD patterns but different structures, particularly when atoms with similar atomic numbers (Z) are\ninterchanged. This insight has guided our methodology, leading us to incorporate supplementary validation\ntechniques to enhance the reliability of our structure determinations.\nAction #1: To acknowledge the related paper on arXiv, the descriptions of their PXRDnet model\nhave been provided in the revised manuscript as follows:\n17\n(Paragraph 2 of Page 5 in the revised Manuscript)\n“Recently, Lipson et al. proposed a generative diffusion model, PXRDnet, to determine the crystal\nstructures from nanocrystalline powder diffraction patterns. This approach addresses the challenge\nof peak broadening effects due to nanosize crystallites, and solved 200 structures of varying\nsymmetry and complexity from simulated nanocrystalline PXRD patterns.”\nComments #2: The problem the current authors tackled was, given a known unit cell and\ncomposition, what are the fractional coordinates of the atoms in the unit cell. This is justified on\nthe basis that the indexing of the diffraction pattern to get the unit cell information is\nstraightforward. This is not completely unreasonable, but in many interesting cases it is actually\nnot super straightforward to get the unit cell for powder patterns, especially when the unit cell is\nlarger (not considered in this work). This is a limitation of the broader applicability of the current\nwork.\nResponse #2: Thank you for the insightful comments. We fully agree that in many interesting cases,\nit is not super straightforward to get the unit cell for powder patterns, especially when the unit cell\nis larger. To improve the clarity of our discussion, we have revised the corresponding statement in\nthe manuscript as follows:\n(Paragraph 2 of Page 6 in the revised Manuscript)\n“Solving crystal structures from PXRD patterns typically begins with determining lattice\nparameters and identifying space group, both of which are relatively straightforward tasks if the\npattern has reasonable resolution and especially when the unit cell is not large.”\nWe are currently working on an improved version of XRDSol to infer the unit cell information\ndirectly from PXRD patterns, rather than providing them as inputs. As far as we currently know,\nlifting such constraints makes it more difficult. Regarding the broader applicability of our model\nXRDSol to larger unit cells, we recognize the current limitations and outline the following three\nmethods for future improvement:\n(1) Enhancing unit cell indexing for large structures: Our model XRDSol can be integrated with\nmore advanced indexing algorithms to improve the determination of lattice parameters for large\nunit cells.\n(2) Dataset generation and model training for large crystal structures: Our model XRDSol\n18\nwas trained on the MP-20 dataset, which is the benchmark dataset for the machine learning\napproach in the material community. The MP-20 dataset consists of crystal structures with up\nto 20 atoms per unit cell. As a result, our model is currently constrained to the structures within\nthis atom number range. To extend its applicability, generating a sufficiently large dataset (on\nthe order of tens of thousands) of crystal structures with large unit cells can enable the model\nto learn and generalize to these more complex cases.\n(3) Incorporating domain knowledge in XRD crystallography: Within our current framework,\nincorporating additional crystallographic domain knowledge (e.g., space group information)\ncan aid in structure determination for larger unit cells. For a given space group, once the special\nand general Wyckoff positions are identified, the model only needs to predict the general\nWyckoff positions. This approach can be particularly beneficial for large crystals, especially\nthose with high symmetry. A very recent work in this direction is WyckoffAUGen (Rees Chang\net, al. Crystal Generative Modeling with Explicit Autoregressive Conditional Likelihoods and\nNontrivial Space Group Stabilizers, ICLR, 2025). The WyckoffAUGen integrates hard space\ngroup constraints into a generative model, which enables more efficient and physically\nplausible crystal structure prediction.\nComments #3: This is a significantly simplifying assumption because it results in the Bragg peaks\nbeing in the right positions and the model simply changes the intensities of the peaks when it atoms\naround. It gives a subjective impression that the model is working better than it is because the Bragg\npeaks are all in the right locations, but close inspection of the figures shows that even \"successful\"\nsolutions have the intensities that don’t match. It is not so clear that a crystallographer would be\nsatisfied with these structure solutions. This is expanded on a bit below.\nResponse #3: Thank you for your insightful comments. We acknowledge that our assumption of\nfixed lattice constants simplifies the structure determination problem by ensuring Bragg peaks are\ncorrectly positioned, with peak intensities adjusted solely through atomic movements (though peak\ndistinction may occur when some atoms occupy special positions).\nIn the experimental XRD patterns, such as those from the ICDD database, peak intensities are\ninfluenced by more than just atom type and positions (as assumed in our approach). Additional\nvariables, including temperature effects, absorption effects, preferred orientation, crystallite size, and\ninstrument settings, also play significant roles. These extra influences may explain why some\npredicted structures look reasonable but still have intensity mismatches. As the reviewer suggests\nin the following comment #5, using Rietveld refinement as a post-processing step can help fine-tune\nthe predicted crystal structures. This can improve how well the simulated XRD patterns of predicted\nstructures match the experimental XRD patterns. We have included the representative refined\nstructure in the comment #5 section below to show these improvements.\nWhile the fixed lattice constant assumption is a simplification, pairing our model with Rietveld\nrefinement offers a practical approach to producing structure solutions that better align with\ncrystallographic standards. We are grateful for your critique, which has helped us refine our\nmethodology and clarify these points.\nComments #4: These authors did a decent job of assessing the outcomes of their studies. In common\nwith most studies in this domain they faced the challenge of determining some way of assessing\nwhether a structure was successfully solved or not. They used standard methods (Rcos of the initial\nand final diffraction pattern) and similarity tools. I wasn’t able to find the MaxDist measure so not\nsure how it works (I looked in the reference that was cited and it doesn’t seem to appear in that\npaper, or I couldn’t find it anyway). This should be addressed.\nResponse #4: We thank the Reviewer’s comments on the assessment of our results. Regarding the\nMaxDist measure, we acknowledge the oversight in providing a clear reference. MaxDist refers to the\nmaximum root-mean-square (RMS) displacement between two structures (Ground truth structure\nand reconstructed structure in this work), normalized by the average free length per atom. The\nmaximum root-mean-square (RMS) displacement is the maximum distance between all paired atomic\nsites between two structures. If the MaxDist falls below a specified site tolerance (stol; stol = 0.5 in this\nwork) (Tian Xie et, al. Crystal Diffusion Variational Autoencoder for Periodic Material Generation,\nICLR, 2022), the two structures are considered as match. Specifically, the MaxDist is calculated by the\nStructureMatcher class in Pymatgen module (Computational Materials Science, 2013, 314-319),\nwhere the fit function compares the MaxDist against a specified site tolerance (stol = 0.5). If the\nMaxDist is less than stol, the fit function returns True, indicating a successful match.\nAction #2: The detailed explanation and reference to the MaxDist metric have been provided in the\nrevised manuscript. Specifically, we have expanded the description of the StructureMatcher usage\nin evaluating the success rate. The revised text now is provided as follows:\n(Paragraph 1 of Page 24 in the revised Manuscript)\n“Furthermore, we used StructureMatcher in Pymatgen (Computational Materials Science, 2013,\n20\n314-319) to evaluate the success rate by calculating the maximum root-mean-square displacement\n(MaxDist) between the structures. The fit function in StructureMatcher compares the normalized\nMaxDist against the specified site tolerance (stol). If the MaxDist is less than stol (stol = 0.5), the\nfit function returns True, indicating a successful match.”\nComments #5: In general, the structure solution field needs better measures of when structures\nare successfully solved. The problem with this is that there is disagreement even among\ncrystallographers about the exact definition. For example, an Rcos of 0.899 sounds pretty good\n(almost 90% good!), but a crystallographer looking at the powder patterns in figure S7 of the\nsupplementary would be pretty surprised with how bad it is. Clearly wrong. Even the Rcos of 0.974\nin figure S2 has a bunch of problems. These could possibly be fixed by a final Rietveld refinement\nstep, which was not done by these authors, but this would normally be part of a minimal\nrequirement from a crystallographer.\nResponse #5: Thank you for your insightful comments. We agree that the field of structure solution\nneeds more robust and universally accepted metrics to determine when a structure is successfully\nsolved. Numerical measures like alone, while useful, often do not fully capture the accuracy of a\nRcos\nsolution. We also agree that Rietveld refinement can help improve the final crystal structures. Fig.