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Neural network-based order parameter for phase transitions and its applications in high-entropy alloys

A preprint version of the article is available at arXiv.


Phase transition is one of the most important phenomena in nature and plays a central role in materials design. All phase transitions are characterized by suitable order parameters, including the order–disorder phase transition. However, finding a representative order parameter for complex systems is non-trivial, such as for high-entropy alloys. Given the strength of dimensionality reduction of a variational autoencoder (VAE), we introduce a VAE-based order parameter. We propose that the Manhattan distance in the VAE latent space can serve as a generic order parameter for order–disorder phase transitions. The physical properties of our order parameter are quantitatively interpreted and demonstrated by multiple refractory high-entropy alloys. Using this order parameter, a generally applicable alloy design concept is proposed by mimicking the natural mixing process of elements. Our physically interpretable VAE-based order parameter provides a computational technique for understanding chemical ordering in alloys, which can facilitate the development of rational alloy design strategies.

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Fig. 1: Neural network architecture of the VAE model.
Fig. 2: VAE analyses on MoNbTaW.
Fig. 3: Comparison of the VAE order parameter with experimental data.
Fig. 4: VAE-informed procedure for HEA design.
Fig. 5: Comparison between our VAE order parameter and conventional order parameters.
Fig. 6: Scalability of the VAE order parameter approach.

Data availability

The sample training and validation data, along with a pre-trained model and VASP input files (version 5.4.4) for Ti38V16Nb23Hf24, are available at Source data are provided with this paper.

Code availability

The VAE model training and order parameter inference codes are available at The training code along with sample data can also be found on our Code Ocean capsule50. The figures are plotted with the notebook at


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This research was sponsored by and used resources of the Oak Ridge Leadership Computing Facility (OLCF), which is a US Department of Energy (DOE) Office of Science User Facility at the Oak Ridge National Laboratory supported by the US DOE under contract no. DE-AC05-00OR22725. This work was also performed in support of the US DOE’s Fossil Energy Crosscutting Technology Research Program, and in part by an appointment to the US DOE Postgraduate Research Program at the National Energy Technology Laboratory (NETL) administered by the Oak Ridge Institute for Science and Education. Neither the United States Government nor any agency thereof, nor any of its employees, nor the support contractor, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

Author information

Authors and Affiliations



The initial project idea was formulated by J.Y. J.Y. and Z.P. designed the experiments. J.Y. performed the data generation, VAE training and order parameter inference. Z.P. performed all DFT calculations, interpreted the total order parameters, proposed the alloy design concept and realized the initial version of the method. J.Y. and Z.P. wrote the manuscript. J.Y., Z.P. and M.C.G. analyzed the results and finalized the manuscript.

Corresponding authors

Correspondence to Junqi Yin or Zongrui Pei.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Computational Science thanks Evert van Nieuwenburg, Sebastian Wetzel and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Handling editor: Jie Pan, in collaboration with the Nature Computational Science team.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–7, Table 1 and Discussion.

Source data

Source Data Fig. 2

VAE embeddings and order parameter, and specific heat data for MoNbTaW.

Source Data Fig. 3

Second-order moment of the VAE order parameter and VAE embeddings for AlxCoCrFeNi, where x = 1, 1.6, and 2.

Source Data Fig. 4

Configurations of eight subsystems of TiVNbHf and the VAE order parameter histogram and element distribution for TiVNbHf alloy design, and specific heat for Ti25V25Nb25Hf25, Ti38V15Nb23Hf24, Ti29V15Nb28Hf28 and Ti28V12Nb30Hf30.

Source Data Fig. 5

SRO, LRO and VAE order parameter, and total order parameter for MoNbTaW.

Source Data Fig. 6

Training and inference time for up to 96 GPUs on the Summit supercomputer.

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Yin, J., Pei, Z. & Gao, M.C. Neural network-based order parameter for phase transitions and its applications in high-entropy alloys. Nat Comput Sci 1, 686–693 (2021).

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