基于机器学习的BiFeO3-PbTiO3-BaTiO3固溶体居里温度预测

doi:10.15541/jim20220080

无机材料学报 ›› 2022, Vol. 37 ›› Issue (12): 1321-1328.DOI: 10.15541/jim20220080 CSTR: 32189.14.10.15541/jim20220080

基于机器学习的BiFeO₃-PbTiO₃-BaTiO₃固溶体居里温度预测

焦志翔¹(), 贾帆豪¹^,²(), 王永晨¹, 陈建国¹, 任伟², 程晋荣¹()

1.上海大学材料科学与工程学院, 上海 200444
2.上海大学物理系, 量子与分子结构国际中心, 上海 200444

收稿日期:2022-02-17 修回日期:2022-03-29 出版日期:2022-12-20 网络出版日期:2022-08-04
通讯作者: 程晋荣, 研究员. E-mail: jrcheng@shu.edu.cn;
贾帆豪, 博士. E-mail: fanhaojia@shu.edu.cn
作者简介:焦志翔(1996-), 男, 硕士研究生. E-mail: jzxxxzj@163.com
基金资助:
水声对抗技术重点实验室开放基金(JCKY2020207CH02);国家自然科学基金(51872180);国家自然科学基金(51672169)

Curie Temperature Prediction of BiFeO₃-PbTiO₃-BaTiO₃ Solid Solution Based on Machine Learning

JIAO Zhixiang¹(), JIA Fanhao¹^,²(), WANG Yongchen¹, CHEN Jianguo¹, REN Wei², CHENG Jinrong¹()

1. School of Materials Science and Engineering, Shanghai University, Shanghai 200444, China
2. Department of Physics, International Center for Quantum and Molecular Structures, Shanghai University, Shanghai 200444, China

Received:2022-02-17 Revised:2022-03-29 Published:2022-12-20 Online:2022-08-04
Contact: CHENG Jinrong, professor. E-mail: jrcheng@shu.edu.cn;
JIA Fanhao, PhD. E-mail: fanhaojia@shu.edu.cn
About author:JIAO Zhixiang (1996-), male, Master candidate. E-mail: jzxxxzj@163.com
Supported by:
Open Fund project of Key Laboratory of Underwater Acoustic Countermeasures Technology(JCKY2020207CH02);National Natural Science Foundation of China(51872180);National Natural Science Foundation of China(51672169)

摘要/Abstract

摘要：

钙钛矿(ABO₃)型压电陶瓷的发展已有几十年历史, 现存有大量数据, 从这些数据中寻找出材料结构与性能之间的关系很有意义。本工作收集了BiFeO₃-PbTiO₃-BaTiO₃钙钛矿型压电陶瓷居里温度(T_c)实验数据, 通过机器学习，构建钙钛矿型压电陶瓷T_c的预测模型。热力学角度, T_c与约合质量符合二次多项式关系, 但偏差较大。选择元素信息、物理量、空间群编号等基础描述符, 利用基于压缩感知原理的SISSO(Sure Independence Screening and Sparsifying Operator)方法进行机器学习, 找出了T_c与成分之间的相关性。比较不同描述符在不同维度上的均方根误差RMSE (Root Mean Square Error), 发现描述符越多、越基础, 维数越大、RMSE越小。同时比较相同个数描述符在同一维度下的RMSE, 用约合质量、A位和B位的离子半径比、A位和B位的未填充电子数比和Ba、Pb、Bi的元素含量等六个描述符构建出最优的四维模型, 其RMSE为0.59 ℃, 最大绝对误差(MaxAE)为1.38 ℃, 外部测试的平均相对误差MRE (Mean Relative Error)为1.00%。结果表明，利用SISSO可以进行有限样本钙钛矿型压电陶瓷T_c的机器学习预测。

关键词: 钙钛矿型压电陶瓷, 机器学习, 居里温度, SISSO

Abstract:

Perovskite (ABO₃) piezoceramics have been developed for several decades, and there are a lot of data available. It is of great significance to find relationships between structure and properties of materials from these data. In this work, experimental data of Curie temperature (T_c) of BiFeO₃-PbTiO₃-BaTiO₃ solid solution of perovskite piezoelectric ceramics was collected to build the model to predict the T_c. From the perspective of thermodynamics, the quadratic polynomial relationship between T_c and reduced mass was introduced but the deviation was relatively large. More descriptors (including element information, physical quantities, space groups number) and SISSO (Sure Independence Screening and Sparsifying Operator) were used for machine learning to find the correlation between T_c and components. Comparing the root mean square error (RMSE) of different descriptors and dimensions, it's found that more descriptors, more fundamental the descriptors are, and larger dimension will result in smaller RMSE to be used. Meanwhile, RMSE of the same number of descriptors in the same dimension are compared. The optimal four-dimensional model is build using six descriptors: reduced mass, the ratio of A- and B-site ion radii, the ratio of A- and B-site unfilled electrons and element contents of Ba, Pb and Bi. RMSE and maximum absolute error (MaxAE) of our model are 0.59 ℃ and 1.38 ℃, respectively. The average relative error (MRE) of external test is 1.00%. Our results indicate that SISSO machine learning based on limited samples is suitable for the predication of T_c of perovskite piezoelectric ceramics.