\nS2 (HoGeAg) and Fig. S7 (NbP) highlight the similarity of powder XRD patterns between incorrect\nICSD structures and our more plausible structures. Since the XRD patterns come from the simulated\nXRD pattern based on incorrect ICSD crystal structures, we sought to obtain the experimental XRD\npatterns of the HoGeAg and NbP to enable more meaningful refinement. However, the HoGeAg was\noriginally determined by neutron diffraction rather than XRD, and the XRD data of NbP from the\noriginal publication in 1954 consists of low-precision peak intensities which are unsuitable for\nrefinement. Therefore, we did not refine these proposed structures by XRDSol in Fig. S2 and Fig.\nS7.\nTo show the benefit of doing Rietveld refinement, we applied Rietveld refinement to an experimental\nXRD pattern from the ICDD database. We chose Li SrGeO (Fig. S13) as an example. The crystal\n2 4\nstructure of Li SrGeO was refined with GSAS-II software (Fig. R2-1b). The refined Li SrGeO\n2 4 2 4\ncrystallizes in the space group P3m1. The refinement converged with factors: R = 14.67%, R =\nwp exp\n10.05%, and χ2 = 2.12 (Fig. R2-1b), indicating a good fit quality between the experimental and\ncalculated diffraction patterns. This represents a substantial improvement over the fit quality without\nRietveld refinement (Fig. R2-1a). We noticed that the R is not sufficiently low. This may be\nwp\nattributed to the fact that the XRD pattern used in the refinement was simulated\n21\nfrom an experimental XRD stick pattern. So the XRD pattern lacks background and noise, which\nare typically present in the experimental XRD data. The absence of these features leads to higher\nR in the refinement process (Powder Diffraction, 2006, 21, 67-70). The refined Li SrGeO structure\nwp 2 4\nexhibits only minor adjustments in atomic coordinates compared to the Li SrGeO proposed by our\n2 4\nXRDSol.\n(a) (b)\nwithout Rietveld refinement with Rietveld refinement\nLi2SrGeO4 Li2SrGeO4\nFig. R2-1 Comparison of observed and calculated XRD patterns for Li 2 SrGeO 4, (a) without and (b)\nwith Rietveld refinement.\nComments #6: I did like that the authors computed the formation energies using DFT and compared\nto the complex hull. It was very reassuring that the model was able to find on hte whole low energy solutions, and indeed some solutions that had lower formation energy than the reported ones. This\ngives good confidence that the diffusion model is learning structures that are energetically stable\n(it was trained on stable structures).\nResponse #6: Thank you for the feedback on our DFT calculations and energy above convex hull\nanalysis. We appreciate your recognition of the ability of our model XRDSol to identify low-energy\nsolutions.\nComments #7: They tested against experimental data with solved structures from ICDD and the\nmodel performed well in most cases. This is good.\nResponse #7: We thank your positive comment regarding our model’s performance against\nexperimental ICDD data.\nComments #8: They also used it to propose structure solutions from some experimental patterns\n22\nthat didn’t have structure solutions proposed in the ICDD and the models performed well. It wasn’t\ncompletely clear the inputs in this case. Was the stoichiometry known and input? And the number\nof formula units in the unit cell? This needs to be placed more prominently.\nResponse #8: Thank you for the comments. The experimental XRD patterns in the ICDD database\nthat we used for structure proposals contained known lattice parameters, stoichiometry, and the\nnumber of formula units per unit cell. However, the corresponding atomic coordinates are not\navailable. The input for solving the crystal structures used in this work includes the target PXRD\npatterns, lattice parameters, number and type of atoms. The lattice parameters refer to the unit cell\nparameters (a, b, c, α, /3, and γ). The lattice parameters define the periodic translation symmetry of\nthe crystal, which differ from the space group. In contrast, space groups provide other symmetry\ninformation, such as reflection, rotation, and mirror symmetries. Our XRDSol uses lattice\nparameters and does not rely on prior knowledge of the space group.\nAction #3: To ensure clarity and highlight the model’s inputs more prominently, the related\ndescription has been provided in the revised manuscript:\n(Paragraph 2 of Page 15 in the revised Manuscript)\n“The input for solving these entries includes known target PXRD patterns, lattice parameters, number\nand type of atoms.”\n(Paragraph 2 of Page 22 in the revised Manuscript)\n“The target PXRD patterns, lattice parameters, number and type of atoms are already known and used\nas the inputs of XRDSol.”\nSome comments on the validity and areas for improvement are:\nComments #9: if the model architecture was built from or inspired by or reused architecture from the\nliterature I encourage the authors to mention this in the methods section where the model is described.\nI would be a bit surprised if it were not built off an existing one or ones because this structure is quite\nsimilar to CDVAE for example (ref 12 in the current manuscript). It is ok to admit where the inspiration\ncame from and what was reused. It gives correct credit to prior workers. This needs to be fixed before\npublication.\nResponse #9: Thank you for the comments. Our model was inspired by CDVAE (Tian Xie et, al.\nCrystal Diffusion Variational Autoencoder for Periodic Material Generation, ICLR, 2022), and its\narchitecture was built upon existing works, including EGNN (PMLR, 2021, 139, 9323-9332),\nCDVAE (Tian Xie et, al. Crystal Diffusion Variational Autoencoder for Periodic Material\nmain text of our original manuscript. In the revised manuscript, we have further explicitly\nmentioned these prior works in the Method section to provide proper credit and clarify the\ninspiration behind our model design.\nComments #10: is it possible to show one or more schematics of the model architecture in a figure?\nthis can be very helpful.\nResponse #10: Thank you for the comments. We have drawn a figure to illustrate the model\narchitecture (Fig. S22, i.e., Fig. R2-2). The related description was added in the revised manuscript.\nForward noising process\nStable structure Random atom positions\n⚫ ● ● ⚫ ● ●\n⚫ ● ● ● ● ●\nS 0 Reverse denoising process St-1 St ST\nSt-1\nPLM\nembedding e\nruta\nef D R X P .taC\nPLM\ne ru\nta\ne f e ru tcu rts\nkcolb\nN N G E\nkcolb\nN N G E\nkcolb\nN N G E\nSt\nPXRD pattern condition EGNN\nReverse denoising process from St to St-1\nFig. R2-2 Schematics workflow of the XRDSol.\nComments #11: In the description of the model the L vector is described but I had to think hard to\nfigure out it was the lattice vectors. Makes sense in hindsight (L) but please add a few words so that\nall the letters are fully defined in the place where they appear.\nResponse #11: Thank you for the comments. We acknowledge that the description of the L matrix\nin the original manuscript could have been clearer. A more detailed description of L is now provided\nas follows:\n(Method section in the revised Manuscript)\n24\n“where 𝑳 represents the lattice vectors of a crystal structure in three-dimensional space, and the lattice\nparameters (a, b, c, α, β, and γ) are used to define the unit cell of the crystal lattice,”\nComments #12: The work would benefit from better baselining. It was baselined against some non\nML approaches that use evolutionary algorithms, For example, and the trained model was much\nfaster (not a huge surprise). But can it be benchmarked against other ML models and simple models\nsuch as nearest neighbor models or something like that?\nResponse #12: Thank you for the valuable feedback. We appreciate the suggestion to compare our\nmodel to other ML approaches. We compared our XRDSol to two recent ML models for crystal\nstructure solutions: CrystalNet (npj Computational Materials, 2024, 10, 209) and\nCrystalyze&StructSnap (Journal of the American Chemical Society, 2024, 146, 44, 30340–30348).\nCrystalNet uses PXRD patterns and chemical composition information as inputs. It achieves a success\nrate of 93.4% on simulated PXRD data of 737 test set, but only for cubic (500 samples) and trigonal\n(237 samples) crystal systems. Its inference time is under a minute per structure.\nCrystalyze&StructSnap is another ML model based on the CDVAE framework. It was tested on 1280\nsamples from the MP-20 test set (out of 9046 total samples) and achieved a success rate of 66.6%.\nWhile the paper of Crystalyze&StructSnap does not provide exact inference times and source code,\nit is inferred to be on the order of seconds to minutes based on the CDVAE framework. In comparison,\nour model achieves a success rate of 82.3% on the full MP-20 test dataset (9046 samples) with an\ninference time of 0.6 seconds per structure. This represents a scale advantage over the two approaches:\nCrystalNet (737 samples) and Crystalyze&StructSnap (1280 samples). While the results of these\nmodels vary depending on the datasets, our XRDSol model shows at least competitive performance\nin terms of success rate and speed. Moreover, our XRDSol has no crystal system restrictions and has\nsolved the largest number of experimental PXRD patterns to date (912 structures, Fig. S11). We\nappreciate your suggestion to refine our comparison. We have added comparisons to these two ML\nmodels in the revised manuscript.\nStrength:\nComments #0:\n- The writing is good and easy to follow.\n- This work is both interesting and novel. Unlike previous methods focused on accurate 3D\nreconstruction, this study explores the use of a diffusion model to approximate spatial structures.\n- The adoption of a graph model to represent the spatial structure of XRD is intuitive and\nnaturally lends itself to being extended and learned by the diffusion model.\n- Compared with numeric methods (e.g., the DFT), the diffusion-based method is efficient with a\nsignificant speedup of at least 10,000x. The estimation accuracy under both the simulation and\nobservation environment is also acceptable for further study of the crystal.