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铅基钙钛矿铁电晶体高临界转变温度的机器学习研究

杨自欣 高章然 孙晓帆 蔡宏灵 张凤鸣 吴小山

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铅基钙钛矿铁电晶体高临界转变温度的机器学习研究

杨自欣, 高章然, 孙晓帆, 蔡宏灵, 张凤鸣, 吴小山

High critical transition temperature of lead-based perovskite ferroelectric crystals: A machine learning study

Yang Zi-Xin, Gao Zhang-Ran, Sun Xiao-Fan, Cai Hong-Ling, Zhang Feng-Ming, Wu Xiao-Shan
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  • 铁电材料由铁电相转化为顺电相的临界温度被称为居里温度, 是铁电材料的一个关键指标. 本文使用固溶体组成元素的基本物理性质等特征对不同组分和配比的铅基钙钛矿铁电固溶体进行了统一的描述, 采用岭回归、支持向量回归、极端随机森林回归等机器学习方法对铅基钙钛矿铁电固溶体的居里温度进行了学习. 使用交叉验证的方法对学习效果进行验证, 得到上述机器学习方法对材料居里温度的预测值与实验值之间的平均误差分别为14.4, 14.7, 16.1 K, 集成三种回归方法优化的模型在交叉验证中测得的平均误差为13.9 K. 在此基础上对超过20万种铅基钙钛矿的居里温度进行了预测, 给出了两种可能具有高居里温度的铁电材料.
    Ferroelectrics undergoes a reversible structural phase from the ferroelectric phase to the paraelectric phase when its temperature exceeds the critical temperature namely Curie temperature Tc. As ferro-paraelectric phase transition is always accompanied by heat-flow, dielectric and pyroelectric anomaly, the value of Tc is extremely important for ferroelectrics. In this paper, the Curie temperature of lead-based perovskite ferroelectric solid solution is studied by machine learning methods including kernel ridge regression (KRR), support vector regression (SVR) and extremely randomized trees regression (ETR). We collect the Tc values of 205 different lead-based perovskites from published experimental papers, both simple perovskites with only one type of B site ion and complex perovskites with up to 5 kinds of ions in B position such as PMN-PFN-PZT are gathered. The diversity of our dataset is guaranteed for the good generalization of our model in perovskite solid solution of different complexity. The features are constructed from the physical and chemical properties of the B site elements in corresponding materials. The weighted-average and variance of the elemental properties are calculated and fed to machine learning models. We use the 5 runs of ten fold cross-validation method to evaluate the machine learning models. The hyperparameters are also chosen carefully with the cross-validation to avoid over fitting. The radial basis function kernel is used in both KRR and SVR. The insensitive error in the SVR is set to be 4 which is comparable to the random error in experiment. From our cross-validation, we find that the mean average errors (MAEs) between the predicted and experimental values of the machine learning methods are 14.4 K, 14.7 K, and 16.1 K, respectively. And the root-mean-square errors (RMSEs) are 22.5 K, 23.4 K, 23.8 K, respectively. After the optimization and the evaluation, our three machine learning models are stacked together by averaging the output of each regression model and thus building an ensemble model. The MAE of the ensemble model is 13.9 K. The RMSE of the ensemble model is 21.4 K. The predicted values keep a correlation coefficient of 0.97 with the experimental values. From the variance reduction in ETR, we derive the importance of our features when determining the Curie temperatures. The five most important factors in our ETR model are " weighted-average thermal conductivity”, " weighted-average conductivity”, " variance of specific heat capacity”, " weighted-average element number”, and " weighted-average relative atomic displacement”. We predict the Curie temperatures higher than those of 200000 types of lead-based perovskites after being trained. Now, we provide two ferroelectric materials that may have high Curie temperatures: 0.02PbMn1/2Nb1/2O3-0.98PbTiO3 (0.02PMN-0.98PT) and 0.02PbGa1/2Nb1/2-0.02PbMn1/2Nb1/2O3-0.96PbTiO3 (0.02PGN-0.02PMN-0.96PT). The predicted Curie temperatures of them are 481 ℃ and 466 ℃, respectively.
      通信作者: 吴小山, xswu@nju.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 118742007)资助的课题
      Corresponding author: Wu Xiao-Shan, xswu@nju.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 118742007)
    [1]

    Liu Y, Li X, Wang Z, He C, Li T, Ai L, Chu T, Pang D, Long X 2013 CrystEngComm 15 1643Google Scholar

    [2]

