Computational Intelligence Approach for Estimating Superconducting Transition Temperature of Disordered MgB2 Superconductors Using Room Temperature Resistivity

Doping and fabrication conditions bring about disorder in MgB 2 superconductor and further influence its room temperature resistivity as well as its superconducting transition temperature (T C ). Existence of a model that directly estimates T C of any doped MgB 2 superconductor from the room temperature resistivity would have immense significance since room temperature resistivity is easily measured using conventional resistivity measuring instrument and the experimental measurement of T C wastes valuable resources and is confined to low temperature regime.This work develops amodel, superconducting transition temperature estimator (STTE), that directly estimates T C of disordered MgB 2 superconductors using room temperature resistivity as input to the model. STTE was developed through training and testing support vector regression (SVR) with ten experimental values of room temperature resistivity and their corresponding T C using the best performance parameters obtained through test-set cross validation optimization technique. The developed STTE was used to estimate T C of different disordered MgB 2 superconductors and the obtained results show excellent agreement with the reported experimental data. STTE can therefore be incorporated into resistivity measuring instruments for quick and direct estimation of T C of disordered MgB 2 superconductors with high degree of accuracy.


Introduction
Superconductor is a material that allows perpetual flow of current as a result of disappearance of its electrical resistivity when it is cooled below a particular temperature called superconducting transition temperature ( ). Several practical applications of superconductors such as magnetic resonance imaging in hospitals, magnetic levitation train, particle accelerators, superconducting quantum interference devices, magnetoencephalography, and filters for mobile communication and their future applications (which include transmission of electricity, fast computing, and high temperature superconducting generator) depend mainly on the value of . Phenomenon of superconductivity was first observed in mercury in 1911 when electrical resistance of pure mercury went to zero around 4 K [1]. Understanding of this behavior was not clearly known until 1957 when three physicists propounded BCS theory which governs the emergence of superconductivity [2]. This theory attributes the emergence of superconductivity in materials to the ability of electrons to pair (formation of cooper pairs) which pave ways for the free movement of electrons in a coordinated manner. Collision of electrons with lattice, other electrons, and defects among others bring about resistivity in materials and this is circumvented in superconductors due to the formation of cooper pairs. Efforts have been made to realize room temperature superconductor and to raise the values of of known superconductors. However, MgB 2 holds a promising future since its can be altered through doping and fabrication conditions [3][4][5].

Applied Computational Intelligence and Soft Computing
The awareness of superconductivity in magnesium diboride (MgB 2 ) in 2001 marked a new advancement that diversifies the applications of superconductors [6]. MgB 2 is a two-band superconductor with several practical applications as a result of its unique properties such as transparency of the grain boundaries which permits flow of current, lower anisotropy, and large coherence length. Its economic affordability as well as its unique properties offers it a special place in practical applications despite its low as compared with high temperature superconductors. The techniques that are usually employed for improving the superconducting properties of MgB 2 include introduction of disorder through chemical doping and several thermomechanical processing techniques which alter its room temperature resistivity [3,5,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In order to improve the superconducting properties of this material (in particular ), we develop STTE for accurate, quick, and direct estimation of of disordered MgB 2 superconductors as an alternative to conventional method which is experimentally intensive. This developed model allows the potential of several dopants to be assessed within short period of time so as to ascertain the specific dopant that improves the superconducting property of MgB 2 superconductor.
The normal state strong electron-phonon scattering which gives rise to the large room temperature resistivity in intermetallic superconductors is associated with the observed high [25]. Low resistivity of MgB 2 at room temperature and temperature just above its could be attributed to its two-band nature characterized with different strength of electron-phonon coupling [25]. Its superconductivity comes from the large coupling in 2D band and MgB 2 demonstrates anisotropy in the normal state resistivity. The proposed SVR based model has been reported as a novel tool of estimating superconducting properties of doped MgB 2 superconductor using few descriptive features [26]. The choice of SVR in building STTE is due to its many unique features which include sound mathematical foundation, nonconvergence to local minima, accurate generalization, and predictive ability when trained with few descriptive features [27].
SVR tackles real life problems using the principles of artificial intelligence in the field of machine learning. It acquires pattern or relation that exists between the target ( ) and descriptor (room temperature resistivity) and adopts the acquired pattern for future estimation of the unknown target with the aid of the descriptors. Its excellent predictive ability is made which is used in tackling several problems in medical field [28], material science [26,[29][30][31][32], and oil and gas industries [33,34], to name but a few. The excellent predictive and generalization ability of SVR in solving numerous problems coupled with the need to have accurate, direct, and effective way of estimating the effects of disorder on of MgB 2 superconductor serves as motivation for carrying out this research work.