\nResponse #0: We truly appreciate the Reviewer for the comments and thorough evaluation of our\nwork. We have addressed the Reviewer’s questions point by point below.\nWeakness&Questions:\nComments #1: (1) Some algorithm details are not clear.\n(a) How is the graph initialized? Is any prior knowledge required? If not, could incorporating prior\nknowledge, such as partial spatial structures, improve prediction accuracy?\n(b) How is the number of diffusion steps determined? Is it consistent across all crystals, or does it\nvary? Would increasing the number of steps consistently enhance performance?\n(c) What is the detailed structure of the model used for the denoising process?\nResponse #1: Thank you for the comments.\n(a) Graph Initialization and Prior Knowledge\nThe graph is initialized based on atomic positions and lattice information. Specifically, the edge\nconstruction follows a fully connected graph, where every atom in the unit cell is connected to\nevery other atom. The edge features are defined as fractional coordinate differences between\nconnected atoms. This method does not require explicit prior knowledge is required beyond the atom\ntypes, atom numbers, and lattice parameters. We agree that incorporating prior knowledge (e.g.,\npartial spatial structures) could potentially enhance prediction accuracy. Partial spatial structures\ncould serve as constraints to guide the model’s predictions and provide physical atomic\n26\ninteractions. To verify this, we conducted a comparative experiment using trigonal crystal system as\nan example. Two versions were tested. One version used our current XRDSol model without any\npredefined partial spatial structures. The other version added prior partial spatial structures using\nWyckoff site properties. In this setup, atom positions at general Wyckoff sites are unknown, but\nthose at special Wyckoff sites were fixed. Thereby, we added partial spatial constraints. As shown\nin Fig. R3-1, using partial spatial structures improved accuracy to 90.5%, which is higher than the\n77.4% achieved without such constraints. This result supports the reviewer’s suggestion and gives\nus a solid direction for improving our model.\n90.5 %\ne 77.4 %\nta\nr\ns\ns\ne\nc\nc\nu\nS\nNo. Constraints Constraints\nFig. R3-1 Success rates of trigonal crystal system with or without constraints.\n(b) Diffusion steps and their impact on the model’s performance\nIn our model XRDSol, the number of diffusion steps is set to 1000, which is commonly used in the\nfield (NIPS, 2020, 33, 6840-6851; CVPR, 2022, 10674-10685). The diffusion steps across all crystal\nstructures are consistent. To assess the impact of the number of diffusion steps on the performance,\nwe conducted experiments comparing different diffusion steps, including 500, 750, 1000, and 1250\nsteps. We investigated the different setups on 5 batches (batch size = 128) in the MP-20 test set. As\nshown in Fig. R3-2, the performance slightly improves with an increase in diffusion steps. However,\nfurther increasing the number of steps leads to a stable performance.\n(c) Detailed structure of the denoising process\nThe detailed structure of the model XRDSol used for the denoising process has been shown in Fig. S22,\ni.e., Fig.R3-3. The reverse denoising process aims to reconstruct the original structure S 0 from S T with\nrandom atomic coordinates. This is achieved using an equivariant graph neural network 𝜙 (EGNN)\n(PMLR, 2021, 139, 9323-9332). The EGNN predicts the noise to construct the desired samples.\nEGNN is a type of Graph Neural Network that satisfies the equivariance constraint. It ensures the\nreflection, translation, rotation, and mirror symmetries of crystal structures. The network operates on a fully\nconnected graph G, where each node represents an atom with coordinates and features. The EGNN\nincludes 3 EGNN blocks, each composed of equivariant convolutional layers. The noisy structure S is\nt\npassed through the EGNN to extract structure features. And the structure features are concatenated with\nthe PXRD features which are derived from the target PXRD pattern by using a neural network. Next,\nthe concatenated features are fed into a neural network to predict the atomic coordinates of This\nSt-1.\nprocess is repeated for 1000 steps and gradually restores the atomic coordinates from S to S with the\nT 0\nguidance of target PXRD patterns.\nStable structure Random atom positions\n⚫ ● ● ⚫ ● ●\n⚫ ● ● ● ● ●\nS0 Reverse denoising process St-1 St ST\nSt-1\nPLM embedding e ruta\nef D R X P .taC\nPLM\ne ru ta\ne f e ru tcu rts\nkcolb\nN N G E\nkcolb\nN N G E\nkcolb\nN N G E\nSt\nPXRD pattern condition EGNN\nReverse denoising process from St to St-1\nFig. R3-3 Schematics workflow of XRDSol.\nComments #2: The diffusion process involves inherent randomness. What is the statistical\nsignificance of the reported results, considering this randomness?\nResponse #2: Thank you for the comments. We agree that the diffusion process in our model XRDSol\ninvolves inherent randomness, which can cause variations in the results. In this work, we have\nconsidered the impact of randomness. To quantitatively assess the impact of this randomness on\ncrystal structure solution from powder XRD patterns, we conducted multiple runs per XRD pattern\nranging from 1 to 50 times on the first batch (128 samples). Then we analyzed the effect of different\nrun counts on the performance of structure determination. As shown in Fig. S20, the XRD matching\nscore (R ) exhibited a rising trend as the number of runs increased from the 1st to the 25th run and\ncos\nthen it stabilized in the range of 25 to 50 runs. The result indicated that after 25 runs, the impact of\nrandomness on the final crystal structure solution becomes minimal. Therefore, for each XRD pattern\nin this work, we performed 25 independent runs using XRDSol to generate 25 potential crystal\nstructures. We then rank these solutions based on R and select the top-ranked solution (highest R )\ncos cos\nas the final crystal structure for the given XRD pattern. This ranking-based selection ensures that the\nreported results are statistically stable and minimally affected by the inherent randomness of the\ndiffusion process.\nsoc\nRuns (times)\nFigure S20. The R are within 50 runs on the first batch (128 samples).\ncos\nComments #3: Figure 2(b) shows examples with relatively low estimation accuracy. What\ndistinguishes these \"bad\" examples from others? Do they share specific characteristics, such as\nbelonging to a similar group? While the paper effectively highlights the success of using the\ndiffusion model for accurate XRD predictions in most cases, it would be helpful to provide a more\ndetailed analysis of these failed examples.\nResponse #3: Thank you for the insightful comments. The “bad” examples, which exhibit relatively\nlow estimation accuracy, tend to share two key characteristics:\n(1) A larger number of atoms in the unit cell: The majority of these bad examples have a larger\nnumber of atoms. The MP-20 dataset used in this work contains compounds with up to 20 atoms\nper unit cell. As shown in Fig. R3-4a, the success rate remains above 96% when the atomic count\nis 7 or fewer. However, as the number of atoms increases, the success rate steadily declines, falling\nbelow 90%, and reaching as low as 50% for compounds with 17 atoms.\n(2) Low symmetry: The accuracy of the diffusion model is also significantly affected by the\nsymmetry of the crystal structure. As the symmetry decreases, the model’s performance tends to\ndegrade (Fig. R3-4b). Specifically, the success rate varies across the seven crystal systems with\ndifferent symmetries (cubic 97.72%, hexagonal 90.33%, trigonal 77.86%, tetragonal 88.02%,\northorhombic 76.32%, monoclinic 68.91%, and triclinic 48.48%). Among these, the cubic crystal\nsystem shows the best performance with a success rate of 97.72%. In contrast, the triclinic system\ndemonstrates the poorest accuracy, with a success rate of only 48.48%.\n30\n(a) (b)\n)%\n(\ne\n(\ne\nta\nta\nr\nr\ns\ns\ns\ns\ne\ne\nc\nc\nc\nc\nu\nu\nS\nS\nNumber of atoms / unit cell\nFig. R3-4 (a) Variation in success rate as a function of atomic count in the unit cell. (b) Success\nrate across the seven crystal systems with different symmetries.\n(Paragraph 2 of Page 19 in the revised Manuscript)\n“In the future, we will focus on overcoming the current limitations in handling large-unit-cell\ncrystals and low-symmetry systems.”\nComments #4: Does the number of cells affect estimation accuracy? Do different space groups\nexhibit similar accuracy trends?\nResponse #4: Thank you for the comments. First, to investigate the influence of the number of cells\non the estimation accuracy, we take the crystal structures with 6 atoms per unit cell as an example\n(Fig. R3-5). Specifically, we compared the success rates between 1×1×1 unit cell and 1×1×2 unit cell\nfor the crystal structures. The success rate of 1×1×1 unit cell achieves 96.70%, while the success rate\nof 1×1×2 unit cell drops to 46.57%. This result indicates that increasing the number of cells (from\n1×1×1 to 1×1×2) leads to a substantial decrease in estimation accuracy. Second, the accuracy varies\nacross different space groups. In this work, we exclude those space groups with 10 or fewer samples\nin the MP-20 test dataset, which can provide a clearer picture of the accuracy trends across different\nspace groups. As shown in Fig. R3-6, the accuracy varies across different space groups. Generally,\nspace groups with higher symmetry tend to exhibit higher estimation accuracy. Specifically, 26 space\ngroups achieved an accuracy of 100%, and 41 space groups had an accuracy greater than 90%.\nHowever, there are 6 space groups with accuracy lower than 50%, namely space groups 2, 19, 46, 51,\n55, and 156. This suggests that symmetry plays a significant role in the model’s ability to predict\ncrystal structures accurately.\n31\n)%\n(\ne\nta\nr\ns\ns\ne\nc\nc\nu\nS\n1×1×1 unit cell 1×1×2 unit cell\nFig. R3-5 Comparison of success rates between 1×1×1 unit cell and 1×1×2 unit cell for the crystal\nstructures with 6 atoms per unit cell.