Modeling of dielectric behavior of polymers nanocomposites using adaptive neuro-fuzzy inference system (ANFIS)

The research presents a theoretical study on modeling of dielectric properties of polymer nanocomposites based on adaptive neuro-fuzzy inference system (ANFIS). The research contributes a better utilizing of ANFIS model in the prediction of dielectric behavior of polymers nanocomposites. In this respect, three different samples are trained (Mg1−xCuxO/PMMA, PPy-DBSA-Y2O3 and PVC/PEMA with [Zn(CF3SO3)2]). Inputs are obtained from earlier experimental studies. ANFIS Takagi–Sugeno type is trained. The model is applied based on weighted average as a defuzzification method. The optimal network structures, which produce the most acceptable results, are implemented in MATLAB. Six ANFIS networks are trained to simulate and predict dielectric permittivity and dielectric loss in terms of nanocomposite weight % (0–0.2%, 0–8% and 10–30% for each sample, respectively) and frequency (10−2–103 kHz). ANFIS simulation results are very close to their targets. Predictions of dielectric properties at nanocomposite weights % that are measured experimentally as a testing step and predictions of other values that are not implicated in the experimental data extent are achieved. Also, predictions of individual points are processed using ANFIS rule viewer. It is found that ANFIS predictions provide excellent results. Three-dimensional illustrations that represent the mapping from frequency and nanocomposite weight% to the dielectric permittivity and dielectric loss are obtained using MATLAB surface viewer. Histogram error plot is obtained to indicate the degree of noisy. Mean error, mean squared error, root-mean-squared error and standard division error are calculated. Their values improve the efficiency of the modeling process. A key goal of this paper is to develop a mechanism to predict the dielectric properties of polymers nanocomposites. In accordance with the modeling results, the ANFIS technique achieves the purpose. It can also form a great link between practical and theoretical domains.

The major aim of the research is to find an accurate and successful way to predict the dielectric behavior of polymers nanocomposites. To achieve the aim, the following hypotheses would apply. The experimental data would utilize as inputs in ANFIS model. After completing the simulation process, it is expected to obtain high-accuracy results and rules that connect inputs and outputs. That will be good in predicting values that have not been experimentally measured. So the contribution of this paper is to use ANFIS in modeling dielectric properties of polymers nanocomposites.
The research is divided into four sections. The first section provides background information and introduces the main idea. The second section reviews basic information about ANFIS model and answers the question how ANFIS model is processed. The third section demonstrates the results, discusses them and compares the ANFIS results with the experimental data. Finally, the fourth section concludes the main targets achieved in the paper.

