# 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}

^{*}

## 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

## References

- Hamdi, R.; Tozri, A.; Dhahri, E.; Bessais, L. Magnetocaloric properties and Landau theory of Dy
_{0.5}(Sr_{1−x}Ca_{x})_{0.5}MnO_{3}(0≤ x≤0.3) manganites at cryogenic temperatures. Chem. Phys. Lett.**2017**, 680, 94–100. [Google Scholar] [CrossRef] - Sellami-Jmal, E.; Ezaami, A.; Cheikhrouhou-Koubaa, W.; Cheikhrouhou, A. Investigation on physical properties in lanthanum vacancy of La0.65Ca0.35MnO3 elaborated at high temperature. J. Magn. Magn. Mater.
**2018**, 465, 762–767. [Google Scholar] [CrossRef] - Ayadi, F.; Ammar, S.; Nowak, S.; Cheikhrouhou-Koubaa, W.; Regaieg, Y.; Koubaa, M.; Monnier, J.; Sicard, L. Importance of the synthesis and sintering methods on the properties of manganite ceramics: The example of La 0.7 Ca 0.3 MnO 3. J. Alloys Compd.
**2018**, 759, 52–59. [Google Scholar] [CrossRef] - Das, K.; Banik, S.; Das, I. Large magnetocaloric effect in geometrically frustrated polycrystalline ErMnO 3 compound at cryogenic temperature. Phys. B Condens. Matter
**2018**, 533, 46–49. [Google Scholar] [CrossRef] - Guorui, X..; Wei, H.; Peiqi, C.; Xiaowei, W. Effect of Nd-substitution on the structural, magnetic and magnetocaloric properties of La0.67-xNdxCa0.13Ba0.2MnO3 manganites. J. Magn. Magn. Mater.
**2019**, 491, 133–142. [Google Scholar] - Owolabi, T.O.; Akande, K.O.; Olatunji, S.O.; Aldhafferi, N.; Alqahtani, A. Support Vector Regression Ensemble for Effective Modeling of Magnetic Ordering Temperature of Doped Manganite in Magnetic Refrigeration. J. Low Temp. Phys.
**2019**, 195, 179–201. [Google Scholar] [CrossRef] - Linh, D.C.; Ha, N.T.; Duc, N.H.; Nam, L.H.G.; Bau, L.V.; An, N.M.; Yu, S.-C.; Tran, D.T. Na-doped La 0.7 Ca 0.3 MnO 3 compounds exhibiting a large magnetocaloric effect near room temperature. Phys. B Condens. Matter
**2018**, 532, 155–160. [Google Scholar] [CrossRef] - Das, S.; Roychowdhury, P.; De, S.; Roy, A.; Chatterjee, S.; De, K. Magnetic and electrical transport of the cation-deficient LaMnO 3: Common origin for both Sr-doping and self-doping effects. Phys. B Condens. Matter
**2018**, 544, 17–22. [Google Scholar] [CrossRef] - Yen, P.D.H.; Dung, N.T.; Tran, D.T.; Yu, S.-C. Magnetic properties and magnetocaloric effect of Sr-doped Pr0.7Ca0.3MnO3 compounds. Curr. Appl. Phys.
**2018**, 18, 1280–1288. [Google Scholar] [CrossRef] - Aziz, F.; Chandra, M.; Das, S.; Prajapat, M.; Mavani, K. Structural and metamagnetic transitions in thin films of Ce-doped Pr0.5Ca0.5MnO3 manganites. Thin Solid Films
**2016**, 615, 338–344. [Google Scholar] [CrossRef] - Sudakshina, B.; Arun, B.; Chandrasekhar, K.D.; Yang, H.D.; Vasundhara, M. Structural and magnetic properties of Nd 0.67 Ba 0.33 MnO 3 manganites with partial replacement of Fe and Cu at Mn-site. Phys. B Condens. Matter
**2018**, 539, 14–20. [Google Scholar] [CrossRef] - Asmira, N.; Ibrahim, N.; Mohamed, Z.; Yahya, A. Effect of Cr 3+ substitution at Mn-site on electrical and magnetic properties of charge ordered Bi 0.3 Pr 0.3 Ca 0.4 MnO 3 manganites. Phys. B Condens. Matter
**2018**, 544, 34–46. [Google Scholar] [CrossRef] - Kataria, B.; Solanki, P.; Pandya, D.; Solanki, P.; Shah, N. Role of antimony in the charge transport mechanisms for La 0.67 Ca 0.33 Mn 1–x Sb x O 3 manganites. Phys. B Condens. Matter
**2018**, 541, 43–49. [Google Scholar] [CrossRef] - Poojary, T.; Babu, P.; Sanil, T.; Daivajna, M.D. Effect of gadolinium dopant on structural, magneto-transport, magnetic and thermo-power of Pr 0.8 Sr 0.2 MnO 3. Solid State Commun.
