# Modeling the Maximum Magnetic Entropy Change of Doped Manganite Using a Grid Search-Based Extreme Learning Machine and Hybrid Gravitational Search-Based Support Vector Regression

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{4+}and Mn

^{3+}ions and A-site disorder are strongly affected by the nature as well as the concentration of dopants [12,13,14]. A-site average ionic radius contributes mainly to magnetic properties of doped manganite through an energy bandwidth of electron alteration that is related to the observed double-exchange interaction, while A-site disorder measures A-site ionic distribution variance [15,16]. The present work developed models through which the maximum magnetic entropy change (MMEC) of manganite compounds could be influenced by the introduction of dopants into the parent manganite compound.

^{4+}and Mn

^{3+}ions to the ferromagnetic feature of these compounds, while the exhibition of magnetic properties is well captured within electron–phonon interaction described in Jahn–Teller formalism [4]. Doping of sites in the parent manganite compound with divalent alkaline–earth ions, monovalent alkaline or any other transition metals induces distortion in the lattice and changes the Mn

^{4+}and Mn

^{3+}ion network that, in turn, alters the magnetocaloric effect as well as the maximum magnetic entropy change (MMEC) of manganite-based compounds. Identifying the relationship between the nature as well as the concentration of the dopants supports the understanding of their physical properties, while the synthesis of new manganite-based compounds having multifunctional applications becomes possible. Some screening approaches to developing and identifying materials with a large magnetocaloric effect have been proposed in the literature [19], [20]. However, machine learning techniques have been identified as effective and efficient tools for determining the influence of dopants on the physical properties (such as magnetic ordering temperature, relative cooling power and magnetocaloric effect) of doped manganite-based compounds [6,21,22,23,24,25]. The proposed grid search-based extreme learning machine (GS-ELM) and hybrid gravitational search-based support vector regression model the relationship between lattice distortion as well as elemental compositions on the MMEC when dopants are incorporated into the crystal structure of the parent manganite-based compounds.

## 2. Mathematical Formulation of the Proposed Models

#### 2.1. Description of the Support Vector Regression Algorithm

#### 2.2. Physical Principles of the Gravitational Search Algorithm

#### 2.3. Mathematical Background of the Extreme Learning Machine

## 3. Dataset Description and Computational Implementation of the Proposed Models

#### 3.1. Dataset Description and Chemical Formula of the Doped Manganite that Can be Incorporated into the Developed Models

^{4+}and Mn

^{3+}ion network. Similarly, since ionic radii of the dopants have a strong influence on the magnetic properties of manganite-based compounds, these ionic radii and the concentrations of dopants serve as the descriptors to the developed GSA-SVR-radii model. This model also employs the value of the applied field as a descriptor. The chemical formula that explains the implementation of the proposed GSA-SVR-radii model on manganite-based compounds is presented in Equation (16).

_{0.6}Ca

_{0.1}Sr

_{0.3}Mn

_{0.8}Fe

_{0.2}O

_{3}compound; the descriptors are the ionic radii of elements Pr, Ca, Sr and Fe, while a, b and c are 0.1, 0.3 and 0.2, respectively. Table 1 shows the results of statistical analysis conducted on the dataset. This is insightful in determining the content and the nature of the dataset employed for modeling and simulation. The correlation coefficients between the descriptors and the target clearly show an insignificant degree of linear relationship between them. Therefore, the need for nonlinear modeling techniques such as those presented in this work becomes necessary.

#### 3.2. Computational Hybridization of the Gravitational Search and Support Vector Regression Algorithms

#### 3.3. Computational Implementation of the Grid Search-Based Extreme Learning Machine

## 4. Results and Discussion

#### 4.1. Searching for the Optimum Hyperparameters of the Developed Models

#### 4.2. Generalization and Predictive Strength Comparison Between the SVR- and ELM-Based Models

#### 4.3. Superiority of the Present Models as Compared to the Existing Models in the Literature

#### 4.4. Investigating the Doping Effect of Fe on the Value of the MMEC of the Pr_{0.6}Ca _{0.1}Sr _{0.3}Mn _{1-x}Fe_{x}O_{3} Manganite Compound Using the Developed GSA-SVR-Crystal Model

