Ab initio structure solutions from nanocrystalline powder diffraction data via diffusion models
Source abstract
Abstract
A major challenge in materials science is the determination of the structure of nanometre-sized objects. Here we present an approach that uses a generative machine learning model based on diffusion processes that are trained on 45,229 known structures. The model factors measured the diffraction pattern as well as the relevant statistical priors on the unit cell of atomic cluster structures. Conditioned only on the chemical formula and the information-scarce finite-sized broadened powder diffraction pattern, we find that our model, PXRDnet, can successfully solve the simulated nanocrystals as small as 10 Å across 200 materials of varying symmetries and complexities, including structures from all seven crystal systems. We show that our model can successfully and verifiably determine structural candidates four out of five times, with an average error among these candidates being only 7% (as measured by the post-Rietveld refinement R-factor). Furthermore, PXRDnet is capable of solving structures from noisy diffraction patterns gathered in real-world experiments. We suggest that data-driven approaches, bootstrapped from theoretical simulation, will ultimately provide a path towards determining the structure of previously unsolved nanomaterials.
Source: openalex_inverted_index · Quality: baseline
Research context
OpenAlex topics: Machine Learning in Materials Science · X-ray Diffraction in Crystallography · Crystallography and molecular interactions
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (72)
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1. Giacovazzo, C. Fundamentals of Crystallography Vol. 7 (Oxford Univ. Press, 2002).
2. Lipson, H. The Study of Metals and Alloys by X-Ray Powder Diffraction Methods (Univ. College Cardiff Press, 1984).
3. Lipson, H. & Beevers, C. The crystal structure of the alums. Proc. R. Soc. Lond. A 148, 664–680 (1935).10.1098/rspa.1935.0040 ↗
4. Billinge, S. J. & Levin, I. The problem with determining atomic structure at the nanoscale. Science 316, 561–565 (2007).10.1126/science.1135080 ↗
5. Dinnebier, R. E. & Billinge, S. J. Powder Diffraction: Theory and Practice (Royal Society of Chemistry, 2008).10.1039/9781847558237 ↗
6. Egami, T. & Billinge, S. J. Underneath the Bragg Peaks: Structural Analysis of Complex Materials (Newnes, 2012).
7. Riesel, E. A. et al. Crystal structure determination from powder diffraction patterns with generative machine learning. J. Am. Chem. Soc. 146, 30340–30348 (2024).10.1021/jacs.4c10244 ↗
8. Li, Q. et al. Powder diffraction crystal structure determination using generative models. Preprint at https://arxiv.org/abs/2409.04727 (2024).
9. Lai, Q. et al. End-to-end crystal structure prediction from powder x-ray diffraction. Adv. Sci. 12, 2410722 (2025).10.1002/advs.202410722 ↗
10. Juhás, P., Cherba, D., Duxbury, P., Punch, W. & Billinge, S. Ab initio determination of solid-state nanostructure. Nature 440, 655–658 (2006).10.1038/nature04556 ↗
11. Juhas, P., Granlund, L., Gujarathi, S. R., Duxbury, P. M. & Billinge, S. J. Crystal structure solution from experimentally determined atomic pair distribution functions. J. Appl. Cryst. 43, 623–629 (2010).10.1107/S002188981000988X ↗
12. Billinge, S. J. & Proffen, T. Machine learning in crystallography and structural science. Acta Cryst. A80, 139–145 (2024).10.1107/S2053273324000172 ↗
13. Choudhary, K. et al. Recent advances and applications of deep learning methods in materials science. npj Comput. Mater. 8, 59 (2022).10.1038/s41524-022-00734-6 ↗
14. Oviedo, F. et al. Fast and interpretable classification of small X-ray diffraction datasets using data augmentation and deep neural networks. npj Comput. Mater. 5, 60 (2019).10.1038/s41524-019-0196-x ↗