ANFIS model: architectures, approaches and learning algorithm
According to fuzzy logic, each member can be a part of cluster with varying degrees of membership. That is opposed to conventional methods where only 0 or 1 can be used to signify the membership. Fuzzy logic depends on conditional statement called IF-THEN rule. These rules improve decision-making capacity in circumstances of uncertainty. By utilizing a decision-making mechanism, a fuzzy inference model can describe uncertainty environment in the form of rules. Thus, it is capable of solving a variety of problems. By analyzing the artificial neural networks technique, it is found that it provides significant benefits, including (1) sample-based learning, (2) classification, correlation and pattern recognition capabilities, (3) dealing with incomplete data and (4) machine learning. The term "neuro-fuzzy" refers to a hybrid artificial intelligence (AI) technique that combines fuzzy logic (FL) with artificial neural networks (ANNs). Among the neuro-fuzzy models, the adaptive neuro-fuzzy inference system (ANFIS) is most used currently. It was proposed in 1992 by J.S Roger [81,82]. ANFIS adjusts its parameters in accordance with training data by using the hybrid learning algorithm. ANFIS acquired the ability of artificial neural networks to learn combined with the ability of fuzzy logic to make decisions. As in artificial neural networks, ANFIS uses learning based on the training data. There are two types of fuzzy interference systems (FIS). Mamdani type creates a control system by using a group of linguistic rules. These rules are obtained from human experience. It has a group of fuzzy output for each rule. Since Mamdani systems rules are easy to understand, they are suited to expert system applications such as medical diagnostics. Takagi-Sugeno is the second type of fuzzy inference systems. It depends on output membership functions (linear or constant) for the inputs. Takagi-Sugeno defuzzification method is more efficient because it uses a weighted average method (Wtaver) to compute the crisp output instead of using a centroid technique [83,84].
The main idea of ANFIS is based on combined configurations, which gains the neural network learning ability and exhibits the benefits of Takagi-Sugeno type. ANFIS architecture is made up of a number of nodes that are arranged in layers to perform unique tasks and act as a rule base for the fuzzy system. See Fig. 1 [85,86]. The premise and the consequence are the two main constituents of ANFIS. During training, ANFIS uses the input-output data pairs. Then, it uses IF-THEN fuzzy rules which are based on the membership functions to identify relations between the premise and the consequence. There are five layers in the architecture of ANFIS model, each of which performs a special job. The fuzzification layer is the first layer. It generates fuzzy clustering from the input data using membership functions. Premise parameters are parameters that are based on the membership function. Equations 1 and 2 have been used to compute the degree of membership of each function. The rule layer is the second layer. The rules of firing strengths (wi) are determined from membership values calculated in the fuzzification layer. Multiplying the parameters of membership yields the firing strengths which can be seen in Eq. 3. The normalization layer is the third layer. It determines the adjusted firing strengths of each rule. As shown in Eq. 4, the adjusted value is the ratio of the ith rules to the sum of total firing strengths. The defuzzification layer is the fourth layer where the weighted rules are calculated for each node. See Eq. 5. The summation layer is the last one. Figure 2 depicts the ANFIS five-layer structure.
where a, b and c are the premise parameters. μ(x) and μ(y) are the membership degrees obtained from this layer.
where w i is the output of the normalization layer and {p i , q i , r i } is the consequence parameter set. The steps of ANFIS could be summarized as follows: The premise part where the input data are compared with the membership functions to get the membership values (fuzzification step). Then determine the firing strength (weight) of each rule and generate consequences using the firing strengths. Finally, aggregate consequences to get the crisp output (defuzzification step) [87][88][89][90]. See the schematic diagram which represents the processing steps of ANFIS model (Fig. 3).