_{0.6}Ca

_{0.1}Sr

_{0.3}Mn

_{1-x}Fe

_{x}O

_{3}manganite compound is presented in Table 5. It was observed that an increase in the concentration of Fe lowers the value of the MMEC. This observation as obtained using the developed GSA-SVR-crystal model conforms to the experimental observation [70]. The effect of the partial substitution of Fe in the Mn site in the Pr

_{0.6}Ca

_{0.1}Sr

_{0.3}Mn

_{1-x}Fe

_{x}O

_{3}manganite compound lowers the value of the MMEC because the incorporation of Fe particles modifies the Mn/Fe-O distance and Mn-O-Mn angle, which subsequently leads to a weakening of the double-exchange interaction at the expense of the superexchange interaction [38].

#### 4.5. Implementation of the Developed Model for Determining the Maximum Magnetic Entropy Change of Different Classes of Doped Manganite

## 5. Conclusions and Recommendations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Comparison between the GSA-SVR-radii and GSA-SVR-Crystal models on the basis of the correlation coefficient.

**Figure 4.**Comparison between the GSA-SVR-radii and GSA-SVR-crystal models on the basis of the mean absolute error.

**Figure 5.**Comparison between the GSA-SVR-radii and GSA-SVR-crystal models on the basis of the root-mean-square error.

**Figure 6.**Comparison between the ELM-GS-radii and ELM-GS-crystal models on the basis of the correlation coefficient.

**Figure 7.**Comparison between the ELM-GS-radii and ELM-GS-crystal models on the basis of the mean absolute error.

**Figure 8.**Comparison between the ELM-GS-radii and ELM-GS-crystal models on the basis of the root-mean-square error.

**Figure 9.**Correlation cross-plot between the measured and estimated MMEC for all developed models using the training set of data.

**Figure 10.**Correlation cross-plot between the measured and estimated MMEC for all developed models using the testing set of data.

**Figure 11.**Comparison between the present and existing ionic radii-based models using the mean absolute error metric.

**Figure 12.**Comparison between the present and existing ionic radii-based models using the root-mean-square error metric.

**Figure 13.**Comparison between the present and existing crystal lattice distortion-based models using the mean absolute error metric.

**Figure 14.**Comparison between the present and existing crystal lattice distortion-based models using the root-mean-square error metric.

Quantity | Minimum | Maximum | Mean | Standard Deviation | Correlation Coefficient |
---|---|---|---|---|---|

R(pm) | 109.8 | 143 | 115.696 | 4.652 | −0.272 |

A(pm) | 0 | 152 | 103.83 | 51.2 | 0.389 |

B(pm) | 0 | 152 | 79.92 | 66.139 | 0.334 |

C(pm) | 0 | 100 | 23.07 | 38.523 | 0.14 |

a | 0 | 0.45 | 0.162 | 0.126 | 0.261 |

b | 0 | 0.45 | 0.161 | 0.169 | 0.441 |

c | 0 | 0.3 | 0.029 | 0.048 | −0.189 |

Distortion along a-axis | 5.429 | 5.553 | 5.488 | 0.033 | −0.557 |

Distortion along b-axis | 5.438 | 13.467 | 6.483 | 1.307 | 0.102 |

Distortion along c-axis | 5.419 | 359.95 | 12.036 | 35.311 | 0.105 |

Applied field (T) | 0.05 | 6 | 3.483 | 1.758 | 0.583 |

MMEC | 0.003 | 7.14 | 2.117 | 1.567 |

**Table 2.**Optimum hyperparameters of support vector regression (SVR)-based models as obtained using a gravitational search algorithm (GSA) and extreme learning machine (ELM)-based models as obtained using grid search (GS).

GSA-SVR-Radii | GSA-SVR-Crystal | ELM-GS-Radii | ELM-GS-Crystal | |
---|---|---|---|---|

Penalty factor | 944.3013 | 592.9928 | ---- | --- |

Epsilon | 0.1003 | 0.3758 | ---- | --- |

Kernel option | 0.8198 | 0.2516 | ----- | ---- |

Number in the population | 70 | 30 | ---- | --- |

Activation function | --- | --- | Hardlim | Sin |

Number of hidden nodes | --- | --- | 58 | 24 |

**Table 3.**Performance measuring parameters for the four developed models at different stages of model development.

Parameters | GSA-SVR-Radii | GSA-SVR-Crystal | ELM-GS-Radii | ELM-GS-Crystal | ||||
---|---|---|---|---|---|---|---|---|