15. Suzuki, Y. et al. Symmetry prediction and knowledge discovery from X-ray diffraction patterns using an interpretable machine learning approach. Sci. Rep. 10, 21790 (2020).10.1038/s41598-020-77474-4 ↗
16. Park, W. B. et al. Classification of crystal structure using a convolutional neural network. IUCrJ 4, 486–494 (2017).10.1107/S205225251700714X ↗
17. Lee, J.-W., Park, W. B., Lee, J. H., Singh, S. P. & Sohn, K.-S. A deep-learning technique for phase identification in multiphase inorganic compounds using synthetic XRD powder patterns. Nat. Commun. 11, 86 (2020).10.1038/s41467-019-13749-3 ↗
18. Aguiar, J., Gong, M. L., Unocic, R., Tasdizen, T. & Miller, B. Decoding crystallography from high-resolution electron imaging and diffraction datasets with deep learning. Sci. Adv. 5, eaaw1949 (2019).10.1126/sciadv.aaw1949 ↗
19. Ziletti, A., Kumar, D., Scheffler, M. & Ghiringhelli, L. M. Insightful classification of crystal structures using deep learning. Nat. Commun. 9, 2775 (2018).10.1038/s41467-018-05169-6 ↗
20. Tiong, L. C. O., Kim, J., Han, S. S. & Kim, D. Identification of crystal symmetry from noisy diffraction patterns by a shape analysis and deep learning. npj Comput. Mater. 6, 196 (2020).10.1038/s41524-020-00466-5 ↗
21. Garcia-Cardona, C. et al. Learning to predict material structure from neutron scattering data. In 2019 IEEE International Conference on Big Data (Big Data) 4490–4497 (IEEE, 2019).10.1109/BigData47090.2019.9005968 ↗
22. Merker, H. A. et al. Machine learning magnetism classifiers from atomic coordinates. iScience 25, 105192 (2022).10.1016/j.isci.2022.105192 ↗
23. Maffettone, P. M. et al. Crystallography companion agent for high-throughput materials discovery. Nat. Comput. Sci. 1, 290–297 (2021).10.1038/s43588-021-00059-2 ↗
24. Szymanski, N. J., Bartel, C. J., Zeng, Y., Tu, Q. & Ceder, G. Probabilistic deep learning approach to automate the interpretation of multi-phase diffraction spectra. Chem. Mater. 33, 4204–4215 (2021).10.1021/acs.chemmater.1c01071 ↗
25. Uryu, H. et al. Deep learning enables rapid identification of a new quasicrystal from multiphase powder diffraction patterns. Adv. Sci. 11, 2304546 (2024).10.1002/advs.202304546 ↗
26. Szymanski, N. J. et al. An autonomous laboratory for the accelerated synthesis of novel materials. Nature 624, 86–91 (2023).10.1038/s41586-023-06734-w ↗
27. Larsen, A. S., Rekis, T. & Madsen, A. Ø. PhAI: a deep-learning approach to solve the crystallographic phase problem. Science 385, 522–528 (2024).10.1126/science.adn2777 ↗
28. Jiao, R. et al. Crystal structure prediction by joint equivariant diffusion. Adv. Neural Inf. Process. Syst. 36, 17464–17497 (2024).
29. Pakornchote, T. et al. Diffusion probabilistic models enhance variational autoencoder for crystal structure generative modeling. Sci. Rep. 14, 1275 (2024).10.1038/s41598-024-51400-4 ↗
30. Antunes, L. M., Butler, K. T. & Grau-Crespo, R. Crystal structure generation with autoregressive large language modeling. Nat. Commun. https://doi.org/10.1038/s41467-024-54639-7 (2024).10.1038/s41467-024-54639-7 ↗
31. Zhao, Y. et al. Physics guided deep learning for generative design of crystal materials with symmetry constraints. npj Comput. Mater. 9, 38 (2023).10.1038/s41524-023-00987-9 ↗
32. Yang, S. et al. Scalable diffusion for materials generation. In Proc. International Conference on Learning Representations 2024 https://openreview.net/pdf?id=wm4WlHoXpC (ICLR, 2024).
33. Xie, T., Fu, X., Ganea, O.-E., Barzilay, R. & Jaakkola, T. Crystal diffusion variational autoencoder for periodic material generation. In Proc. International Conference on Learning Representations 2022 https://openreview.net/pdf?id=03RLpj-tc (ICLR, 2022).