\n2 5 7 9 11 31 51 32 62 33 63 93 34 64 15 55 95 36 56 76 96 17 47 28 78 79 701 411 611 911 221 721 631 931 141 741 941 351 551 061 261 461 761 381 781 191 491 002 502 712 122 422 722\n)%\n(\ne\nta\nr\ns\ns\ne\nc\nc\nu\nS\n1 4 6 8 01 21 41 91 52 13 53 83 24 44 74 25 85 26 46 66 86 07 27 97 68 88 99 901 511 811 121 321 921 731 041 641 841 051 451 651 161 361 661 671 681 981 391 891 402 612 022 322 522 922\nSpace group\nFig. R3-6 Success rates across the 107 space groups.\nComments #5: Is there any way to use DFT or something else to further refine the structure?\nResponse #5: Thank you for the comments. In principle, using DFT to refine structures is feasible, but\nwe do not expect it to yield satisfactory results. Structure optimization using DFT are typically\nperformed at 0 K by minimizing system energy, ignoring temperature effects. Besides, the optimized\nstructures depend on the choice of pseudopotentials, symmetry constraints, and other parameter\nchoices. For instance, the commonly used PBE functional underestimates van der Waals interactions\nin layered structures, potentially leading to errors in interlayer spacing. Additionally, for ferromagnetic\nelements like Fe, Co, or Ni, different spin states can alter bond lengths. For cases where DFT cannot\nconverge to a spin state consistent with the experiment, it will lead to incorrect bond lengths. Due to\nthese limitations, we chose not to use DFT for further structural refinement.\nrefinement can account for experimental effects (e.g., preferred orientation) while DFT cannot directly\nincorporate these real-world factors. We applied Rietveld refinement to an experimental XRD pattern\nfrom the ICDD database. As a representative example, we chose Li SrGeO (Fig. S13) proposed by\n2 4\nXRDSol in the ICDD database to demonstrate that the Rietveld refinement can further refine the\nstructures. The crystal structure of Li SrGeO was refined with GSAS-II software (Fig. R3-7b). The\n2 4\nrefined Li SrGeO crystallizes in the space group P3m1. The refinement converged with factors: R =\n2 4 wp\n14.67%, R = 10.05%, and χ2 = 2.12 (Fig. R3-7b), indicating a good fit quality between the\nexp\nexperimental and calculated diffraction patterns. This represents a substantial improvement over the fit\nquality without Rietveld refinement (Fig. R3-7a). We noticed that the R is not sufficiently low. This\nwp\nmay be attributed to the fact that the XRD pattern used in the refinement was simulated from an\nexperimental XRD stick pattern. So the XRD pattern lacks background and noise, which are typically\npresent in the experimental XRD data. The absence of these features leads to higher in the\nRwp\nrefinement process (Powder Diffraction, 2006, 21, 67-70). The refined Li SrGeO structure exhibits\n2 4\nonly minor adjustments in atomic coordinates compared to the starting Li SrGeO proposed by our\n2 4\nXRDSol.\n(a) (b)\nwithout Rietveld refinement with Rietveld refinement\nLi2SrGeO4 Li2SrGeO4\nFig. R3-7 Comparison of observed and calculated XRD patterns for Li 2 SrGeO 4, (a) without and (b) with\nRietveld refinement.\nComments #6: It would be beneficial to include more details about the diffusion model to enhance\nclarity for the materials science community.\nResponse #6: Thank you for the comments. A more detailed description of the diffusion model is\nnow provided in the revised manuscript as follows:\n33\n(Method section in the revised Manuscript)\n“The diffusion model is a generative model inspired by non-equilibrium thermodynamics. It\nincludes a forward noising process and a reverse denoising process (Fig. S22). During the\nforward process, random Normal distribution noise is gradually added to the input data 𝑥0\nvia T steps, thereby establishing a Markov chain 𝑥 0 → 𝑥 1 → ⋯ → 𝑥 𝑡 → ⋯ → 𝑥 𝑇. This\nnoising process can be described by the transition distribution as 𝑞(𝒙 𝑡 |𝒙 𝑡−1 ) = 𝒩(𝒙 𝑡\n|𝛼 𝑡|𝑡−1𝒙 𝑡−1 , 𝜎 𝑡|𝑡−1\n2 𝐈) , where the\nparameter 𝛼 𝑡 controls the amount of 𝑥0 retained and 𝜎 𝑡 controls the amount of noise added.\nFor XRDSol, we perform diffusion on the atomic coordinates of the given structure 𝑺, which\ndefines a Markov chain 𝑺0 → 𝑺1 → ⋯ → 𝑺 𝑡 → ⋯ → 𝑺 𝑇. Due to the periodicity of stable\ncrystal structures, we add wrapped Normal distribution noise 𝒩 𝑊 on the atomic coordinates\nto better model the periodic process. We define our forward process via transition distribution\nas 𝑞(𝑺 𝑡 |𝑺 𝑡−1 ) = 𝒩 𝑊 (𝑺 𝑡 |𝛼 𝑡 |𝑡 −1𝑺 𝑡 −1, 𝜎 𝑡|𝑡−1\n2 𝐈). During the reverse denoising process, the equivariant graph neural\nnetwork (EGNN) 𝜙 predicts the noise to construct the desired samples. EGNN is a type of Graph\nNeural Network that satisfies the equivariance constraint. In this work, the graph is initialized by a\nfully connected graph 𝐺 based on atomic positions and lattice parameters. Each node is endowed with\ncoordinates as well as features. The edge construction follows a fully connected graph 𝐺 , where\nevery atom in the unit cell is connected to every other atom, forming a fully connected graph. The\nedge features are defined as fractional coordinate differences between connected atoms. In this setting,\nEGNN consists of 3 EGNN blocks, each composed of equivariant convolutional layers. The noisy\nstructure S is passed through the EGNN to extract structure features. And the structure features are\nt\nconcatenated with the PXRD features which are derived from the target PXRD pattern by using a\nneural network. Next, the concatenated features are fed into a neural network to predict the atomic\ncoordinates of S . This process is repeated for 1000 steps and gradually restores the atomic\nt-1\ncoordinates from S to S with the guidance of target PXRD patterns. We define the reverse denoising\nT 0\nprocess via transition distribution as 𝑝(𝑺 𝑡−1 |𝑺 𝑡 ) = 𝒩 𝑊 (𝑺 𝑡−1 |𝜙(𝑺 𝑡 ), 𝜎 𝑡 →𝑡−1\n2 𝐈). To solve\ncrystal structures conditioned on given XRD patterns 𝐶, we input the XRD condition along with\natomic coordinates into the neural network to guide the correct placement of atoms. The final\ndenoising process was defined as 𝑝 (𝑺 𝑡−1 |𝑺 𝑡 , 𝐶) = 𝒩 𝑊 (𝑺 𝑡−1 |𝜙(𝑺 𝑡 , 𝐶), 𝜎 𝑡→𝑡−1\n2\n𝐈).”\nRemarks on code availability:\nComments #7: The code is open-source with all data available, with good instructions to reproduce\nthe results.\nResponse #7: Thank you for the evaluation of our code.\nComments #0: I co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training in peer review and\nto provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse #0: Thank you for the co-reviewed comments.\nRemarks on code availability:\nComments #1: The code is clean and organized. The instructions are clean to install the software.\nThe data is also available.\nResponse #1: We appreciate your evaluation on the code and instructions.\n(Black italic: Reviewer’s comments; Black type: Our response; Blue type: Key revisions)\nResponses to Reviewer #1:\nComments #0: This revised manuscript details a particularly fast ML informed tool for predicting\natomic positions given a PXRD pattern and unit cell data. As was noted by multiple reviewers,\nthis paper exists in a somewhat strange intermediate space between computer science and\nchemistry. From a chemistry perspective, it is not written in a way to convince the field that this\nwould be useful, although the point made about autonomous materials discovery and the\nparticularly time efficient nature of this tool is a compelling one that should be centered in the\nintroduction (rather than a general overview to crystallography). Although I find the manuscript\nimproved upon revision in terms of clarity and better defining the tool developed, by not\nsignificantly revising the manuscript to address the source of reviewer comments, it still seems\nlike a computational tool was developed first and a chemical problem was searched for second.\nIt remains unclear what chemical problem is actually being solved, and the question remains how\nthe authors envision it will be used? Can the authors include a demonstration/provide an example\nof how it can be implemented completely autonomously? Or how the process of correcting a\ndatabase would be performed (it is not stated how the potentially incorrect ICSD structures were\nidentified, or can an entire database be screened at once?). I find this issue embodied by the lack\nof chemical analysis presented in the discussion section, which is too speculative in ways that are\nnot specific enough to bridge the disconnect.\nResponse #0: We truly appreciate your thorough evaluation and the opportunity to clarify the\nchemical relevance of our work. We fully recognize the importance of clearly articulating the\nchemical problem. To address concerns regarding the lack of chemical focus, here we provide\na detailed description of how our XRDSol can be used for the inspection and correction of\nentries in large-scale inorganic structural databases.\nDemonstration of the inspection and correction of structural databases (e.g., ICSD)\nIn response to the reviewer’s request for a concrete example of a quality control step for entries\nin structural databases, we have added more details in the Results and Supplementary\nInformation that describe a complete workflow for screening and correcting potentially\nincorrect database entries.\nIn the original manuscript, we indeed conducted a systematic screening and correction of the\nordered structures of the ICSD database. The workflow is shown in Fig. S7a, i.e., Fig. R1-1a. We\nbegan by collecting ordered structures from ICSD (Table S2, i.e., Table R1), which\n1\nare available through the Materials Project database. All of them are ordered structures (49283\nstructures), and with DFT-calculated energy above the convex hull (E ) available. We screened\nhull\nthese entries for ones with poor thermodynamic stability (high E ). We set the E threshold to\nhull hull\n0.1 eV/atom, which is a commonly used criterion for assessing thermodynamic stability. Crystal\nstructures with high E values (> 0.1 eV/atom) were identified as high-energy structures that are\nhull\npotentially erroneous.