**2018**, 275, 35–42. [Google Scholar] [CrossRef] - Makni-Chakroun, J.; M’Nassri, R.; Cheikhrouhou-Koubaa, W.; Koubaa, M.; Chniba-Boudjada, N.; Cheikhrouhou, A. Effect of A-site deficiency on investigation of structural, magnetic and magnetocaloric behaviors for (LaSr)-lacunar manganites. Chem. Phys. Lett.
**2018**, 707, 61–70. [Google Scholar] [CrossRef] - Akça, G.; Çetin, S.K.; Güneş, M.; Ekicibil, A. Magnetocaloric properties of (La1−xPrx)0.85K0.15MnO3 (x=0.0, 0.1, 0.3 and 0.5) perovskite manganites. Ceram. Int.
**2016**, 3, 3–10. [Google Scholar] [CrossRef] - Selmi, A.; M’Nassri, R.; Cheikhrouhou-Koubaa, W.; Boudjada, N.C.; Cheikhrouhou, A. Effects of partial Mn-substitution on magnetic and magnetocaloric properties in Pr0.7Ca0.3Mn0.95X0.05O3 (Cr, Ni, Co and Fe) manganites. J. Alloys Compd.
**2015**, 619, 627–633. [Google Scholar] [CrossRef] - Vadnala, S.; Asthana, S. Magnetocaloric effect and critical field analysis in Eu substituted La0.7-xEuxSr0.3MnO3(x = 0.0, 0.1, 0.2, 0.3) manganites. J. Magn. Magn. Mater.
**2018**, 446, 68–79. [Google Scholar] [CrossRef] - Bocarsly, J.D.; Levin, E.E.; Garcia, C.A.C.; Schwennicke, K.; Wilson, S.D.; Seshadri, R. A Simple Computational Proxy for Screening Magnetocaloric Compounds. Chem. Mater.
**2017**, 29, 1613–1622. [Google Scholar] [CrossRef] [Green Version] - Zarkevich, N.; Johnson, D.D. Reliable thermodynamic estimators for screening caloric materials. J. Alloys Compd.
**2019**, 802, 712–722. [Google Scholar] [CrossRef] [Green Version] - Owolabi, T.; Akande, K.O.; Olatunji, S.O.; Alqahtani, A.; Aldhafferi, N. Estimation of Curie temperature of manganite-based materials for magnetic refrigeration application using hybrid gravitational based support vector regression. AIP Adv.
**2016**, 6, 105009. [Google Scholar] [CrossRef] [Green Version] - Owolabi, T.; Oloore, L.E.; Akande, K.O.; Olatunji, S.O. Modeling of magnetic cooling power of manganite-based materials using computational intelligence approach. Neural Comput. Appl.
**2017**, 31, 1291–1298. [Google Scholar] [CrossRef] - Owolabi, T.; Akande, K.O.; Olatunji, S.O.; Alqahtani, A.; Aldhafferi, N. Incorporation of GSA in SBLLM-based neural network for enhanced estimation of magnetic ordering temperature of manganite. J. Intell. Fuzzy Syst.
**2017**, 33, 1225–1233. [Google Scholar] [CrossRef] - Owolabi, T.; Akande, K.O.; Olatunji, S.O.; Alqahtani, A.; Aldhafferid, N. Modeling of Curie temperature of manganite for magnetic refrigeration application using manual search and hybrid gravitational-based support vector regression. Soft Comput.
**2017**, 22, 3023–3032. [Google Scholar] [CrossRef] - Owolabi, T.; Akande, K.O.; Olatunji, S.O.; Aldhafferi, N.; Alqahtani, A. Ensemble-Based Support Vector Regression with Gravitational Search Algorithm Optimization for Estimating Magnetic Relative Cooling Power of Manganite Refrigerant in Magnetic Refrigeration Application. J. Supercond. Nov. Magn.