34. Zeni, C. et al. A generative model for inorganic materials design. Nature https://doi.org/10.1038/s41586-025-08628-5 (2025).10.1038/s41586-025-08628-5 ↗
35. Guo, G. et al. Towards end-to-end structure determination from X-ray diffraction data using deep learning. npj Comput. Mater. 10, 209 (2024).10.1038/s41524-024-01401-8 ↗
36. Kjær, E. T. et al. DeepStruc: towards structure solution from pair distribution function data using deep generative models. Digit. Discov. 2, 69–80 (2023).10.1039/D2DD00086E ↗
37. Kjær, E. et al. MLstructureMining: a machine learning tool for structure identification from X-ray pair distribution functions. Digit. Discov. 3, 908–918 (2024).10.1039/D4DD00001C ↗
38. Pan, T. et a. CrysFormer: protein structure determination via Patterson maps, deep learning, and partial structure attention. Struct. Dyn. https://doi.org/10.1063/4.0000252 (2024).10.1063/4.0000252 ↗
39. Pan, T., Jin, S., Miller, M. D., Kyrillidis, A. & Phillips, G. N. A deep learning solution for crystallographic structure determination. IUCrJ 10, 487–496 (2023).10.1107/S2052252523004293 ↗
40. Kwon, H. et al. Spectroscopy-guided discovery of three-dimensional structures of disordered materials with diffusion models. Preprint at https://arxiv.org/abs/2312.05472 (2023).
41. Juhás, P., Granlund, L., Duxbury, P., Punch, W. & Billinge, S. The Liga algorithm for ab initio determination of nanostructure. Acta Cryst. A64, 631–640 (2008).10.1107/S0108767308027591 ↗
42. Song, Y. & Ermon, S. Generative modeling by estimating gradients of the data distribution. In 33rd Conference on Neural Information Processing Systems (NeurIPS 2019) https://papers.neurips.cc/paper_files/paper/2019/file/3001ef257407d5a371a96dcd947c7d93-Paper.pdf (NeurIPS, 2019).
43. Jain, A. et al. Commentary: the Materials Project: a materials genome approach to accelerating materials innovation. APL Mater. 1, 011002 (2013).10.1063/1.4812323 ↗
44. Kreyling, W. G., Semmler-Behnke, M. & Chaudhry, Q. A complementary definition of nanomaterial. Nano Today 5, 165–168 (2010).10.1016/j.nantod.2010.03.004 ↗
45. Lathi, B. P. & Green, R. A. Linear Systems and Signals Vol. 2 (Oxford Univ. Press, 2005).
46. Sivia, D. S. Elementary Scattering Theory: for X-Ray and Neutron Users (Oxford Univ. Press, 2011).10.1093/acprof:oso/9780199228676.001.0001 ↗
47. David, W. I., Shankland, K., McCusker, L. B. & Baerlocher, C. Structure Determination from Powder Diffraction Data Vol. 13 (Oxford Univ. Press, 2006).10.1093/acprof:oso/9780199205530.001.0001 ↗
48. Merchant, A. et al. Scaling deep learning for materials discovery. Nature 624, 80–85 (2023).10.1038/s41586-023-06735-9 ↗
49. Cheetham, A. K. & Seshadri, R. Artificial intelligence driving materials discovery? Perspective on the article: scaling deep learning for materials discovery. Chem. Mater. 36, 3490–3495 (2024).10.1021/acs.chemmater.4c00643 ↗
50. Leeman, J. et al. Challenges in high-throughput inorganic materials prediction and autonomous synthesis. PRX Energy 3, 011002 (2024).10.1103/PRXEnergy.3.011002 ↗
51. Szymanski, N. J. et al. Toward autonomous design and synthesis of novel inorganic materials. Mater. Horiz. 8, 2169–2198 (2021).10.1039/D1MH00495F ↗
52. Farrow, C. L. & Billinge, S. J. Relationship between the atomic pair distribution function and small-angle scattering: implications for modeling of nanoparticles. Acta Cryst. A65, 232–239 (2009).10.1107/S0108767309009714 ↗
53. IUCr. Crystallographic databases and related resources (International Union of Crystallography, accessed 15 April 2025); https://www.iucr.org/resources/data/databases
54. Coelho, A. A. TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic objects written in C++. J. Appl. Cryst. 51, 210–218 (2018).10.1107/S1600576718000183 ↗
55. Jiao, R., Huang, W., Liu, Y., Zhao, D. & Liu, Y. Space group constrained crystal generation. In Proc. International Conference on Learning Representations 2024 https://openreview.net/pdf?id=jkvZ7v4OmP (ICLR, 2024).