Results and discussion
Modeling has great contributions in the practical domain, as it is useful in obtaining relations and rules that link the inputs and outputs of the studied phenomenon, as well as in predicting values that have not been measured in practice. Also, modeling is useful in predicting values out of the permissible range of practical devices, which avoids the defect of lack abilities of laboratories. Depending on the prediction results, the experiment is carried out using less number of samples with high precision, which saves time and effort. Despite the important role of modeling and the existence of several researches related to this issue, there is a gap in the literature and an unexplored area that has not been sufficiently addressed. The new contribution is the utilization of modeling to predict the dielectric properties of polymers nanocomposites, especially ANFIS model with hybrid benefits: high accuracy, learning ability, stability in results and ease of use. Therefore, the current paper has been dedicated to study the mentioned point in details.
To model, simulate and predict the dielectric coefficients of three different samples of polymers nanocomposites based on ANFIS, specific steps are followed. Experimental data samples characterized by their importance in several areas are selected. See Table  1 which introduces the selected polymers nanocomposites under study. Modeling is processed using MATLAB (R2017a). Six ANFIS networks are designed to complete the modeling process for the aforementioned polymer nanocomposites in Table 1. The experimental data are employed as inputs into ANFIS model. The number of membership functions (MF) and their type for both inputs and output are specified for each ANFIS network. Also, the number of epochs is adjustable to achieve the minimum training error. After retraining many times with different specifications, the optimal ANFIS network that presents the least possible training error is reached. The optimal ANFIS architecture using MATLAB (R2017a) application designer is shown in Fig. 4. As shown in Fig. 4, the optimal ANFIS network consists of two inputs, one output, two MF for both inputs and output, and four fuzzy rules. The  characteristics and specifications for the optimal ANFIS networks are presented and summarized in Table 2. Observing Table 2, it is found that the optimal ANFIS networks share a set of characteristics. They are Takagi-Sugeno type, employ weight average (Wtaver) as defuzzification method, number of nodes (21), number of linear parameters (12), number of nonlinear parameters (8) and total parameters (20). Table 3 summarizes the number of training data pairs, number of training epochs, the minimal training root-meansquared error (RMSE), the minimal training mean squared error (MSE), the mean error and the mean standard error (Std error) for each optimal ANFIS network. ANFIS simulation and prediction results are processed and presented in Figs. 5a-c, 6a-c and Table 4. Figure 5a-c shows the ANFIS trained results for dielectric permittivity ε of aforementioned polymers nanocomposites, respectively. The simulated ANFIS results are represented by the solid line curves. The experimental data are represented by symbols. Also, the predicted ANFIS results are represented by dashed curves. Analyzing Fig. 5, the simulation solid lines and the experimental data symbols are congruent, which indicates the results accuracy. Two types of predictions are processed: prediction for experimentally measured values and prediction for values that are not measured. It is noted that the first type of prediction is processed as a testing step to compare the predicted ANFIS results with the targets (the experimental data). Predictions for experimentally measured values are processed for 0.15%, 4% and 25% polymer nanocomposite weights of a, b and c, respectively, showing a great accordance with the experimental data. The second type of predictions is processed for {0.12%, 0.25%, 0.3% and 0.35%}, {1%, 3% and 5%} and {17%, 28%, 35% and 45%}, respectively. It is clear that they give acceptable results. Figure 6a-c represents the ANFIS simulated and predicted results for dielectric loss ε of aforementioned polymers nanocomposites, respectively. According to Fig. 6, ANFIS simulation results and the experimental data have a similar behavior. Predictions as informative step are processed for 0.15%, 4% and 25%, respectively, while predictions for unmeasured values are processed for {0.12%, 0.18%, 0.22% and 0.24%}, {1%, 3%, 7% and 10%} and {17%, 28%, 35% and 45%}, respectively. Prediction results prove the excellent ability and great efficiency of ANFIS in modeling dielectric properties of polymer nanocomposites. ANFIS simulated and predicted results for individual points are obtained using ANFIS rule viewer and presented in Table 4. Using the MATLAB Surface Viewer, three-dimensional illustrations that represent the mapping from frequency and nanocomposites weight % to the dielectric permittivity and dielectric loss are obtained. They represent two-input one-output case. In order to view the dependency of the output on the two inputs, the surface viewer results are introduced in Figs. 7a-c and 8a-c for samples a, b and c, respectively. Figures 9a-c and 10a-c show a comparison between the experimental data and the ANFIS output. They showed the extent of the closeness and complete compatibility between ANFIS results and their targets for all trained data points. A histogram plot is a tool for determining the degree of noise in a given dataset, allowing one to ensure that the average error is within reasonable limits. As shown in Fig. 11, the minimum and maximum Eur. Phys. J. Plus (2022) 137:384 The ANFIS error e t , mean error e t , the mean squared error (MSE), root-mean-squared error (RMSE) and the standard deviation error (Std error) are calculated using Eqs. (7)- (11).
e t e t /n where n is the number of experimental data points.

Conclusion
In accordance with the research objectives, ANFIS is utilized in modeling dielectric properties of polymers nanocomposites. Based on the ANFIS modeling results, it is found that the optimal network structure is Takagi-Sugeno type. It is composed of two inputs (nanocomposite weight % and frequency), one output (dielectric permittivity or dielectric loss). Six ANFIS networks are trained to model the experimental data. They have 21 nodes, 20 parameters, 2 membership functions and 4 fuzzy rules. Gauss-mf and linear-mf are the membership functions for the inputs and the output, respectively. The mean errors are calculated for each ANFIS network. The mean errors changed between 10 −10 and 10 −7 at epoch from 200 to 2000. ANFIS simulated and predicted results are discussed in detail. They showed a great performance of ANFIS in modeling dielectric properties of polymers nanocomposites.  Data availability Data supporting the findings of this study are available from the corresponding author (R. A. Mohamed) on request.

Conflict of interest
The author has no conflicts of interest to declare that are relevant to the content of this article.
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