**2018**, 32, 2107–2118. [Google Scholar] [CrossRef] - Vapnik, V.N. Statistical Learning Theory; Wiley-Interscience: New York, NY, USA, 1998. [Google Scholar]
- Oloore, L.; Owolabi, T.; Fayose, S.; Adegoke, M.; Akande, K.; Olatunji, S. Modeling of semiconductors refractive indices using hybrid chemometric model. Model. Meas. Control. A
**2018**, 91, 95–103. [Google Scholar] [CrossRef] - Owolabi, T.; Akande, K.O.; Olatunji, S.O. Estimation of average surface energies of transition metal nitrides using computational intelligence technique. Soft Comput.
**2016**, 21, 6175–6182. [Google Scholar] [CrossRef] - Zhang, H.-R.; Zhang, Y.; Dai, D.-B.; Cao, M.; Shen, W.-F. Modelling and optimization of the superconducting transition temperature. Mater. Des.
**2016**, 92, 371–377. [Google Scholar] [CrossRef] - Cai, C.; Xiao, T.; Tang, J.; Huang, S. Analysis of process parameters in the laser deposition of YBa2Cu3O7 superconducting films by using SVR. Phys. C Supercond.
**2013**, 493, 100–103. [Google Scholar] [CrossRef] - Motamedi, S.; Shamshirband, S.; Hashim, R.; Petković, D.; Roy, C. RETRACTED: Estimating unconfined compressive strength of cockle shell–cement–sand mixtures using soft computing methodologies. Eng. Struct.
**2015**, 98, 49–58. [Google Scholar] [CrossRef] - Behnood, A.; Verian, K.P.; Modiri-Gharehveran, M. Evaluation of the splitting tensile strength in plain and steel fiber-reinforced concrete based on the compressive strength. Constr. Build. Mater.
**2015**, 98, 519–529. [Google Scholar] [CrossRef] - Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S. GSA: A Gravitational Search Algorithm. Inf. Sci.
**2009**, 179, 2232–2248. [Google Scholar] [CrossRef] - Huang, G.-B.; Zhu, Q.-Y.; Siew, C.-K. Extreme learning machine: Theory and applications. Neurocomputing
**2006**, 70, 489–501. [Google Scholar] [CrossRef] - Yan, C.; Qi, J.; Ma, J.; Tang, H.; Zhang, T.; Li, H. Determination of carbon and sulfur content in coal by laser induced breakdown spectroscopy combined with kernel-based extreme learning machine. Chemom. Intell. Lab. Syst.
**2017**, 167, 226–231. [Google Scholar] [CrossRef] - Owolabi, T.; Gondal, M. Development of hybrid extreme learning machine based chemo-metrics for precise quantitative analysis of LIBS spectra using internal reference pre-processing method. Anal. Chim. Acta
**2018**, 1030, 33–41. [Google Scholar] [CrossRef] - Wang, Y.G.; Cao, F.; Yuan, Y. A study on effectiveness of extreme learning machine. Neurocomputing
**2011**, 74, 2483–2490. [Google Scholar] [CrossRef] [Green Version] - Owolabi, T. Modeling the magnetocaloric effect of manganite using hybrid genetic and support vector regression algorithms. Phys. Lett. A
**2019**, 383, 1782–1790. [Google Scholar] [CrossRef] - Boser, B.E.; Guyon, I.M.; Vapnik, V.N. A training algorithm for optimal margin classifiers. In Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory, Pittsburgh, PA, USA, 27–29 July 1992; pp. 144–152. [Google Scholar]
- Basak, D.; Pal, S.; Patranabis, D.C. Support Vector Regression. Neural Inf. Process. Lett. Rev.
**2007**, 11, 203–224. [Google Scholar] - Peng, J.; Li, L. Support vector regression in sum space for multivariate calibration. Chemom. Intell. Lab. Syst.
**2014**, 130, 14–19. [Google Scholar] [CrossRef] - Aldhafferi, N.; Owolabi, T.O.; Akande, K.O.; Olatunji, S.O.; Alqahtani, A. Support vector regression approach of modeling the superconducting transition temperature of borocarbide-based superconductors. ARPN J. Eng. Appl. Sci.