Description of the Proposed Model
SVR follows a learning algorithm formulated from support vector machines which was originally proposed by Vapnik for classification purposes [35]. It uses loss function that controls the smoothness of the response of SVR and the number of support vectors which ultimately affect the generalization capacity as well as the complexity of the model. The loss function represents the maximum error (deviation of the target vectors tolerated by the model). Equation (1) represents a linear decision function in which ⟨ , ⟩ indicates a dot product in space : where ∈ (input space) and ∈ . Minimization of Euclidean norm ‖ ‖ 2 is necessary since flatness of the defined decision equation is desired. The optimization problem for the regression is presented in The assumption entailed in the optimization problem presented in (2) is the existence of a function that controls the error in all training pairs in such a way that the error is less than . In order to build up a robust system that caters for external constraints, slack variables ( and ) are introduced in the optimization problem and presented in [27] The regularization factor introduced in (3) trades off the frequency of error and the complexity of decision rule by altering the size of slack variables. The optimization problem is solved by using Lagrangian multipliers ( , , , and ) to transform the problem to dual space representation. The constraint equations are multiplied by the multipliers and the result is taken away from ‖ ‖ 2 (objective function). The resulting Lagrangian is illustrated in Applied Computational Intelligence and Soft Computing In order to obtain the solution of the optimization problem, the saddle points of the Lagrangian function are obtained by equating the partial derivative of Lagrangian with respect to , , , and to zero which gives rise to (5) as described in [29]: By putting (5) in (4), we have The estimation problem presented in (1) is finally represented by (8) after incorporating and obtained from (6): Kernel function ( , ) represents a nonlinear mapping function that maps nonlinear regression problem to high dimensional feature space where linear regression is performed. The choice of kernel function depends on the nature of the problem and could be linear, polynomial, Gaussian, or sigmoid kernel function. During the course of optimizing the SVR model, SVR variables such as the regularization factor ( ), epsilon ( ), hyperparameter ( ), and kernel option were defined and adjusted until the model attained optimum performance.
The training stage of the proposed model entails acquisition of pattern that exists between the descriptor and the target which could further be generalized by the model so as to accurately estimate unknown target with the aid of the descriptor. High correlation coefficient, low root mean square error, and low mean absolute error signify the accuracy and efficiency of the model. In the case of STTE developed in this research work, the SVR model was taken through training and testing stages and then used to estimate of several MgB 2 superconductors.
where and represent error (difference between the experimental and estimated ) and number of data points, respectively. (exp) and (est) , respectively, represent the experimental and the estimated , while (exp) and (est) represent their mean value, respectively.
In the case of the developed model, high correlation coefficient of 100%, low root mean square error of 1.29, and low absolute error of 0.279 were obtained during the testing phase of the model.

Description of the Dataset.
The dataset that was employed in modeling STTE comprises a total number of ten experimental values of room temperature resistivity and the corresponding of disordered MgB 2 superconductors. The adopted dataset was drawn from the literatures [36][37][38] and presented in Table 1.
Statistical analysis was carried out on the dataset and the results of the analysis are presented in Table 2.
Deduction from the values of the mean, median, standard deviation, minimum, and maximum gives insight into the dataset as well as the level of its discrepancies. The correlation coefficient between and room temperature resistivity was calculated and found negatively correlated. Meanwhile,

Computational Methodology.
The modeling and simulations involved in this work were conducted within the computing environment of MATLAB. The dataset presented in Table 1 was randomized prior to the commencement of the modeling. The randomization was carried out purposely to enhance computational efficiency and to ensure consistency. The dataset was further divided into training and testing set in the ratio of 8 to 2. The best values of SVR hyperparameters (i.e., regularization factor, kernel option, lambda, and epsilon) were used for generating support vectors through training dataset and the testing set of data was used to validate the model. The developed model (STTE) was then used to estimate the effect of starch doping, nano-silicon carbide, and preparation condition on of MgB 2 superconductors and the obtained values are compared with the experimental results.

Optimization Strategy.
The optimization strategy employed during the development of STTE was the testset cross validation technique. In this case, the values of correlation coefficient, root mean square error, and mean absolute error were monitored in every run of the training and testing dataset for a group of parameters (regularization factor, kernel option, epsilon, and hyperparameter). In the course of searching through all the possible values of the parameters in a given range, the best performance measures were identified with the corresponding values of parameters for the fixed set of features. The whole process involves searching for the initial kernel option from the pool of the available kernel options and the identification of the best values of the parameter and so as to identify the corresponding performance measures. The best values of the hyperparameters were further used to train SVR model. The values of hyperparameters that ensure optimum performance of the model are presented in Table 3.

Development of the Model (STTE).
The STTE was developed using randomly selected ten experimental values of room temperature resistivity with their corresponding . Excellent correlation coefficient of 100% was obtained in the  course of testing the developed model prior to its usage. Low values of 1.29 and 0.279 for the root means square and mean absolute errors were, respectively, recorded. The pairwise comparison of the experimentally reported and estimated values is depicted in Figure 1.
The actual (experimental value) and estimated values of obtained during the development stage of STTE are presented in Table 4. The percentage errors were also calculated in order to observe the percentage deviation of the estimated values.

Application of the Developed Model (STTE) in Estimating
Disordered MgB 2 Superconductors. The developed STTE was used to estimate of different disordered MgB 2 superconductors and the obtained values are compared with the experimental data. The utilization of the developed model involves feeding the model with the value of the room temperature resistivity and the model employs this value to generate the corresponding .

Effect of Starch Doping on
of MgB 2 Superconductor Using the Developed Model. Addition of starch to MgB 2 superconductor alters its transition temperature as reported in the literature [4]. In order to assess the estimation strength of the developed model, the model was used to estimate the effect of starch doping on of MgB 2 superconductor and the results are compared with the experimental values. Figure 3 shows the reduction in the transition temperature of MgB 2 superconductor when 1% concentration of starch was added.  Figure 2).

Effect of Nano-Silicon Carbide and Graphene on
of MgB 2 Superconductor Using the Developed Model. The effect of nano-silicon carbide (nano-SiC) and graphene on of MgB 2 superconductor is also investigated and presented in Figure 3. The results of the developed model also agree well with the experimental values [39]. Introduction of graphene to MgB 2 already doped with 5% nano-SiC has little effect on as depicted in Figure 3.  superconductor was estimated and presented in Figure 4. The rate of heating has significant effect on reducing of MgB 2 superconductor, especially, when the rate was increased to 30 K/min as depicted in the graph. The results of the developed model show good agreement with the reported experimental values [40].

Conclusion and Recommendation
SVR was used to develop STTE through training and testing using test-set cross validation optimization technique with the aid of ten experimental values of room temperature resistivity and their corresponding . The developed model (STTE) was then used to estimate the effect of starch doping, nano-silicon carbide, graphene, and preparation condition