# Predicting Perovskite Performance with Multiple Machine-Learning Algorithms

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{3}with matching compatible sizes of A-site and B-site cations. In this work, four different prediction models of machine learning algorithms, including support vector regression based on radial basis kernel function (SVM-RBF), ridge regression (RR), random forest (RF), and back propagation neural network (BPNN), are established to predict the formation energy, thermodynamic stability, crystal volume, and oxygen vacancy formation energy of perovskite materials. Combined with the fitting diagrams of the predicted values and DFT calculated values, the results show that SVM-RBF has a smaller bias in predicting the crystal volume. RR has a smaller bias in predicting the thermodynamic stability. RF has a smaller bias in predicting the formation energy, crystal volume, and thermodynamic stability. BPNN has a smaller bias in predicting the formation energy, thermodynamic stability, crystal volume, and oxygen vacancy formation energy. Obviously, different machine learning algorithms exhibit different sensitivity to data sample distribution, indicating that we should select different algorithms to predict different performance parameters of perovskite materials.

## 1. Introduction

_{3}perovskite composite oxides have attracted great interest [3,4,7,8,9,10,11,12,13,14,15]. Research has focused on the development of new perovskite materials to improve activity, selectivity, and stability, as well as the development of advanced manufacturing techniques to reduce their cost while ensuring their reliability, safety, and reproducibility [14,15,16]. In ABO

_{3}perovskite oxides, the A site is the rare earth or alkaline earth metal ions, which usually stabilize the structure, while the B site is occupied by the smaller transition metal ions [17]. Through partial substitution of A and B sites, multi-component perovskite compounds can be combined [16]. When A or B sites are partially replaced by other metal ions, anion defects or B sites at different valences can be formed. This improves the properties of the compounds but does not fundamentally change the crystal structure [17]. This kind of composite oxide has gas sensitivity, oxidation catalytic property, conductivity, oxygen permeability, and other properties. In addition, its structure and performance are closely related to the composition of the system [17]. Perovskite-type oxides can form compounds through partial doping of metal ions at A and B sites on the basis of maintaining stable crystal structure, as well as controlling the elements and valence states so that the performance of perovskite materials present rich diversity [18,19,20].

_{3}A

_{2}D

_{3}O

_{12}garnets and ABO

_{3}perovskites with low mean absolute errors (MAEs) [34]. Wei et al. developed machine learning models to predict the thermodynamic phase stability of perovskite oxides using a dataset of more than 1900 DFT-calculated perovskite. The results showed that that error is within the range of errors in DFT formation energies relative to elemental reference states when compared to experiments and, therefore, may be considered sufficiently accurate to use in place of full DFT calculations [35]. Using different machine learning algorithms, the formation energy, thermodynamic stability, crystal volume, and oxygen vacancy formation energy of perovskite materials were predicted [36,37].

## 2. Principles and Methods

#### 2.1. Regression Prediction of Support Vector Machines

#### 2.2. Random Forest

_{1}, θ

_{2}, …, and θ

_{k}use each training set to generate the corresponding decision tree {T(x, θ

_{1})}, {T(x, θ

_{2})}, …, and {T(x, θ

_{k})}.

_{i}(i = 1, 2, …, n) of the dependent variable, and the predicted value $\widehat{u}$ of a single decision tree is shown in Equation (3).

_{i}(x, θ

_{t})(t = 1, 2, … k) of the decision tree, as shown in Equations (4) and (5).

#### 2.3. Ridge Regression

#### 2.4. BP Neural Network

_{1}, X

_{2}, …, and X

_{n}represent the input value of the BP neural network, and Y

_{1}, Y

_{2}, …, and Y

_{n}represent the output value of the BP neural network [56].

#### 2.5. Performance Evaluation

^{2}) were used to observe and measure the prediction accuracy of the model and to compare the performance differences of different models. The smaller MAE and MSE are, the larger R

^{2}is, and the closer to 1 is, indicating that the prediction effect of the model is better [42,49,57]. Their formula is as follows:

_{j}is the true value, $\widehat{{y}_{j}}$ is the predicted value, and $\overline{y}$ is the average value.

## 3. Model Construction

_{3}perovskite high-throughput data sets were obtained, and four characteristic performance parameters including formation energy, thermodynamic stability, crystal volume, and oxygen vacancy formation energy in the original material data set were going to predict [58].

^{2}were used to evaluate the model effect.