**2018**, 13, 1900–1905. [Google Scholar] - Owolabi, T.; Akande, K.O.; Olatunji, S.O. Estimation of surface energies of hexagonal close packed metals using computational intelligence technique. Appl. Soft Comput.
**2015**, 31, 360–368. [Google Scholar] [CrossRef] - Owolabi, T.; Suleiman, M.A.; Adeyemo, H.; Akande, K.O.; Alhiyafi, J.; Olatunji, S.O. Estimation of minimum ignition energy of explosive chemicals using gravitational search algorithm based support vector regression. J. Loss Prev. Process. Ind.
**2019**, 57, 156–163. [Google Scholar] [CrossRef] - Adeyemo, H.B.; Owolabi, T.O.; Suleiman, M.A.; Akande, A.J.; Fayose, S.; Olatuji, S.O. Heliyon Hybrid chemometric approach for estimating the heat of detonation of aromatic energetic compounds. Heliyon
**2019**, 5, 1–8. - Owolabi, T. Determination of the Velocity of Detonation of Primary Explosives Using Genetically Optimized Support Vector Regression. Propellants Explos. Pyrotech.
**2019**, 1–12. [Google Scholar] [CrossRef] - Yildiz, B.S.; Lekesiz, H.; Yildiz, A.R.; Yıldız, B.S. Structural design of vehicle components using gravitational search and charged system search algorithms. Mater. Test.
**2016**, 58, 79–81. [Google Scholar] [CrossRef] - Sabri, N.M.; Puteh, M.; Mahmood, M.R. A review of gravitational search algorithm. Int. J. Adv. Soft Comput. Appl.
**2013**, 5. [Google Scholar] - Owolabi, T.; Gondal, M.A. A hybrid intelligent scheme for estimating band gap of doped titanium dioxide semiconductor using crystal lattice distortion. Comput. Mater. Sci.
**2017**, 137, 249–256. [Google Scholar] [CrossRef] - Huang, G.-B.; Chen, L. Convex incremental extreme learning machine. Neurocomputing
**2007**, 70, 3056–3062. [Google Scholar] [CrossRef] - Owolabi, T.; Gondal, M.A. Quantitative analysis of LIBS spectra using hybrid chemometric models through fusion of extreme learning machines and support vector regression. J. Intell. Fuzzy Syst.
**2018**, 1–10. [Google Scholar] [CrossRef] - ElGhoul, A.; Krichene, A.; Boudjada, N.C.; Boujelben, W. Rare earth effect on structural, magnetic and magnetocaloric properties of La0.75Ln0.05Sr0.2MnO3 manganites. Ceram. Int.
**2018**, 44, 12723–12730. [Google Scholar] [CrossRef] - Koubaa, M.; Regaieg, Y.; Koubaa, W.C.; Cheikhrouhou, A.; Ammar-Merah, S.; Herbst, F. Magnetic and magnetocaloric properties of lanthanum manganites with monovalent elements doping at A-site. J. Magn. Magn. Mater.
**2011**, 323, 252–257. [Google Scholar] [CrossRef] - Nedelko, N.; Lewińska, S.; Pashchenko, A.; Radelytskyi, I.; Diduszko, R.; Zubov, É.E.; Lisowski, W.; Sobczak, J.; Dyakonov, K.; Ślawska-Waniewska, A.; et al. Magnetic properties and magnetocaloric effect in La0.7Sr0.3−xBixMnO3 manganites. J. Alloys Compd.
**2015**, 640, 433–439. [Google Scholar] [CrossRef] - Arun, B.; Athira, M.; Akshay, V.R.; Sudakshina, B.; Mutta, G.R.; Vasundhara, M. Investigation on the structural, magnetic and magnetocaloric properties of nanocrystalline Pr-deficient Pr1−xSrxMnO3−δmanganites. J. Magn. Magn. Mater.
**2018**, 448, 322–331. [Google Scholar] [CrossRef] - Saleh, J.A.; Sarsari, I.A.; Kameli, P.; Salamati, H. Influence of Al-doping on the structural, magnetic, and electrical properties of La0.8Ba0.2Mn1−xAlxO3(0 ≤ x ≤ 0.25) manganites. J. Magn. Magn. Mater.