\nDue to the architectural constraints of our XRDSol model (limited to primitive cells\ncontaining 20 atoms or fewer), we filtered these high-energy structures accordingly, yielding\n3,122 candidate structures for re-evaluation. For each of them, we simulated the powder\nXRD patterns based on the available structure and extracted the lattice parameters and the\nnumber and type of atoms. They are used as inputs for XRDSol. XRDSol then predicted the\natomic coordinates to reconstruct the crystal structures. Among the resulting predictions,\n1429 proposed structures differ from the original ICSD structures (Table R1). Within this subset,\n588 structures exhibited good agreement with the target simulated XRD patterns (R > 0.9)\ncos\nand were selected to perform further DFT calculations. Only those candidates that exhibit both\nlow and high XRD matching quality were retained as plausible corrected structures for\nEhull\nmanual inspection. Finally, 39 high-confidence corrections that we report in the original\nmanuscript are identified (Table R1).\nIt is worth noting that within the 588 structures with good XRD agreement, some still have\nE values exceeding 0.1 eV/atom. Although these might be classified as unstable by strict\nhull\nthermodynamic standards, they may still represent valid metastable or experimentally realizable\nphases. Nevertheless, to uphold rigorous crystallographic standards, we conducted manual\ninspections and extensive literature reviews. As a result, only the 39 most reliable corrected\nstructures are presented in the manuscript. The remaining candidates, while promising, will\nrequire further experimental validation or manual inspection to confirm their correctness and\nchemical plausibility.\nAnother issue worth noting is that we used simulated PXRD patterns based on available\nICSD entries. Ideally, it would be better if we could use the original PXRD patterns, which are\nunavailable from ICSD, since the simulated PXRD pattern may also be problematic if the\nstructures are potentially incorrect.\nCase study: CsIO (ICSD-33665)\n3\nan example (Fig. S7b, i.e., Fig. R1-1b). In this early study, CsIO was reported to be a\n3\n̅\ncovalent perovskite material with a cubic perovskite structure in the Pm3 m space group, where Cs\noccupies the six-coordinate B-site. The calculated E of this original structure (ICSD-\nhull\n2\n33665) is extremely high at 1.97 eV/atom, which strongly suggests an incorrect atomic\narrangement. We simulated the corresponding XRD pattern and extracted the lattice parameters\nand the number and type of atoms based on the originally reported CsIO structure. We used these\n3\nas inputs to XRDSol. 5 atoms (1 Cs, 1 I, and 3 O atoms) with random atomic coordinates were\ninitialized in a primitive cell (a = b = c = 4.67 Å, α = β = γ = 90°). XRDSol then predicted atomic\ncoordinates to reconstruct the CsIO crystal structure. In the CsIO structure proposed by XRDSol,\n3 3\nthe larger Cs cations are moved to the twelve-coordinate A-sites. Subsequent DFT calculations\nconfirmed the thermodynamic stability of this solution with a significantly lower E of 0.02\nhull\neV/atom. Moreover, the structure solution is very close to the experimentally resolved single-\ncrystal structure of CsIO (Chemistry of Materials, 2018, 1136-1145). The\n3\n̅\nstudy clarified that the earlier assignment of Pm3 m space group in 1928 using powder X-ray\ndiffraction was inaccurate. Thus, this case demonstrated XRDSol’s capability to identify and\ncorrect some suspicious structure reports.\nFurthermore, we believe our work goes beyond merely providing a new tool for chemistry\nor materials science. Rather, this study is inherently interdisciplinary, spanning chemistry,\nmaterials science, and computer science. As AI for chemistry and materials science is an emerging\nand rapidly evolving field, the terminology in this interdisciplinary domain is less mature and\nwell-defined compared to established fields like crystallography. Therefore, we strive to describe\nour work in language accessible to researchers across all these disciplines. We have also included\na more detailed discussion in the Introduction section of the main text about the advancements of\nAI tools in crystal structure generation and PXRD analysis to better position our own work within\nthe context of community progress as follows:\n(Introduction in the revised Manuscript)\n“Recently, machine learning approaches, particularly generative diffusion models, have provided\na new paradigm for materials structure generation. Xie et al. built the Crystal Diffusion\nVariational Autoencoder (CDVAE), which employs score matching to generate realistic crystal\nstructures by learning from the data distribution of known stable materials (Tian Xie et, al. Crystal\nDiffusion Variational Autoencoder for Periodic Material Generation, ICLR, 2022) . Without any\npredefined prototypes, CDVAE shows its ability to generate valid and diverse crystal structures\nthat captures the physical inductive bias of material stability.”\n“Equivariant diffusion can effectively preserve reflection, translation, rotation, and mirror\nsymmetries, enabling generation of physically reasonable crystal structures. Such equivariant\ndenoising models were employed to generate inorganic crystal structures and molecules, and\nshowed remarkable performance. Besides generating crystal structures without constraints,\nconditioning diffusion models allow crystal generation with desired chemistry, mechanical,\n3\nelectronic, or magnetic properties, paving the way towards generative AI-assisted materials\ndesign paradigms.”\n“Recently, Lipson et al. proposed a generative diffusion model, PXRDnet, to determine the\ncrystal structures from nanocrystalline powder diffraction patterns. This approach addresses the\nchallenge of peak broadening effects due to nanosized crystallites, and solved 200 structures of varying\nsymmetry and complexity from simulated nanocrystalline PXRD patterns.\nConsequently, the conditional equivariant diffusion model holds tremendous potential for\ndetermining atomic coordinates under PXRD pattern conditions.”\nTable R1-1. The statistics of solved entries in ICSD.\nStructures Number\nOrdered structures in the ICSD database 49283\nPotentially incorrect structures with unit cell ≤ 20 atoms (Ehull > 0.1 eV/atom) 3122\nXRDSol proposed structures that differ from the original ICSD structures 1429\nXRDSol proposed structures that show good XRD match (Rcos > 0.9) 588\nXRDSol proposed structures that show low energy (Ehull < 0.1 eV/atom) 39\n(a) Workflow of correcting potentially incorrect ICSD structures\nI o C rd S e D re d d s a tr t u a c b tu a r s e e s D\nM\nF\na\nT\nte\nc\nri\na\na\nl\nl\nc\ns\nu\np\nla\nro\nti\nj\no\ne\nn\nc t E\nH\nhu\ns i\nll\ng t\n> r h u\n0\n- c\n.\ne\n1\nt n u\ne\ne r\nV\ne r\n/\ns g\nat\ny\no m simu\nIn\nla\np\nt\nu\ne\nt\nd XRD,\npr\nc\ne X\no\nd\no\nR i\nr\nc\nd\nt D\ni\na\nn\nS t\na\no o\nt\nm\ne\nl\ns i c\nX\nD\nR\nF\nD\nT c\nm\na\na\nlc\ntc\nu\nh\nla tion Pla\ng\nu\no\ns lo\no\ni w\nd\nb e l\nX\ne r\nR\ne s\nD\nn t e r\nm\nr u g\na\nc y\nt\nt ,\nc\nu\nh\nr es\nlattice parameters,\ntype and number of atoms.\n(b) Example: Correcting CsIO3 structure in ICSD\nPlausible CsIO3 structure\nInput lower energy: Ehull = 0.02 eV/atom\ngood XRD match: Rcos = 0.999\n).u\nC E s hu IO\n(\nll\nh\n= 3\ni g\n( 1 I\nh\n. C 9\n-e\nS 7\nn\nD e\ne\n- V\nr\n3\ng\n/ 3 a\ny\nt\n) o 6 m 5)\n.a ( ytis n e tn\nI\nSimulated XRD\np c re\nX\no d o\nR\ni r c d\nD\nt in a\nS\na to\no\nte m\nl\ns i c ).u .a\n( ytis\nn e\nRe\nR\nc\nc\no\nS\no\nn\ns\ni m\ns\n=\nt\nu\nr\ns\nt\nu\nl\na\nc\nt\nt\ne\ne\nd\nd\nX\nX\nR\nR\nD\nD\ntn\nCs  2 Theta (deg.)\nDFT calculation:\nEhull = 0.02 eV/atom\natoms Chemistry of Materials, 2018,\nO  3 lattice CsIO3 (this work) 1136-1145\nFig. R1-1 (a) Workflow of XRDSol as the inspection and correction for entries in ICSD. (b)\nIllustration using CsIO (ICSD-33665) from the ICSD database as an example.\n3\nComments #0: I co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nResponse #0: We thank the Reviewer for the co-reviewed comments.\nComments #0: The authors have worked hard to address my comments, but I still feel that they\nfall somewhat short. I am not sure if they have a professional crystallographer among the authors,\nbut tha paper still lacks insights that I would expect from such a person, and since the paper is\nreally about developing a tool for crystallographers to do their crystallography, this is a\nsignificant shortcoming. A reading of the comments of the other referees and the author responses\ndoesn't do anything to allay this impression. With that said, the paper is greatly improved from\nthe original version. I focus primarily on responses to my comments.\nResponse #0: We thank the Reviewer for the thoughtful and constructive feedback. We\nacknowledge the previous limitations in crystallographic depth. We have followed your advice\nand actively sought collaborations with a professional crystallographer, who has since joined\nas a co-author. With his expertise, we have further revised the manuscript to strengthen its\ncrystallographic rigor. We hope these improvements address your concerns, and we will\nrespond to your concerns point by point below.\nComments #1: I appreciate their efforts to place their work in context of the PXRDNet paper\nthat was mentioned. This is mostly handled well, though I note that the version that has now\nappeared in print does have a post-validation step which invalidates one of their comments in\nthe rebuttal.\nResponse #1: Thank you for your insightful comments. We acknowledge that the published\nversion of PXRDNet includes a post-validation step.