## 4. Results and Discussion

^{2}were used to evaluate the model, and the results are shown in Figure 3. It can be seen that the R

^{2}value of RF is the highest, which is 0.7231, and the values of MAE and MSE are the lowest, which are 0.3731 and 0.2449, respectively. RF has the best prediction effect on the formation energy. For the stability prediction, the R

^{2}value of SVM-RBF is 0.8081, and the MAE and MSE are 0.2074 and 0.0898, respectively, which are the best for the stability prediction. For crystal volume prediction, the R

^{2}value of BPNN is the largest, which is 0.9372, and the MAE and MSE are the smallest, which are 0.4134 and 0.4679, respectively. For the prediction of oxygen vacancy formation energy, the evaluation indexes of SVM-RBF and RF are similar, and the prediction effect is better than that of RR and BPNN.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Prediction performance of machine learning model on test set, (

**a**) prediction performance of formation energy, (

**b**) prediction performance of stability, (

**c**) prediction performance of volume, and (

**d**) prediction performance of formation energy of oxygen vacancy.

**Figure 4.**The fitting diagram based on the predicted value of SVM-RBF and the calculated value of DFT is (

**a**) the fitting diagram of formation energy, (

**b**) the fitting diagram of stability, (

**c**) the fitting diagram of volume, and (

**d**) the fitting diagram of oxygen vacancy formation energy. Red points work as reference points, which is ideal values obeyed y = x.

**Figure 5.**Fitting diagram based on RF predicted value and DFT calculated value: (

**a**) fitting diagram for formation energy, (

**b**) fitting diagram for stability, (

**c**) fitting diagram for volume, and (

**d**) fitting diagram for formation energy of oxygen vacancy. Red points work as reference points, which is ideal values obeyed y = x.

**Figure 6.**The fitting diagram based on the predicted value of RR and the calculated value of DFT is (

**a**) the fitting diagram of formation energy, (

**b**) the fitting diagram of stability, (

**c**) the fitting diagram of volume, and (

**d**) the fitting diagram of formation energy of oxygen vacancy. Red points work as reference points, which is ideal values obeyed y = x.

**Figure 7.**The fitting diagram based on the predicted value of BPNN and the calculated value of DFT is (

**a**) the fitting diagram of formation energy, (

**b**) the fitting diagram of stability, (

**c**) the fitting diagram of volume, and (

**d**) the fitting diagram of formation energy of oxygen vacancy. Red points work as reference points, which is ideal values obeyed y = x.

No. | Property | Type | Unit | Description |
---|---|---|---|---|

1 | Radius A | number | ang | Shannon ionic radius of atom A. |

2 | Radius B | number | ang | Shannon ionic radius of atom B. |

3 | Formation energy | number | eV/atom | Formation energy as calculated by equation of the distortion with the lowest energy. |

4 | Stability | number | eV/atom | Stability as calculated by equation of the distortion with the lowest energy. |

5 | Volume per atom | number | A^{3}/atom | Volume per atom of the relaxed structure. |

6 | Band gap | number | eV | PBE band gap obtained from the relaxed structure. |

7 | a | number | ang | Lattice parameter a of the relaxed structure. |

8 | b | number | ang | Lattice parameter b of the relaxed structure. |

9 | c | number | ang | Lattice parameter c of the relaxed structure. |

10 | alpha | number | deg | α angle of the relaxed structure. |

11 | beta | number | deg | β angle of the relaxed structure. |

12 | gamma | number | deg | γ angle of the relaxed structure. |

13 | Vacancy energy | number | eV/O atom | Thermodynamic stability was assessed using an energy convex hull construction. |

Property | Method | Evaluation Index | ||
---|---|---|---|---|

MAE | MSE | R^{2} | ||

Formation energy | SVM-RBF | 0.5104 | 0.4016 | 0.5607 |

RF | 0.3731 | 0.2449 | 0.7231 | |

RR | 0.5822 | 0.5109 | 0.4574 | |

BPNN | 0.4744 | 0.3514 | 0.6091 | |

Stability | SVM-RBF | 0.2074 | 0.0898 | 0.8081 |

RF | 0.2023 | 0.0895 | 0.7792 | |

RR | 0.2465 | 0.1078 | 0.7263 | |

BPNN | 0.2239 | 0.0993 | 0.7808 | |

Volume per atom | SVM-RBF | 0.4626 | 0.7085 | 0.9042 |

RF | 0.4442 | 0.6271 | 0.9195 | |

RR | 1.8019 | 5.0720 | 0.3205 | |

BPNN | 0.4134 | 0.4679 | 0.9372 | |

Vacancy energy | SVM-RBF | 1.8631 | 6.7088 | 0.6631 |

RF | 1.8742 | 7.0501 | 0.6562 | |

RR | 2.3823 | 9.9980 | 0.5265 | |

BPNN | 2.0144 | 6.7663 | 0.6651 |

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**MDPI and ACS Style**

Li, R.; Deng, Q.; Tian, D.; Zhu, D.; Lin, B.
Predicting Perovskite Performance with Multiple Machine-Learning Algorithms. *Crystals* **2021**, *11*, 818.
https://doi.org/10.3390/cryst11070818

**AMA Style**

Li R, Deng Q, Tian D, Zhu D, Lin B.
Predicting Perovskite Performance with Multiple Machine-Learning Algorithms. *Crystals*. 2021; 11(7):818.
https://doi.org/10.3390/cryst11070818

**Chicago/Turabian Style**

Li, Ruoyu, Qin Deng, Dong Tian, Daoye Zhu, and Bin Lin.
2021. "Predicting Perovskite Performance with Multiple Machine-Learning Algorithms" *Crystals* 11, no. 7: 818.
https://doi.org/10.3390/cryst11070818