56. Lin, P. et al. Equivariant diffusion for crystal structure prediction. In Forty-first International Conference on Machine Learning 1–24 (PMLR, 2024).
57. Luo, Y., Liu, C. & Ji, S. Towards symmetry-aware generation of periodic materials. In 37th Conference on Neural Information Processing Systems (NeurIPS 2023) https://openreview.net/pdf?id=Jkc74vn1aZ (NeurIPS, 2023).
58. Ong, S. P. et al. Python Materials Genomics (pymatgen): a robust, open-source python library for materials analysis. Comput. Mater. Sci. 68, 314–319 (2013).10.1016/j.commatsci.2012.10.028 ↗
59. Toby, B. in International Tables for Crystallography Volume G: Definition and Exchange of Crystallographic Data 258–269 (Springer, 2006).10.1107/97809553602060000742 ↗
60. Shannon, C. Communication in the presence of noise. Proc. IRE 37, 10–21 (1949).10.1109/JRPROC.1949.232969 ↗
61. Kingma, D. P. & Welling, M. Auto-encoding variational Bayes. Preprint at https://arxiv.org/abs/1312.6114 (2013).
62. Yang, L. et al. Diffusion models: a comprehensive survey of methods and applications. ACM Comput. Surv. 56, 105 (2023).
63. Gasteiger, J., Groß, J. & Günnemann, S. Directional message passing for molecular graphs. In Proc. International Conference on Learning Representations 2020 https://openreview.net/pdf?id=B1eWbxStPH (ICLR, 2020).
64. Xie, T. & Grossman, J. C. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. Phys. Rev. Lett. 120, 145301 (2018).10.1103/PhysRevLett.120.145301 ↗
65. Song, Y., & Ermon, S. Generative modeling by estimating gradients of the data distribution. In 33rd Conference on Neural Information Processing Systems (NeurIPS 2019) https://openreview.net/pdf?id=B1lcYrBgLH (NeurIPS, 2019).
66. Gasteiger, J., Becker, F. & Günnemann, S. GemNet: universal directional graph neural networks for molecules. In 35th Conference on Neural Information Processing Systems (NeurIPS 2021) https://proceedings.neurips.cc/paper_files/paper/2021/file/35cf8659cfcb13224cbd47863a34fc58-Paper.pdf (NeurIPS, 2021).
67. Huang, G., Liu, Z., Van Der Maaten, L. & Weinberger, K. Q. Densely connected convolutional networks. In Proc. IEEE Conference on Computer Vision and Pattern Recognition 4700-4708 (IEEE, 2017).10.1109/CVPR.2017.243 ↗
68. Loshchilov, I. & Hutter, F. SGDR: stochastic gradient descent with warm restarts. In Proc. International Conference on Learning Representations 2017 https://openreview.net/pdf?id=Skq89Scxx (ICLR, 2017).
69. Goodfellow, I., Bengio, Y. & Courville, A. Deep Learning (MIT Press, 2016).
70. Williams, R. J. & Zipser, D. A learning algorithm for continually running fully recurrent neural networks. Neural Comput. 1, 270–280 (1989).10.1162/neco.1989.1.2.270 ↗
71. Guo, G. & Saidi, T. gabeguo/cdvae_xrd: public release. Zenodo https://doi.org/10.5281/zenodo.15022544 (2025).10.5281/zenodo.15022544 ↗
72. Ioffe, S. & Szegedy, C. Batch normalization: accelerating deep network training by reducing internal covariate shift. In 32nd International Conference on Machine Learning 448–456 (ICML, 2015).