    Wu J, Chang Y, Yang B, Zhang S, Sun Y, Guo F, Cao W 2015 J. Mater. Sci.: Mater. Electron 26 1874Google Scholar

    [3]

    Zhao Y Q, Ma Q R, Liu B, Yu Z L, Yang J, Cai M Q 2018 Nanoscale 10 8677Google Scholar

    [4]

    Ding Y F, Zhao Q Q, Yu Z L, Zhao Y Q, Liu B, He P B, Zhou H, Li K, Yin S F, Cai M Q 2019 J. Mater. Chem. C 7 7433Google Scholar

    [5]

    Yu Z L, Ma Q R, Liu B, Zhao Y Q, Wang L Z, Zhou H, Cai M Q 2017 J. Phys. D 50 465101Google Scholar

    [6]

    Zhang T, Jiang Y, Song Z, Huang H, He Y, Fang Z, Weng H, Fang C 2019 Nature 566 475Google Scholar

    [7]

    Umehara M, Stein H S, Guevarra D, Newhouse P F, Boyd D A, Gregoire J M 2019 NPJ Comput. Mater. 5 34Google Scholar

    [8]

    Iwasaki Y, Takeuchi I, Stanev V, et al. 2019 Sci. Rep. 9 2751Google Scholar

    [9]

    Aykol M, Hegde V I, Hung L, Suram S, Herring P, Wolverton C, Hummelshøj J S 2019 Nat. Commun 10 1Google Scholar

    [10]

    Balachandran P V, Kowalski B, Sehirlioglu A, Lookman T 2018 Nat. Commun. 9 1668Google Scholar

    [11]

    Zhai X, Chen M, Lu W 2018 Comput. Mater. Sci. 151 41Google Scholar

    [12]

    Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, Duchesnay É 2011 J. Mach. Learn. Res. 12 2825

    [13]

    Chen H, Fan C 2010 Mater. Lett. 64 654Google Scholar

    [14]

    Fang B, Shan Y, Tezuka K, Imoto H 2005 J. Mater. Sci. 40 6445Google Scholar

    [15]

    Gao Y Q, Xu H Q, Wu Y J, He T H, Xu G S, Luo H S 2001 Jpn. J. Appl. Phys. 40 4998Google Scholar

    [16]

    Guo Y P, Xu H Q, Luo H S, Xu G S, Yin Z W 2001 J. Cryst. Growth 226 111Google Scholar

    [17]

    Hao H, Zhang S, Liu H, Shrout T R 2009 J. Appl. Phys. 105 024104Google Scholar

    [18]

    He C, Li X, Wang Z, Liu Y, Shen D, Li T, Long X, Ye Z G 2012 CrystEngComm 14 4407Google Scholar

    [19]

    Wang L, Liang R, Mao C, Du G, Wang G, Dong X 2013 Ceram. Int. 39 8571Google Scholar

    [20]

    Yasuda N, Ohwa H, Hasegawa D, Hayashi K, Hosono Y, Yamashita Y, Iwata M, Ishibashi Y 2000 Jpn. J. Appl. Phys. 39 5586Google Scholar

    [21]

    Zhang S, Lebrun L, Jeong D Y, Randall C A, Zhang Q, Shrout T R 2003 J. Appl. Phys. 93 9257Google Scholar

    [22]

    Bing Y, Ye Z 2002 Proceedings of the 13th IEEE International Symposium on Applications of Ferroelectrics, Nara, Japan, June 1, 2002 pp447−450

    [23]

    Li X, Wang Z, He C, Liu Y, Long X, Han S, Pan S 2015 Mater. Lett. 143 88Google Scholar

    [24]

    Rajan K K, Zhang M J, Lim L C 2005 Jpn. J. Appl. Phys. 44 264Google Scholar

    [25]

    Salak A N, Shilin A D, Bushinski M V, Olekhnovich N M, Vyshatko N P 2000 Mater. Res. Bull. 35 1429Google Scholar

    [26]

    Salak A N, Vyshatko N P, Ferreira V M, Olekhnovich N M, Shilin A D 2003 Mater. Res. Bull. 38 453Google Scholar

    [27]

    Yamashita Y, Harada K 1997 Jpn J. Appl. Phys. 36 6039Google Scholar

    [28]

    Wang J, Tang X G, Chan H L W, Choy C L, Luo H 2005 Appl. Phys. Lett. 86 152907Google Scholar

    [29]

