Energetic thermo-physical analysis of MLP-RBF feed-forward neural network compared with RLS Fuzzy to predict CuO/liquid paraffin mixture properties

ABSTRACT Dynamic viscosity of novel generated Copper Oxide (CuO)/Liquid Paraffin nanofluids is obtained experimentally for various temperatures and concentrations. To optimize the empirical process and for cost-efficiency, Feed-Forward Neural Networks (FFNNs) were modeled and compared with Recursive Least Squares (RLS) Fuzzy model. To prepare CuO/ liquid paraffin nanofluids, CuO nanoparticles are dispersed within paraffin. Then an input-target dataset containing 30 input-target pairs is available for , and . Based on the empirical results, two types of FFNNs are examined and compared with RLSF model to predict CuO/liquid paraffin nanofluids. To evaluate the best optimization methods of nanofluid viscosity, Multi-Layer Feed forward (MLF), Radial Basis Function (RBF), and RLSF are compared and discussed. The MLF network provides a global approximation while the RBF acts more locally, further, RLSF provides a better fit. On the contrary, the RBF network has better properties from the generalization and noise rejection points of view. Also, RBF networks can be applied in an online manner. Further, three curves of RLS Fuzzy model by Parabola2D, ExtremeCum, and Poly2D models were fitted on the empirical data and compared. The ExtremeCum model showed the least margin of error and can be employed to predict the data.


Introduction
One of the main incentives for the application of nanofluids in the industry is the thermal conductivity of the nanoparticles, which results in increased heat transfer. It causes a significant improvement in the conduction of the mixture. However, nanoparticles Brownian motions can cause the activation of the nanofluid convention. Besides, because of the hybrid nanoparticles' lower concentration, the costs are reduced, and the efficiency is increased. Additionally, different kinds of base liquids can be studied depending on the operational conditions (Abdalla et al., 2010;Rajpal, 2014;Santos et al., 2013). Hence, it seems necessary that one should have an acceptable level of knowledge regarding nanofluid thermophysical properties, particularly for the new kinds of nanofluid. The heat transfer rate and thermal efficiency are among the CONTACT Xifeng Wang dweisky@yeah.net; Shahab S. Band shamshirbands@yuntech.edu.tw advantages of encouraging applying the nanofluid. Various studies have reported these advantages. Because of the high-cost empirical works on nanofluid, some studies have utilized numerical approaches. A correlation is proposed in these works. It is between the favorite subjects of researchers in the field of composites and nanofluids (Akhlaghi & Kompany-Zareh, 2005;Bagwari et al., 2015;Ghasemi & Karimipour, 2018). The experimental studies have some problems, and some specific matters should be taken into account. Hence, it is suggested to use the numerical approaches to predict fluid properties at the optimal condition. Generally, the ANN method is utilized for estimating the thermo-physical characteristics according to the empirical data. Because of the universal approximation property of the ANN approach, it is considered the most leading regression approach.