\nComments #2: In Response 3 the authors attempt a justification of the reason for mismatched\nBragg peak intensities in some of their solutions. They mention experimental things such as\ntemperature and absorption effects. However, these cannot explain the differences in relative\nBragg peak intensities that are a characteristic of atoms being in incorrect positions that I was\nreferring to in my original comment, and so they have not successfully addressed my comment,\nunfortunately and this issue still remains. They did work hard to incorporate a Rietveld post-\nprocessing step, at least on one representative sample. As one might expect, it did result in an\nimprovement of metrics like Rwp. However, they did not select one of the patterns that had\nproblematic relative peak intensities. It is therefore not quite clear whether this would have\ncorrected the problematic cases that showed success by the MaxDist (thanks for fixing the\ndescription of this in the manuscript) measure, but which were almost certainly not actually correct\nin some key way based on a visual assessment of the diffraction pattern. I also find their rebuttal\non the basis that their data are from simulated data so the Rietveld will give a lower\nattempt to evade the question rather than address it head on as a serious scientist would seek\nto do. All in all, these responses reinforce an impression that the crystallography expertise in\nthe group is lacking. The authors have worked hard on this work and it deserves better. In\nsummary, the authors did not address the key point of my original objection in comment #5 with\nany of their responses.\nFor reference, we also paste your previous comment #3 and comment #5 here.\nPrevious Comments #3: This is a significantly simplifying assumption because it results in the\nBragg peaks being in the right positions and the model simply changes the intensities of the\npeaks when it atoms around. It gives a subjective impression that the model is working better\nthan it is because the Bragg peaks are all in the right locations, but close inspection of the\nfigures shows that even \"successful\" solutions have the intensities that don’t match. It is not so\nclear that a crystallographer would be satisfied with these structure solutions. This is expanded\non a bit below.\nPrevious Comments #5: In general, the structure solution field needs better measures of when\nstructures are successfully solved. The problem with this is that there is disagreement even among\ncrystallographers about the exact definition. For example, an Rcos of 0.899 sounds pretty good\n(almost 90% good!), but a crystallographer looking at the powder patterns in figure S7 of the\nsupplementary would be pretty surprised with how bad it is. Clearly wrong. Even the Rcos of 0.974\nin figure S2 has a bunch of problems. These could possibly be fixed by a final Rietveld refinement\nstep, which was not done by these authors, but this would normally be part of a minimal\nrequirement from a crystallographer.\nResponse #2: Thank you for your insightful comments. We now better understand the core of\nyour original comment: namely, that in some of our solutions, the “relative Bragg peak\nintensities” differ significantly from those in the reference data, in a way that suggests not just\nexperimental noise (e.g., temperature effects, absorption effects, preferred orientation,\ncrystallite size, and instrument settings), but may due to incorrect atomic positions. We\nacknowledge that our previous response did not fully address this concern, and we appreciate\nthe opportunity to clarify.\nFirst, we explain why we did not perform Rietveld refinement for the previous Figure S7 and\nS2 (corresponding to the HoAgGe and NbP in the main text Figure 3), and why visual assessments\nof these diffraction patterns look bad. Both entries were documented in ICSD, and DFT\ncalculations suggested poor thermodynamic stability for these two structures (high E ,\nhull\n0.17 eV/atom for HoAgGe, 0.6 eV/atom for NbP), indicating that the documented structures\nmight be incorrect. Ideally, we should attempt to solve these structures using the original\nexperimental PXRD patterns. However, ICSD is a structural database that only provides\nstructure information, but no diffraction data is available. We also looked into the original\nliterature (Journal of Alloys and Compounds, 1998, 92-98 and Acta Chemical Scandinavica,\n1954, 226-239). We found that for HoAgGe, only neutron diffraction data is documented, while\nPXRD data is not available. For NbP, the PXRD pattern is relatively low quality, as the\nexperiment was performed more than 70 years ago. Below is the original diffraction data for\nNbP (Acta Chemical Scandinavica, 1954, 226-239).\nWe observed that all intensities I are multiples of 5. After consulting with professional\nobs\ncrystallographers’ opinions, we believe this may be because early experiments did not use digitized\nintensity recordings, but rather researchers estimated diffraction intensities visually. Therefore, we\ngave up the attempt to perform Rietveld refinement on the original PXRD pattern.\nAs we explained, since the ICSD lacks diffraction data, we adopted an alternative strategy.\nFor structures in the ICSD that we deemed suspicious (with extremely high E based on DFT\nhull\ncalculations), we hypothesized the existence of different but more reasonable structures that have\nsimilar PXRD patterns to the suspicious structures documented in the ICSD. In this case, we can\nfirst calculate a PXRD pattern based on the documented structure in the ICSD and then use our\nXRDSol to solve it. Since these calculated PXRD patterns are based on a suspicious crystal\nstructure documented in the ICSD, which may be incorrect, rather than on\n8\nexperimentally measured PXRD patterns, we believe that performing Rietveld refinement in\nthis case is not reasonable, even from a crystallographic perspective.\nNow we address your concern about whether these inconsistencies in peak intensities\noriginate from incorrect atomic positions. In other words, whether the solutions we provided\nare indeed correct. We mainly rely on two evaluation methods. First, DFT calculations\nconfirmed that our proposed structures in Figure 3 have much lower energy compared to the\npreviously documented ones. Second, we conducted a comprehensive literature review and\ndirectly compared our proposed crystal structures with those reported in the literature. This\ndirectly addresses your concern about whether our proposed structures have incorrect\natomic positions.\nWe now use root mean square deviation (RMSD) of atomic positions, which is widely\naccepted in crystallography as a metric, to directly compare our proposed structure with some\nrecently validated structures. As can be seen, the structures given by our model are almost identical\nto these correct structures. Therefore, in most cases, we have replaced XRD pattern comparison\nwith direct structure comparisons. To provide a more rigorous analysis, we now present detailed\ncase-by-case discussions of the total present 6 ICSD corrected structures.\n(1) HoAgGe: HoGeAg was originally reported in 1998 based on neutron diffraction (ICSD-\n86732, Journal of Alloys and Compounds, 1998, 92-98), but the reported structure has an E of\nhull\n0.17 eV/atom. This E is extremely high for an intermetallic compound, indicating poor\nhull\nthermodynamic stability. The original work did not provide an experimental PXRD pattern. The\nsolution proposed by XRDSol is thermodynamically stable (E = 0 eV/atom). Recently, Zhao\nhull\net al. (Science, 2020, 1218-1223) determined the crystal structure of HoGeAg using high-\nconfidence single-crystal XRD at 15 K. Although the lattice parameters reported in that study (a\n= 4.1864, c = 7.0862Å) differ slightly from ours (a = 4.1136, c = 7.3431Å) which was measured\nin ambient conditions, the atomic coordinates predicted by XRDSol exhibit good agreement with\nthe experimental structure, with an RMSD of 0.04 Å. Instead of comparing simulated XRD\npatterns of the incorrect ICSD structure and our solution, we present a structural comparison in\nFig. S17, i.e., Fig. R2-1. This highlights the difference between the incorrect ICSD structure\n(Fig. R2-1a) and XRDSol prediction (Fig. R2-1c), and demonstrates the consistency between\nthe XRDSol prediction and the experimentally validated structure (Science, 2020, 1218-1223)\n(Fig. R2-1b).\n9\n(a) ICSD-86732 (b) Zhao et al. (c) This work\n(1998) (2020)\nPublished Data figures Redacted\nHo Ag Ge\nFig. R2-1 Structural comparison of HoGeAg. (a) ICSD-reported structure (ICSD-86732, Journal\nof Alloys and Compounds, 1998, 92-98); (b) Recently reported experimental structure (Science,\n2020, 1218-1223); (c) Structure proposed by this work, consistent with (b).\n(2) SrPbF : SrPbF (ICSD-25629) was first synthesized and characterized by Hoppe in 1958\n6 6\n(Zeitschrift für anorganische und allgemeine Chemie, 1958, 251-263). It was originally proposed\nto crystallize in space group P4 /mmc. However, the original report noted that the presence of\n2\nseveral unindexed peaks in the XRD diffraction pattern which were attributed to another\nunknown phase. First principles calculations reveal that the original structure has a high E of\nhull\n0.16 eV/atom. The crystal structure proposed by XRDSol is predicted to be a stable compound\n(E = 0 eV/atom). In 2018, the compound characterized by Hoppe in 1958 was experimentally\nhull\nrevisited, and its space group was corrected to be P4 /mcm (Zeitschrift für anorganische und\n2\nallgemeine Chemie, 2018, 1721-1726), where Sr is eight-coordinated instead of ten-coordinated\nby F. This revisited structure perfectly matches the PXRD profile obtained from the\nexperiment with no unindexed peaks, suggesting its correctness. In the original manuscript\n(Fig. S3), we compared simulated XRD patterns of the incorrect 1958 ICSD structure and our\ncorrected XRDSol structure, which led to unmatched peaks due to structural discrepancies. We\nnow present a direct structural comparison among the 1958 ICSD structure, the 2018 reported\nstructure, and our XRDSol prediction (Fig. S18, i.e., Fig. R2-2). The structure proposed by our\nXRDSol (Fig. R2-2c) closely matches that of the 2018 revisited structures (RMSD = 0.08 Å)\n(Fig. R2-2b).