**2018**, 465, 339–347. [Google Scholar] [CrossRef] - Choudhary, Y.; Mangavati, S.; Patil, S.; Rao, A.; Nagaraja, B.; Thomas, R.; Okram, G.; Kini, S.G. Effect of rare-earth substitution at La-site on structural, electrical and thermoelectric properties of La0.7−xRExSr0.3MnO3 compounds (x = 0, 0.2, 0.3; RE = Eu, Gd, Y). J. Magn. Magn. Mater.
**2018**, 451, 110–120. [Google Scholar] [CrossRef] - Liu, Z.; Lin, W.; Zhou, K.; Yan, J. Effect of Cu doping on the structural, magnetic and magnetocaloric properties of La0.7Sr0.25Na0.05Mn1−xCuxO3 manganites. Ceram. Int.
**2018**, 44, 2797–2802. [Google Scholar] [CrossRef] - Ben Khlifa, H.; Othmani, S.; Chaaba, I.; Tarhouni, S.; Cheikhrouhou-Koubaa, W.; Koubaa, M.; Cheikhrouhou, A.; Hlil, E. Effect of K-doping on the structural, magnetic and magnetocaloric properties of Pr0.8Na0.2–K MnO3 (0 ≤ x ≤ 0.15) manganites. J. Alloys Compd.
**2016**, 680, 388–396. [Google Scholar] [CrossRef] - Iqbal, M.; Khan, M.N.; Khan, A.A. Structural, magnetic, magnetocaloric and critical behavior studies in the vicinity of the paramagnetic to ferromagnetic phase transition temperature in LaMnO3+δcompound. J. Magn. Magn. Mater.
**2018**, 465, 670–677. [Google Scholar] [CrossRef] - Snini, K.; Ben Jemaa, F.; Mohamed, E.; Hlil, E. Structural, magnetic and magnetocaloric investigations in Pr0.67Ba0.22Sr0.11Mn1-xFexO3 (0 ≤ x ≤ 0.15) manganite oxide. J. Alloys Compd.
**2018**, 739, 948–954. [Google Scholar] [CrossRef] - Tejaswini, B.P.; Daivajna, M.D. Thrupthi Structural, electrical, magnetic and thermal properties of Pr0.8-xDyxSr0.2MnO3 with (x = 0, 0.2 and 0.25). J. Alloys Compd.
**2018**, 741, 97–105. [Google Scholar] [CrossRef] - Hira, U.; Sher, F. Structural, magnetic and high-temperature thermoelectric properties of La0.4Bi0.4Ca0.2Mn1−Co O3 (0 ≤ x ≤ 0.3) perovskites. J. Magn. Magn. Mater.
**2018**, 452, 64–72. [Google Scholar] [CrossRef] - Elyana, E.; Mohamed, Z.; Kamil, S.; Supardan, S.; Chen, S.; Yahya, A. Revival of ferromagnetic behavior in charge-ordered Pr0.75Na0.25MnO3 manganite by ruthenium doping at Mn site and its MR effect. J. Solid State Chem.
**2018**, 258, 191–200. [Google Scholar] [CrossRef] - Ghodhbane, S.; Tka, E.; Dhahri, J.; Hlil, E. A large magnetic entropy change near room temperature in La0.8Ba0.1Ca0.1Mn0.97Fe0.03O3 perovskite. J. Alloys Compd.
**2014**, 600, 172–177. [Google Scholar] [CrossRef] - Selmi, A.; M’Nassri, R.; Cheikhrouhou-Koubaa, W.; Boudjada, N.C.; Cheikhrouhou, A. Influence of transition metal doping (Fe, Co, Ni and Cr) on magnetic and magnetocaloric properties of Pr0.7Ca0.3MnO3 manganites. Ceram. Int.
**2015**, 41, 10177–10184. [Google Scholar] [CrossRef] - Wang, Z.; Jiang, J. Magnetic entropy change in perovskite manganites La0.7A0.3MnO3 La0.7A0.3Mn0.9Cr0.1O3 (A = Sr, Ba, Pb) and Banerjee criteria on phase transition. Solid State Sci.