A multi-layer feed-forward can anticipate the relationships between the independent and dependent variables. Among the optimization methods, the artificial neural network (ANN) has been applied in numerous studies for predicting the nanofluid characteristics, particularly for its viscosity and thermal conductivity. Besides, the acceptable accuracy of this approach motivates the researchers to construct newer ANN models for better consistency with the available physical conditions of a nanofluid flow Dehghani et al., 2019;Karimipour et al., 2018;Sedeh et al., 2019;Teng et al., 2010). Current progress in machine learning (ML) suggests novel tools to excerpt novel understandings from large data sets and to achieve small data sets more efficiently Ali Ahmadi & Ahmadi, 2016;Amedi et al., 2016;Moosavi et al., 2019). Scientists in nanoscience are investigating with these tools to tackle demands in many fields. Further, ML's progress of nanoscience, nanoscience supports the foundation for neuromorphic computing hardware to develop the implementation of ML algorithms: (i) employing ML to analyze and extract novel understandings from large nanoscience data sets, (ii) employing ML to advance material discovery, containing the use of active learning to guide empirical design, and (iii) the nanoscience of memristive devices to realize hardware tailored for ML (Ahmadi, Ebadi, Marghmaleki et al., 2014, Ahmadi, Ebadi, & Yazdanpanah, 2014Ahmadi, Bahadori et al., 2015;Ahmadi, Pouladi et al., 2015;Nguyen, Ghorbani et al., 2020;Shafiei et al., 2014).
Cost efficiency is the act of saving money by optimizing a process. This is done by decreasing experimental costs and improving efficiencies across the process (Abidi et al., 2021;Du et al., 2020;Nguyen, Rizvandi et al., 2020;Sun et al., 2021). Liu et al. (2019) presented a new approach of 'Recursive Least Squares Fuzzy' neural network. They optimized the thermal conductivity measurement of Graphene Oxide/Water nanofluid by R-Squared of 0.99. Alsarraf,  used the neural network to optimized the thermal conductivity measurement of Graphene/Water nanofluid by R-Squared of 0.99. Xu et al. (2020) used Levenberg Marquardt (LM) algorithm to optimize the viscosity measurement process of Graphene Oxide/Water nanofluid by R-Squared of 0.997 for RMPs of 10 and 100. Li et al. (2021) presented a novel neural network algorithm (Orthogonal Distance Regression (ODR)) to optimize the thermal conductivity measurement of Carbon Nanotube-Titanium Dioxide/Water-Ethylene Glycol nanofluid by R-Squared of 0.9999. Malekahmadi et al. (2021) compared two kinds of neural network algorithms, LM and ODR, to find the best optimization method for thermal conductivity measurement of Carbon Nanotube-Hydroxyapatite/Water nanofluid. They reported that the ODR is a better method to model the process and to reduce the cost of experiments.
For neural networks models, Ebtehaj et al. (2018) by hybrid decision tree (DT) technique with multilayer perceptron (MLP) and radial basis function (RBF) artificial neural networks, indicated that it is possible to extract matrices to provide applicable models. Also, Ebtehaj and Bonakdari (2016) optimized Multilayer Perceptron (MLP) network with three different training algorithms, including variable learning rate (MLP-GDX), resilient back-propagation (MLP-RP) and Levenberg-Marquardt (MLPLM) to apply in the practical application.