\n10\n(a) ICSD-25629 (b) Bandemehr et al. (c) This work\n(1958) (2018)\nPublished Data Figures Redacted\nSr Pb F\nFig. R2-2 Structural comparison of SrPbF . (a) ICSD-reported structure (ICSD-25626,\nby Bandemehr et al. (Zeitschrift für anorganische und allgemeine Chemie, 2018, 1721-1726);\n(c) Structure proposed by this work, consistent with (c).\nHalf-Heusler compounds (NbSnRh and HoPdSb): NbSnRh and HoPdSb are both Half-\nHeusler. In this structure type, three equal molar elements form three interpenetrating face-\ncentered cubic sublattices, and only one element is eightfold coordinated. The originally\nreported NbSnRh (ICSD-105220, Metallurgical Transactions, 1970, 3159-3162) and HoPdSb\n(ICSD-108547, Journal of the Less Common Metals, 1980, 25-28) exhibit high E of 0.704\nhull\nand 0.276 eV/atom, respectively. These extremely high E values indicate poor\nhull\nthermodynamic stability. XRDSol proposed different site assignments for the two Half-Heusler\ncompounds. The structures predicted by XRDSol have much lower E (0 eV/atom for both),\nhull\nsuggesting the solutions from XRDSol are more plausible. As shown in Fig. R2-3, the solutions\nby XRDSol also match the recently reported results (Advanced Materials, 2023, 2210788), with\nRMSD values of nearly 0 Å in both cases, respectively.\nNb Sn Rh\n(a) ICSD-105220 (b) Schmidt et al. (c) This work\n(1970) (2023)\nPublished Data Figures Redacted\nHoSbPd\nHo Sb Pd\n(d) ICSD-108547 (e) Schmidt et al. (f) This work\n(1980) (2023)\nPublished Data Figures Redacted\nFig. R2-3 Structural comparison of (a-c) NbSnRh and (d-f) HoPdSb. (a) ICSD-reported\nstructure (ICSD-105220, Metallurgical Transactions, 1970, 3159-3162); (b) Recently reported\nstructure (Advanced Materials, 2023, 2210788); (c) Structure proposed by this work, consistent\nwith (b). (d) ICSD-reported structure (ICSD-108547, Journal of the Less Common Metals, 1980,\n25-28); (e) Recently reported structure (Advanced Materials, 2023, 2210788); (f) Structure\nproposed by this work, consistent with (e).\nCsIO : CsIO (ICSD-33665) is a covalent perovskite material. It was first reported in 1928 with\n3 3\n̅\na cubic perovskite structure (space group Pm3 m), where Cs occupies the six-coordinate B-site\n(Fig. S19a, i.e., Fig. R2-4a). However, the calculated E of this original structure is extremely\nhull\nhigh at 1.97 eV/atom. In contrast, our solution from XRDSol exhibits a much lower E of 0.02\nhull\neV/atom. The larger Cs cations are moved to twelve-coordinate A-sites, and the original ICSD\nstructure incorrectly swapped the positions of Cs and I atoms. Our structure solution (Fig. S19c,\ni.e., Fig. R2-4c) is very close to the recently synthesized and experimentally characterized\nstructure (Fig. S19b, i.e., Fig. R2-4b) from a single crystal (Chemistry of Materials, 2018, 1136-\n1145), with an RJ'VISD of 0.28 Å. Although RJ'VISD is slightly larger due to minor deviations\nin atomic positions caused by different extents of distortions, a direct comparison between the\nreported structure (Fig. R2-4b) and our predicted structure (Fig. R2-4c) demonstrates the\ncorrectness of our structure.\n12\n(a) ICSD-33665 (b) Zhang et al. (c) This work\n(1928) (2018)\nPublished Data Figures Redacted\nCs I O\nFig. R2-4 Structural comparison of CsIO . (a) ICSD-reported structure (ICSD-33665,\ncharacterized based on a single crystal sample (Chemistry of Materials, 2018, 1136-1145); (c)\nStructure proposed by this work, consistent with (b).\nNbP: NbP was first characterized by PXRD in 1954 and reported as centrosymmetric with\nI4/amd space group (ICSD-76027, Acta Chemical Scandinavica, 1954, 226-239). It contains a\n1\nvery uncommon four-fold square planar coordination for Nb and P (Fig. S20a, i.e., Fig. R2-\n5a). DFT calculation reveals an extremely high of 0.6 eV/atom, further raising doubts about\nEhull\nthe validity of this structure. In contrast, XRDSol suggests a non-centrosymmetric structure with\na different space group I4md, where Nb is in a trigonal prismatic coordination environment with\n1\nsix P atoms (Fig. S20c, i.e., Fig. R2-5c). DFT calculation confirms this proposed structure is stable\n(E = 0 eV/atom), and it agrees with the previous literature report with an RMSD value of 0.07\nhull\nÅ (Inorganic Chemistry, 1996, 845-849) (Fig. S20b, i.e., Fig. R2-\n5b).\n(a) ICSD-76027 (b) Xu et al. (c) This work\n(1996)\n(1954)\nPublished Data Figures Redacted\nNb P\nFig. R2-5 Structural comparison of NbP. (a) ICSD-reported structure (ICSD-76027); (b)\nPreviously reported experimental structure from a single crystal sample (Inorganic Chemistry,\n13\n1996, 845-849); (c) Structure proposed by this work, consistent with\n(b). Comparison with ICDD XRD patterns\nThis category also includes structure comparisons. To rigorously assess the accuracy of the\natomic coordinates predicted by XRDSol, we compare our proposed structures to\nexperimentally determined structures reported in the literature. Our predicted structures show\ngood agreement in atomic arrangement. This supports the validity of our solutions. We now\npresent a total of 5 structures in the original manuscript.\n(NH )CoF : (NH )CoF was originally reported in 1970 without atomic coordinates. XRDSol\n4 3 4 3\nsolved its structure as a cubic perovskite (Fig. S23b, i.e., Fig. R2-6b), with cobalt at the B-site\nand ammonium at the A-site. Nitrogen is tetrahedrally coordinated with hydrogen, forming the\ncorrect ammonium structure. The structure has a low E of 0.03 eV/atom. It also aligns well\nhull\nwith the structure later reported from single-crystal XRD by Siebeneichler et al. with an RMSD\nvalue of 0.28 Å (The Journal of Chemical Physics, 2020, 104501) (Fig. R2-6a and b). This\nvalidates the plausibility of our solution.\n(a) Siebeneichler et al. (b) This work\n(solved by SCXRD)\nPublished Data Figure Redacted\nH N C o F\nFig. R2-6 Structural comparison of (NH )CoF . (a) Previously reported experimental structure\n4 3\nfrom a single crystal (The Journal of Chemical Physics, 2020, 104501); (b) Structure proposed\nby this work, consistent with (a).\nLi SrGeO : Li SrGeO was characterized in 1986 as a tetragonal phase with I-42m space group\n2 4 2 4\nbut without atomic coordinates. The XRDSol predicts a stable Sr-analog of Li CaGeO phase\n2 4\n(E = 0 eV/atom). Our solution agrees with the recently reported structure by Griesemer et al.\nhull\nwith an RMSD of 0.08 Å (Fig. S25a and b, i.e., Fig. R2-7a and b). Therefore, our proposed\nstructure is highly plausible.\n14\n(a) Griesemer et al. (b) This work\n(2021)\nPublished Data Figure Redacted\nLi Sr Ge O\nFig. R2-7 Structural comparison of Li SrGeO . (a) Previously reported structure (Physical\n2 4\nReview Materials, 2021, 105003); (b) Structure proposed by this work, consistent with (a).\nMnO : Natural ramsdellite MnO was documented by ICDD without atomic coordinates. And\n2 2\nthe structure solved by XRDSol has a low of 0.009 eV/atom. The structure proposed by\nEhull\nXRDSol is consistent with the literature report by Post et al. with an RMSD of 0.1 Å (American\nMineralogist, 2004, 969-975) (Fig. S27, i.e., Fig. R2-8). We believe the proposed structure is\nhighly plausible.\n(a) Post et al. (b) This work\n(2004)\nPublished Data Figure Redacted\nMn O\nFig. R2-8 (a) Previously reported experimental structure (American Mineralogist, 2004, 969-\n975); (b) Structure proposed by this work, consistent with (a).\nNaSrPO : NaSrPO is a natural olgite mineral discovered in 1980. Early XRD experiments\n4 4\nrevealed that it belongs to the hexagonal lattice system and was originally assumed to have the P3\nspace group. Lattice parameters were extracted from diffraction patterns, but without atomic\ncoordinates. XRDSol provided a solution with a different space group P3m1. Despite that the\nreconstructed pattern matches most of the major peaks, strong intensity deviation was observed.\nThis is probably due to the strong texture of the sample and potentially associated impurity phases.\nDFT calculation confirms that the structure is highly plausible with a low E of 0.01 eV/atom. We\nhull\nconfirmed that our proposed structure agrees with a recent report on olgite crystal\n15\nstructure, except that one lattice site might be co-occupied by Na or Sr (Canadian Mineralogist,\n2005, 1521-1526).\nSr Ca TaO N: Sr Ca TaON is an oxynitride perovskite material, in which complex anion\n0.5 0.5 2 0.5 0.5 2\norder behavior may exist. The A-site is co-occupied by Sr and Ca (Journal of the Korean\nCeramic Society, 2020, 432-439). In contrast, in the structure proposed by XRDSol, two A-sites\nin a supercell are occupied by Sr and Ca, respectively. For the anion-sites, XRDSol proposed\nan ordered trans-type anion configuration, in which nitrogen occupies axial sites while oxygen\noccupies the equatorial sites. Remarkably, this is in excellent agreement with recent neutron\ndiffraction results (ACS nano, 2017, 3860-3866).\nWe hope that we have convinced you that we are doing our utmost to be serious about the\nvalidity of our solutions. The approach we have adopted is neither a lazy one nor an attempt to\nevade the question. It is our firm belief that the reliability of results is more important than\nwhether the paper is published. Therefore, we have confidence in allowing our solutions to\nundergo rigorous questioning or challenging. Additionally, we sincerely thank you, as a\ncrystallography expert, for your critical review and questioning of our solutions, which greatly\nbenefits our results and algorithms.\nComments #3: Further, on the point of the use of structureMatcher/MaxDist to assess structures:\nThis metric has not been validated in any meaningful way by the crystallographic community, and\nwhilst it is widely used by the ML community, this is an issue that needs to be urgently addressed.\nthis is not the fault of the current authors, and they are following practices used by others in\nthis regard, but with respect to the specific scientific claims of the paper, this measure is so flawed as\nto make them highly dubious. Increasing the crystallographic domain knowledge in the team\nwould, again, greatly help.