**2013**, 18, 36–41. [Google Scholar] [CrossRef] - Selmi, A.; M’Nassri, R.; Cheikhrouhou-Koubaa, W.; Boudjada, N.C.; Cheikhrouhou, A. The effect of Co doping on the magnetic and magnetocaloric properties of Pr0.7Ca0.3Mn1−xCoxO3 manganites. Ceram. Int.
**2015**, 41, 7723–7728. [Google Scholar] [CrossRef] - Oumezzine, E.; Hcini, S.; Hlil, E.-K.; Dhahri, E.; Oumezzine, M. Effect of Ni-doping on structural, magnetic and magnetocaloric properties of La 0.6 Pr 0.1 Ba 0.3 Mn 1− x Ni x O 3 nanocrystalline manganites synthesized by Pechini sol–gel method. J. Alloys Compd.
**2014**, 615, 553–560. [Google Scholar] [CrossRef] - Mahjoub, S.; Baazaoui, M.; M’Nassri, R.; Rahmouni, H.; Boudjada, N.C.; Oumezzine, M. Effect of iron substitution on the structural, magnetic and magnetocaloric properties of Pr0.6Ca0.1Sr0.3Mn1−xFexO3 (0 ≤ x ≤ 0.075) manganites. J. Alloys Compd.
**2014**, 608, 191–196. [Google Scholar] [CrossRef] - Mleiki, A.; Othmani, S.; Cheikhrouhou-Koubaa, W.; Koubaa, A.; Cheikhrouhou, A.; Hlil, E.K. Effect of praseodymium doping on the structural, magnetic and magnetocaloric properties of Sm0.55Sr0.45MnO3 manganite. J. Alloys Compd.
**2015**, 645, 559–565. [Google Scholar] [CrossRef] - Kossi, S.E.; Ghodhbane, S.; Dhahri, J.; Hlil, E.K. The impact of disorder on magnetocaloric properties in Ti-doped manganites of La0.7Sr0.25Na0.05Mn(1-x)TixO3 (0≤x ≤0.2). J. Magn. Magn. Mater.
**2015**, 395, 134–142. [Google Scholar] [CrossRef] - Taboada-Moreno, C.; Jesús, F.S.-D.; Pedro-García, F.; Cortés-Escobedo, C.; Betancourt-Cantera, J.; Ramírez-Cardona, M.; Bolarín-Miró, A. Large magnetocaloric effect near to room temperature in Sr doped La0.7Ca0.3MnO3. J. Magn. Magn. Mater.
**2020**, 496, 165887. [Google Scholar] [CrossRef] - Xiao, G.; He, W.; Yang, T.; Huang, G.; Wang, T.; Huang, J. Effect of Co-doping on structural, magnetic and magnetocaloric properties of La 0.67 Ca 0.13 Ba 0.2 Mn 1-x Co x O 3 (x = 0, 0.02, 0.04, 0.06, 0.08, 0.1) manganites. Curr. Appl. Phys.
**2019**, 19, 424–435. [Google Scholar] [CrossRef] - Bouzaiene, E.; Dhahri, A.; Dhahri, J.; Hlil, E.; Bajahzar, A. Effect of A-site-substitution on structural, magnetic and magnetocaloric properties in La0.7Sr0.3 Mn0.9Cu0.1O3 manganite. J. Magn. Magn. Mater.
**2019**, 491, 165540. [Google Scholar] [CrossRef] - Hassayoun, O.; Baazaoui, M.; Laouyenne, M.; Hosni, F.; Hlil, E.; Oumezzine, M.; Farah, K. Magnetocaloric effect and electron paramagnetic resonance studies of the transition from ferromagnetic to paramagnetic in La0.8Na0.2Mn1-xNixO3 (0≤x≤0.06). J. Phys. Chem. Solids
**2019**, 135, 109058. [Google Scholar] [CrossRef] - Mahjoub, S.; M’Nassri, R.; Baazaoui, M.; Hlil, E.; Oumezzine, M. Tuning magnetic and magnetocaloric properties around room temperature via chromium substitution in La0.65Nd0.05Ba0.3MnO3 system. J. Magn. Magn. Mater.
**2019**, 481, 29–38. [Google Scholar] [CrossRef] - Pecharsky, V.K.; Gschneidner, K.A. Magnetocaloric effect from indirect measurements: Magnetization and heat capacity. J. Appl. Phys.
**1999**, 565. [Google Scholar] [CrossRef]

**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