Numerical method
Artificial Neural Networks (ANNs) are frequently used for curve fitting and function estimation. The ANNs are very successful in this regard due to their iterative process, by which the precision of the outputs is continuously improved. Since the ANNs have nonlinear activation functions, they can approximate any arbitrary function without knowing the investigated data. Despite effective applications of the ANN in the curve fitting and function approximation fields, it is not easy to select the proper type of ANNs in practice. In this paper, two types of Feed-Forward Neural Networks (FFNNs) are examined and compared with RLS Fuzzy model to predict CuO/liquid paraffin nanofluids.
Multi-Layer Perceptrons (MLPs) are the most popular type of FFNNs for curve fitting, pattern recognition, and clustering. In this method, any neuron in a specified layer collects weighted outputs of all the neurons in the previous layer. The MLP networks usually have one or two hidden layers and one output layer. In any layer, the summation of the weighted inputs is mapped by an activation function to the output space. The MLP networks usually have 'sigmoid' and 'linear' neurons. Once the outputs of the networks are obtained, they can be compared with the targets. One can use a training algorithm to minimize the Mean Squared Error (MSE) between the outputs and targets. There are a variety of back-propagation training algorithms with dissimilar schemes and characteristics. The MLP neuron can be mathematically described as follows: in which x and y are the input and output, respectively. Also, w and b indicate the network weight and bias, respectively. Also, the 'sigmoid' activation function can be represented as follows: A Radial Basis Function (RBF) network consists of only one hidden layer and one output layer. The hidden layer of the RBF networks is different from the MLP ones. In the hidden layer, the Euclidean distance between the weight and input vectors are obtained. Therefore, the number of inputs and weights of the hidden layer are identical. Then, the output of the previous step is multiplied by the bias value. Finally, the net value is mapped by an activation function to the output space. The RBF network usually employs 'radbas' and 'linear' transfer functions. The training algorithm of the RBF is different from that of the MLP. The training is usually performed in a two-phase process where the centers and scaling parameters of the RBF layer are fixed in the first step, and the weights of the output layer are fixed in the second step. The RBF neuron can be mathematically described as follows: Also, the 'radbas' activation function can be represented as follows: The architecture of two-layer MLP and RBF networks are illustrated schematically in Figures 1 and 2. In these figures, the first activation functions f 1 represent the 'sigmoid' and 'radbas' for the MLP and RBF networks, respectively. Also, the second activation functions f 2 are 'linear' for both cases. Also, Figure 3 is illustrated the RLS Fuzzy architecture that Layer 1 is the inputs, Layer 2 is the 'IF part', Layer 3 is the 'Rules + Norm', Layer 4 is 'THEN part', and Layer 5 is the output.
It can be observed that the MLP has simpler architecture. The MLP is widely used for diverse types of problems, while the RBF is less popular. Also, there are essential differences between the performances of the MLP and RBF networks. According to (Santos et al., 2013), the MLP network provides a global approximation while the RBF acts more locally. On the contrary, the RBF network has better properties from the generalization and noise rejection points of view. Also, RBF networks can be applied in an online manner. Due to these properties, it is not theoretically possible to select the network having the best performance. Hence, for any specific real problem, the performance of the MLP and RBF should be numerically compared (Abdalla et al., 2010;Akhlaghi & Kompany-Zareh, 2005;Bagwari et al., 2015;Rajpal, 2014;Santos et al., 2013). In this study, the performance of the  MLP, RBF and RLS Fuzzy are compared for the problem of prediction CuO/ liquid paraffin nanofluids.
Data points selection, that used to train the models, depends both on the complexity of the problem (the unknown underlying function that best relates the input variables to the output variable) and on the complexity of chosen algorithm (the algorithm used to inductively learn the unknown underlying mapping function from specific examples). Thus, the number of data points are selected based on previous related studies which are at least 30 data points to train the model (Alsarraf et al., 2020b;Liu et al., 2019;Xu et al., 2020).
The parameters of ANN, RBF, and RLSF are the learning algorithm, number of hidden layers and neurons in the hidden layer. The selection of each parameter was based on the following: For the learning algorithm, the selection was based on the size of the training data (which is 30 in this study), Accuracy and/or Interpretability of the output (Linear Regression could easily be understood how any individual predictor is associated with the response while the flexible models give higher accuracy at the cost of low interpretability), Speed or Training time (Higher accuracy typically means higher training time, however, the training time was not an important issue in this study), Linearity (not always is the data is linear, so other algorithms are required which can handle high dimensional and complex data structures), and Number of features  (The dataset may have a large number of features that may not all be relevant and significant, however, all the datasets in this study are significant).
For the number of inputs, with respect to the number of neurons comprising this layer, this parameter is completely and uniquely determined once the shape of training data is determined. Thus, the inputs are 30 datasets.
Like the Input layer, every NN has exactly one output layer. Determining its size (number of neurons) is completely determined by the chosen model configuration.
While for the hidden layers, one hidden layer is sufficient for the large majority of problems. However, the optimal size of the hidden layer is usually between the size of the input and size of the output layers. To trim network size (by nodes not layers) to improve computational performance and sometimes resolution performance, is removing nodes from the network during training by identifying those nodes which, if removed from the network, would not noticeably affect network performance (weights very close to zero in the trained matrix).
For the neurons in the hidden layer, the selection was based on the following equation which usually prevent over-fitting: While N i is the number of input neurons, N o is the number of output neurons, N s is the number of samples in training data set, and α is an arbitrary scaling factor usually 2-10.