Key words: perovskite piezoelectric ceramics, machine learning, Curie temperature, SISSO

中图分类号:

TQ174

焦志翔, 贾帆豪, 王永晨, 陈建国, 任伟, 程晋荣. 基于机器学习的BiFeO₃-PbTiO₃-BaTiO₃固溶体居里温度预测[J]. 无机材料学报, 2022, 37(12): 1321-1328.

JIAO Zhixiang, JIA Fanhao, WANG Yongchen, CHEN Jianguo, REN Wei, CHENG Jinrong. Curie Temperature Prediction of BiFeO₃-PbTiO₃-BaTiO₃ Solid Solution Based on Machine Learning[J]. Journal of Inorganic Materials, 2022, 37(12): 1321-1328.

图/表 10

图1 ABO3钙钛矿结构示意图

Fig. 1 Schematic diagram of ABO3 perovskite structure

表1 基础描述符及其物理意义

Table 1 Basic descriptor and related physical meaning

Feature	Name	Physical attributes
μ	Reduced mass	Reduced mass of atoms
A/B	Ionic radius	Shannon ion radius
A/B_C	Covalent radius	The covalent bond radius of an atom
A/B_E	Electronegativity	Electronegativity of atoms
A/B_NV	NValence	The number of electrons of unfilled orbitals
A/B_NU	NUfilled	The number of electrons of unfilled orbitals
A/B_S	Space group numbering	The serial number of the element's space group in the space group table
Ti,Fe,Ba,Pb,Bi	Element content	Element content ratio of metal ions

图2 Tc与μ和μ*A/B的关系

Fig. 2 Relationship between Tc and μ or μ*A/B

图3 Tc与候选描述符的相关性

Fig. 3 Correlations between the Tc and the candidate descriptors

图4 不同运算方式下μ和μ*A/B随维度变化的RMSE和MaxAE

Fig. 4 RMSE and MaxAE of μ and μ*A/B varied with dimension under different operation modes (a, c) use operation mode 1; (b, d) use operation mode 2; (a, b) use the descriptor μ; (c, d) use the descriptor μ*A/B

图5 采用两个描述符(a)和三个描述符(b)拟合结果的RMSE和MaxAE随维度的变化

Fig. 5 RMSE and MaxAE of two descriptors (a) and three descriptors (b) as a function of dimension

图6 描述符的变化和模型维度对拟合结果RMSE和MaxAE的影响

Fig. 6 Effects of the changes of descriptor and model dimension on the RMSE and MaxAE (a) μ and A/B are the first two descriptors, and the third descriptor changes; (b) μ, A/B, Ba, Pb, and Bi are the first five descriptors, and the sixth descriptor changes; (c) μ and A/B are the first two descriptors and the third to seventh descriptors are introduced one by one. (d) Effect of the change of the dimension of the model fitted by six finally selected descriptors (μ, A/B, A/B_NU, Ba, Pb, and Bi) on RMSE and MaxAE

表2 调整描述符参数后的测试集

Table 2 The test sets after adjusting descriptor parameters

Sample	T_c/℃ (y)	μʹ (x₁)	A/Bʹ (x₂)	A/B_NU (x₃)	Ba (x₄)	Pb (x₅)	Bi (x₆)
Bi_0.62Pb_0.23Ba_0.15Fe_0.62Ti_0.38O₃	547	0.2821	0.3882	0.5036	0.0750	0.1150	0.3100
Bi_0.6Pb_0.25Ba_0.15Fe_0.6Ti_0.4O₃	540	0.2690	0.4606	0.5000	0.0750	0.1250	0.3000
Bi_0.54Pb_0.31Ba_0.15Fe_0.54Ti_0.46O₃	502	0.2295	0.6789	0.4897	0.0750	0.1550	0.2700

表3 外部验证集结果

Table 3 Results of external verification set

Sample	Experim- ental T_c/℃	Predi- ction T_c/℃	Abso- lute error/℃	Rela- tive- error/%
Bi_0.62Pb_0.23Ba_0.15Fe_0.62Ti_0.38O₃	547	544.91	2.09	0.38
Bi_0.6Pb_0.25Ba_0.15Fe_0.6Ti_0.4O₃	540	536.12	3.88	0.72
Bi_0.54Pb_0.31Ba_0.15Fe_0.54Ti_0.46O₃	502	492.52	9.48	1.89

图7 训练集(黑色方块)和测试集(红色圆形)的实验Tc与预测Tc

Fig 7 Experimental Tc and prediction Tc for training set (black square) and test set (red circle)

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基于机器学习的BiFeO₃-PbTiO₃-BaTiO₃固溶体居里温度预测

Curie Temperature Prediction of BiFeO₃-PbTiO₃-BaTiO₃ Solid Solution Based on Machine Learning

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