\nResponse #3: Thank you for your insightful comments. We agree that the use of\nstructureMatcher/MaxDist has not been systematically validated or accepted by the\ncrystallographic community as a standard for structural assessment. We consulted crystallography\nexperts and re-evaluated our evaluation strategy. To address this concern, we have revised our\nmanuscript to incorporate RMSD, a metric that is broadly recognized and commonly used by the\ncommunity. We calculated the RMSD values for 9,046 structures solved by XRDSol in the MP-20\ntest set. The distribution of RMSD is shown in Fig. S4, i.e., Fig. 2-9. 72.15% of the structures\nachieved an RMSD less than 0.1 Å, and 78.13% achieved an RMSD less than 0.3 Å. Among the\nstructures with RMSD < 0.1 Å, the mean RMSD is 0.0215 Å, the median RMSD is 0.0129 Å, the\nfirst quartile is 0.0024 Å, and the third quartile is 0.0325 Å. These results demonstrate that XRDSol\ncan reproduce most crystal structures with high fidelity\n16\nin the MP-20 test set. The description of RMSD has been provided in the revised manuscript.\ny\ntis\nn\ne\nD\n> 0.3\nRMSD (Å)\nFig. R2-9 Distribution of RMSD for 9046 structures solved by XRDSol in the MP-20 test\ndataset, reflecting the reconstruction quality of the crystal structures.\nComments #4: I appreciate the efforts of the authors to increase the clarity of the presentation.\nAlso, they report at making a better effort of acknowledging the inspiration for their work (e.g.,\nCDVAE) in the methods section, but I still find this inadequate. I would rather expect to see it high\nup in the paper (e.g., the Introduction) and with a bit more fanfare. There is no harm in a full\nbodied celebration of the work of others. It does not diminish in any way the developments made by\nthe authors. Perhaps they feel pressure to publish in a high profile journal, and they feel they need to\ntry and make their work stand out to accomplish that. But this is bad for science in general, and\ncounter-productive for the authors as it makes it harder for others to cite their work if it doesn't do\na fair job of explaining how it fits in to the broader community.\nResponse #4: Thank you for your comments. We fully agree that the prior materials generative\nmodels (e.g., CDVAE) deserve more prominent recognition as a key source of inspiration for our\nwork. We are grateful for the foundational work of CDVAE, which significantly influenced our\nthinking in designing a diffusion-based structure generation model. While we have cited CDVAE\n(Reference 12) in both the Introduction and Method sections of our original manuscript, we\nrealize that this did not sufficiently highlight the foundational role of CDVAE. To clearly\narticulate how the CDVAE inspired the design and development of our model, we have now\nrevised the Introduction section to explicitly and prominently describe the contribution of\nCDVAE as follows:\n(Introduction in the revised Manuscript)\n“Recently, machine learning approaches, particularly generative diffusion models, have\nprovided a new paradigm for materials structure generation12-15. Xie et al. built the Crystal\nDiffusion Variational Autoencoder (CDVAE), which employs score matching to generate\n17\nrealistic crystal structures by learning from the data distribution of known stable materials12.\nWithout any predefined prototypes, CDVAE shows its ability to generate valid and diverse\ncrystal structures that captures the physical inductive bias of material stability. Equivariant\ndiffusion can effectively preserve reflection, translation, rotation, and mirror symmetries12-16,\nenabling generation of physically reasonable crystal structures. Such equivariant denoising\nmodels were employed to generate inorganic crystal structures12,14,15 and molecules13,16, and\nshowed remarkable performance. Besides generating crystal structures without constraints,\nconditioning diffusion models allow crystal generation with desired chemistry, mechanical,\nelectronic, or magnetic properties, paving the way towards generative AI-assisted materials\ndesign paradigms.”\nComments #5: I thank the authors for their efforts at better benchmarking. Just a note that I am\nmore interested in benchmarking prediction performance, in part as a way to understand how\nthe model is working, where it is getting information from, and so on, and less interested in\ncomputation time. It is clear that a trained ML model will work in production much quicker\nthan any regression method, but getting an answer is of greater interest in structure solution\nrather than wall-clock time in most applications and so understanding the performance is more\nimportant than how quickly it gets there. Overall, given the challenges of not really having good\nsuccess metrics in general, the benchmarking work that the authors have done is about as good\nas we can hope for.\nResponse #5: We thank the reviewer for the valuable feedback. The concerns regarding\nbenchmarking prediction performance have been addressed in our revised responses to\nComments #2 and #3, where we provide detailed analyses using meaningful metrics (e.g.,\nRMSD) in the crystallographic community to evaluate the accuracy and quality of the predicted\nstructures.\nSome small points:\nComments #6: In one response the authors mention that better performance can be obtained by\nincorporating \"'domain knowledge (e.g., space group information)\". This an incorrect use of\nthe term 'domain knowledge' which refers to the general knowledge of a domain expert rather\nthan incorporating additional information in a particular input. I sugget to just delete mention\nof domain knowledge at that point.\nResponse #6: Thank you for pointing this out. We agree that the term “domain knowledge” is\nmisused. We have removed the term and now refer directly to “space group information” as\nadditional input.\nextinction here'? Again, bolstering their crystallography domain expertise might help this\npaper.\nResponse #7: Thank you for pointing this out. We acknowledge that the term “peak distinction”\nwas incorrectly used. Wha we originally intended to express was indeed “peak extinction”.\nUpon further discussion with our co-auther, a crystallographer, we realized that the term\n“systematic absence” is more precise and widely accepted in the crystallography community.\nWe appreciate the opportunity to clarify.\nComments #8: In methods they describe how they run the model 25 times with different random\nseeds and pick the best solution. This is fine, it is fairly standard practice, but then they call it\n'Top-1', but I find this a bit misleading. I think more normally this would be called 'Top-25'.\nClearly you measure the model performance with the best model of your sample and report that\nperformance, but I think of 'top-1' as how well the model performs at a guess of one sample, Top-\n10 when you let it guess 10 times and take the best one, and so on. So I would think that what the\nauthors are doing is better described as Top-25. They justify very nicely why they do that, so it is\nno criticism of the approach, just the naming.\nResponse #8: Thank you for pointing this out. We agree that the term “Top-25” better reflects\nour evaluation protocol, which involves running the model 25 times and selecting the best\nresult. To avoid misunderstanding, we have revised the term “Top-1” to “Top-25” in the\nrevised manuscript.\nComments #9: They call their approach \"end-to-end\" on page 3 but requiring knowledge of\nthe lattice parameters suggests that this is more mid-to-end than end-to-end. I recommend\nsimply to remove this statement. It doesn't add much.\nResponse #9: Thank you for your insightful observation. We agree that the term “end-to-end”\ncould be misleading, as our approach requires the unit cell parameters as input. We have\nremoved the phrase “end-to-end”.\nStrength:\nComments #0: I appreciate the authors’ detailed responses. My main concerns, including the\ngraph initialization, diffusion step selection & potential randomness, and failure case analysis,\nhave been well addressed.\nResponse #0: We truly appreciate the Reviewer’ positive feedback. We are glad that our\nrevisions have addressed your main concerns regarding graph initialization, diffusion step\nselection, potential randomness, and failure case analysis.\nComments #1: To further improve clarity, I suggest the authors consider simplifying the\npresentation of the diffusion model in the final revision. Including a clear formulation of the\nforward and reverse steps, perhaps via a concise pseudo-code for one noising and denoising\nstep, would help a lot.\nResponse #1: Thank you for the helpful suggestion. In the revised manuscript, we have added\nconcise pseudo-code (Table S6, i.e., Table R2-1) to illustrate the noising and denoising steps.\nTable R2-1 Pseudo-code of XRDSol for crystal structure solutions using powder XRD data.\nInput: PXRD pattern 𝐶 , initial structure 𝑆 = \n0\n[𝐿, 𝐴, 𝑃\n0\n}, diffusion steps T, schedule [𝛼𝑡, 𝜎𝑡}\n𝑓𝐶 ← MLP(𝐶)\nNoising process:\nfor t = 1 to T do\n𝜀𝑡~ 𝒩𝑤 (0, I)\n𝑆𝑡← 𝛼𝑡∙ 𝑆𝑡−1 + 𝜎𝑡∙ 𝜀𝑡\nend for\nDenoising process:\n𝑆𝑇 ← [𝐿, 𝐴, 𝑃𝑇}\nfor t = T to 1 do\n𝑓𝑠𝑡𝑟𝑢𝑐𝑡 ← EGNN (𝑆𝑡)\n𝑓𝑐𝑎𝑡 ← concat(𝑓𝑠𝑡𝑟𝑢𝑐𝑡, 𝑓𝐶)\n𝜀𝑡˄← 𝜑(𝑓𝑐𝑎𝑡)\n𝑆𝑡−1 ← 𝑆𝑡 − 𝜎𝑡∙ 𝜀𝑡˄\nend for\nOutput: 𝑆 = [𝐿, 𝐴, 𝑃 }\n0 0\nComments #2: The code is clear and runnable with good instructions to reproduce the results.\nResponse #2: Thank you for the evaluation of our code.\nComments #0: I co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training in peer review\nand to provide appropriate recognition for Early Career Researchers who co-review\nmanuscripts.\nResponse #0: Thank you for the co-reviewed comments.\nRemarks on code availability:\nComments #1: The code is runnable with clear instructions.\nResponse #1: We appreciate your evaluation of the code and instructions.\n22"}]

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