    Grinberg I, Rappe A M 2007 Phase Trans. 80 351Google Scholar

    [30]

    Balachandran P V, Shearman T, Theiler J, Lookman T 2017 Acta Crystallogr. Sect. B 73 962Google Scholar

    [31]

    周志华 2016 机器学习 (北京: 清华大学出版社) 第133−137页

    Zhou Z H 2016 Machine Learning (Beijing: Tsinghua Univer-sity Press) pp133−137 (in Chinese)

    [32]

    Geurts P, Ernst D, Wehenkel L 2006 Machine Learning 63 3Google Scholar

    [33]

    Hoerl A E, Kennard R W 1970 Technometrics 12 55Google Scholar

    [34]

    Welling M https://www.ics.uci.edu/~welling/classnotes/papers _class/Kernel-Ridge.pdf [2019-6-7]

    [35]

    Lin C J https://www.csie.ntu.edu.tw/~cjlin/talks/kuleuven_svm.pdf [2019-6-7]

    [36]

    Bergstra J, Bengio Y 2012 J. Mach. Learn. Res. 13 281

    [37]

    Louppe G, Wehenkel L, Sutera A, Geurts P 2013 Under-standing Variable Importances in Forests of Randomized Trees (Advances in Neural Information Processing Systems) pp431−439

  • 图 1  支持向量回归中的超参数ε的优化及支持向量数目分析

    Fig. 1.  Optimization of hyperparameters in support vector regression and the analysis of the number of support vectors

    图 2  不同模型权重的融合实验结果

    Fig. 2.  Performance of ensembled model with different base model weight.

    图 3  三种机器学习模型及其集成模型对材料的预测值与实验值的比较

    Fig. 3.  Comparison of prediction and experimental values of three machine learning models and their ensemble models.

    图 4  ETR模型中最重要的5项特征

    Fig. 4.  Five most important features in ETR.

    图 5  集成机器学习模型对PGN-PMN-PT固溶体的居里温度的预测

    Fig. 5.  Prediction of Curie temperature of PGN-PMN-PT solid solution by ensemble machine learning model

    表 1  本文三种机器学习方法所采用的超参数

    Table 1.  Hyperparameters of the three machine learning methods in this study.

    模型超参数
    KRR$\lambda $ = 0.001, $\gamma $ = 0.005
    SVR$\gamma $ = 0.02, ε = 4, C = 900
    ETR n = 108, depth = 20, min_samples = 2
    下载: 导出CSV

    表 2  本文所使用的机器学习方法的评估及与其他研究者工作的对比

    Table 2.  Evaluation of machine learning methods in this paper and the comparison with other works.

    模型对比
    KRRSVRETR集成
    模型
    文献[10]文献[11]
    MAE/K14.414.716.113.930.221.2
    RMSE/K22.523.423.821.428.7
    相关系数0.960.960.960.970.85
    下载: 导出CSV
  • [1]

    Liu Y, Li X, Wang Z, He C, Li T, Ai L, Chu T, Pang D, Long X 2013 CrystEngComm 15 1643Google Scholar

    [2]

    Wu J, Chang Y, Yang B, Zhang S, Sun Y, Guo F, Cao W 2015 J. Mater. Sci.: Mater. Electron 26 1874Google Scholar

    [3]

    Zhao Y Q, Ma Q R, Liu B, Yu Z L, Yang J, Cai M Q 2018 Nanoscale 10 8677Google Scholar

    [4]

    Ding Y F, Zhao Q Q, Yu Z L, Zhao Y Q, Liu B, He P B, Zhou H, Li K, Yin S F, Cai M Q 2019 J. Mater. Chem. C 7 7433Google Scholar

    [5]

    Yu Z L, Ma Q R, Liu B, Zhao Y Q, Wang L Z, Zhou H, Cai M Q 2017 J. Phys. D 50 465101Google Scholar

    [6]

    Zhang T, Jiang Y, Song Z, Huang H, He Y, Fang Z, Weng H, Fang C 2019 Nature 566 475Google Scholar

    [7]

    Umehara M, Stein H S, Guevarra D, Newhouse P F, Boyd D A, Gregoire J M 2019 NPJ Comput. Mater. 5 34Google Scholar

    [8]

    Iwasaki Y, Takeuchi I, Stanev V, et al. 2019 Sci. Rep. 9 2751Google Scholar

    [9]

    Aykol M, Hegde V I, Hung L, Suram S, Herring P, Wolverton C, Hummelshøj J S 2019 Nat. Commun 10 1Google Scholar