Training Testing | Training Testing | Training Testing | Training Testing | |||||

CC | 0.9545 | 0.958 | 0.9317 | 0.9044 | 0.83777 | 0.961358 | 0.88462 | 0.914984 |

RMSE (J/Kg K) | 0.4655 | 0.462 | 0.577 | 0.767 | 0.85233 | 0.285445 | 0.715575 | 0.46994 |

MAE (J/Kg K) | 0.1766 | 0.376 | 0.4094 | 0.5299 | 0.586574 | 0.426137 | 0.51419 | 0.670479 |

GA-SVR_Ionic (Existing) [38] | GA-SVR-Lattice (Existing) [38] | GSA-SVR-Radii (This Work) | GSA-SVR-Crystal (This Work) | ELM-GS-Radii (This Work) | ELM-GS-Crystal (This Work) | |
---|---|---|---|---|---|---|

MAE (J/Kg K) | 0.304311 | 0.578056 | 0.216517 | 0.433503 | 0.526348 | 0.50534 |

RMSE (J/Kg K) | 0.590673 | 0.890676 | 0.464783 | 0.619666 | 0.785806 | 0.706786 |

**Table 5.**Comparison between the results of the developed GSA-SVR-crystal model and the measured values of the MMEC for different classes of manganite-based compounds.

Doped Manganite-Based Compounds | Measured MMEC (J/KgK) | Estimated MMEC Using GSA-SVR-Crystal (J/KgK) | Absolute Error |
---|---|---|---|

La_{0.7}Ca_{0.25}Sr_{0.05}MnO_{3} | 1.65 [73] | 1.5601 | 0.0899 |

La_{0.67}Ca_{0.13} Ba_{0.2}Mn_{0.9}Co _{0.1}O_{3} | 1.93 [74] | 1.5141 | 0.4159 |

La_{0.6}Bi_{0.1}Sr_{0.3}Mn_{0.9}Cu _{0.1}O_{3} | 3.39 [75] | 3.9562 | 0.5662 |

La_{0.8}Na_{0.2}Mn_{0.94}Ni _{0.06}O_{3} | 3.70 [76] | 4.1572 | 0.4572 |

La_{0.065}Nd_{0.05}Ba_{0.3}Mn_{0.85}Cr_{0.15}O_{3} | 3.55 [77] | 3.1356 | 0.4144 |

Pr_{0.6}Ca _{0.1}Sr _{0.3}MnO_{3} | 3.64 [70] | 3.8108 | 0.1708 |

Pr_{0.6}Ca _{0.1}Sr _{0.3}Mn_{0.975} Fe_{0.025}O_{3} | 3.53 [70] | 3.9058 | 0.3758 |

Pr_{0.6}Ca _{0.1}Sr _{0.3}Mn_{0.95} Fe_{0.05}O_{3} | 3.7 [70] | 3.845 | 0.145 |

Pr_{0.6}Ca _{0.1}Sr _{0.3}Mn_{0.925} Fe_{0.075}O_{3} | 3.12 [70] | 3.5018 | 0.3818 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shamsah, S.M.I.; Owolabi, T.O. Modeling the Maximum Magnetic Entropy Change of Doped Manganite Using a Grid Search-Based Extreme Learning Machine and Hybrid Gravitational Search-Based Support Vector Regression. *Crystals* **2020**, *10*, 310.
https://doi.org/10.3390/cryst10040310

**AMA Style**

Shamsah SMI, Owolabi TO. Modeling the Maximum Magnetic Entropy Change of Doped Manganite Using a Grid Search-Based Extreme Learning Machine and Hybrid Gravitational Search-Based Support Vector Regression. *Crystals*. 2020; 10(4):310.
https://doi.org/10.3390/cryst10040310

**Chicago/Turabian Style**

Shamsah, Sami M. Ibn, and Taoreed O. Owolabi. 2020. "Modeling the Maximum Magnetic Entropy Change of Doped Manganite Using a Grid Search-Based Extreme Learning Machine and Hybrid Gravitational Search-Based Support Vector Regression" *Crystals* 10, no. 4: 310.
https://doi.org/10.3390/cryst10040310