    [10]

    Balachandran P V, Kowalski B, Sehirlioglu A, Lookman T 2018 Nat. Commun. 9 1668Google Scholar

    [11]

    Zhai X, Chen M, Lu W 2018 Comput. Mater. Sci. 151 41Google Scholar

    [12]

    Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, Duchesnay É 2011 J. Mach. Learn. Res. 12 2825

    [13]

    Chen H, Fan C 2010 Mater. Lett. 64 654Google Scholar

    [14]

    Fang B, Shan Y, Tezuka K, Imoto H 2005 J. Mater. Sci. 40 6445Google Scholar

    [15]

    Gao Y Q, Xu H Q, Wu Y J, He T H, Xu G S, Luo H S 2001 Jpn. J. Appl. Phys. 40 4998Google Scholar

    [16]

    Guo Y P, Xu H Q, Luo H S, Xu G S, Yin Z W 2001 J. Cryst. Growth 226 111Google Scholar

    [17]

    Hao H, Zhang S, Liu H, Shrout T R 2009 J. Appl. Phys. 105 024104Google Scholar

    [18]

    He C, Li X, Wang Z, Liu Y, Shen D, Li T, Long X, Ye Z G 2012 CrystEngComm 14 4407Google Scholar

    [19]

    Wang L, Liang R, Mao C, Du G, Wang G, Dong X 2013 Ceram. Int. 39 8571Google Scholar

    [20]

    Yasuda N, Ohwa H, Hasegawa D, Hayashi K, Hosono Y, Yamashita Y, Iwata M, Ishibashi Y 2000 Jpn. J. Appl. Phys. 39 5586Google Scholar

    [21]

    Zhang S, Lebrun L, Jeong D Y, Randall C A, Zhang Q, Shrout T R 2003 J. Appl. Phys. 93 9257Google Scholar

    [22]

    Bing Y, Ye Z 2002 Proceedings of the 13th IEEE International Symposium on Applications of Ferroelectrics, Nara, Japan, June 1, 2002 pp447−450

    [23]

    Li X, Wang Z, He C, Liu Y, Long X, Han S, Pan S 2015 Mater. Lett. 143 88Google Scholar

    [24]

    Rajan K K, Zhang M J, Lim L C 2005 Jpn. J. Appl. Phys. 44 264Google Scholar

    [25]

    Salak A N, Shilin A D, Bushinski M V, Olekhnovich N M, Vyshatko N P 2000 Mater. Res. Bull. 35 1429Google Scholar

    [26]

    Salak A N, Vyshatko N P, Ferreira V M, Olekhnovich N M, Shilin A D 2003 Mater. Res. Bull. 38 453Google Scholar

    [27]

    Yamashita Y, Harada K 1997 Jpn J. Appl. Phys. 36 6039Google Scholar

    [28]

    Wang J, Tang X G, Chan H L W, Choy C L, Luo H 2005 Appl. Phys. Lett. 86 152907Google Scholar

    [29]

    Grinberg I, Rappe A M 2007 Phase Trans. 80 351Google Scholar

    [30]

    Balachandran P V, Shearman T, Theiler J, Lookman T 2017 Acta Crystallogr. Sect. B 73 962Google Scholar

    [31]

    周志华 2016 机器学习 (北京: 清华大学出版社) 第133−137页

    Zhou Z H 2016 Machine Learning (Beijing: Tsinghua Univer-sity Press) pp133−137 (in Chinese)

    [32]

    Geurts P, Ernst D, Wehenkel L 2006 Machine Learning 63 3Google Scholar

    [33]

    Hoerl A E, Kennard R W 1970 Technometrics 12 55Google Scholar

    [34]

    Welling M https://www.ics.uci.edu/~welling/classnotes/papers _class/Kernel-Ridge.pdf [2019-6-7]

    [35]

    Lin C J https://www.csie.ntu.edu.tw/~cjlin/talks/kuleuven_svm.pdf [2019-6-7]

    [36]

    Bergstra J, Bengio Y 2012 J. Mach. Learn. Res. 13 281

    [37]

    Louppe G, Wehenkel L, Sutera A, Geurts P 2013 Under-standing Variable Importances in Forests of Randomized Trees (Advances in Neural Information Processing Systems) pp431−439

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出版历程
  • 收稿日期:  2019-06-18
  • 修回日期:  2019-08-20
  • 上网日期:  2019-11-01
  • 刊出日期:  2019-11-05

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