Materials
To prepare CuO/liquid paraffin nanofluids, CuO nanoparticles are dispersed within paraffin. All nanomaterial were obtained with the pureness of 99.91%. Furthermore, paraffin with large pureness was acquired from Merck Company, Germany. To obtain standard clean conditions, deionized water was applied to clean Entirely glaswares of the laboratory (Ghasemi & .

Nanofluid preparation
As mentioned in our previous research (Dehghani et al., 2019), stirring is used with 500 rpm for about 30 min to disperse 1 g of CuO nanoparticles within 19 g of paraffin. To avoid nanoparticles accumulation, the sonication waves with a maximum amplitude of 60% for three separate steps of 20 min is applied to suspension (Dehghani et al., 2019). The specific amount of stock nanofluid was applied. In the next step, to obtain the total mass of nanofluid to reach 20 g, the paraffin was applied to the nanofluid.

The experimental viscosity evaluation
To assure the results, the evaluation was repeated three times in an identical situation, and deviation was evaluated by applying Equation (6) Karimipour et al., 2018;Sedeh et al., 2019): where R μ,i represents the ratio of dynamic viscosity of a fluid with nanoparticles to clean basefluid, R μ is average relative dynamic viscosity, and n stands for numbers of repetition, which is three in this work. To perform the uncertainty analysis, we determined the variation of significant parameters and the variation of thermal circulator bath accuracy the precise electric balance, and viscometer are ±0.005°C, ±0.0003 gr, and ±1%, respectively (Rajpal, 2014). Equation (7) was applied to evaluate the uncertainty of tests (Teng et al., 2010): According to our results, the uncertainty of the dynamic viscosity of nanofluid was approximately 8.4%.

Results and discussion
In the conducted experiments, the dynamic viscosity of CuO/liquid paraffin nanofluids is obtained for different temperatures and nanoparticle mass fractions. An input-target dataset containing 30 input-target pairs is available for T = 25, 35, 40, 50, 55, 70( • C), and ϕ = 0.1, 0.5, 1.0 , 3.0, 5.0 (%). Figure 4 presents the impact of dynamic viscosity of CuO/paraffin. According to obtained results, viscosity significantly increases as the concentration is raised. Indeed, this is mainly due to the presence of solid particles within nanofluid. Subsequently, viscosity is enhanced. Our results also demonstrated that viscosity declines meaningfully when the temperature is increased. The micro-convection of nanoparticles in basefluid is the main reason for these phenomena. Viscosity reduces 59, 64, 67, 68 and 69% at CuO nanoparticle concentration of 0.1, 0.5, 1.0, 3.0 and 5.0 wt%; while temperature varies from 25 to 70°C. The values of the dynamic viscosity are illustrated in Figure 5. Moreover, nanofluid dynamic viscosity is shown in Figure 6. The MLP and RBF networks are applied. Regression curves are presented in Figure 7. The numerical outputs of the models are plotted against the experimental targets. It can be seen that both networks have acceptable performance. The error plots are shown in Figure 8(a) and (b). Results are depicted in Figure 9 for the numerical data from both methods. It can be observed that both networks result in identical predictions.
Finally, the networks are examined at non-trained inputs throughout the investigated intervals. The results are presented in Figure 10. The results of the MLP network have better smoothness. It can be concluded that the MLP network has better generalization while the RBF network has better precision. In a small training dataset, the MLP network may lead to better results, while the RBF is suggested for a large training dataset. The RLS Fuzzy model curve fitting method applied on the empirical data based on three different models, 'Parabola2D', 'ExtremeCum' and 'Poly2D'. As can be seen in Figure 11, Parabola2D model did not fit completely on the data, hence, ExtremeCum and Poly2D models did fit completely and can be used to predict the data.
Three original formulas derived from these three curves. As can be seen, the second Equation (ExtremeCum) has better R-squared (almost 1), and can be employed to predict the data.
Fitted Curve: Parabola2D   While the T 0 is 25°C and T is the Temperature from 25 to 70°C. Figure 12 shows Margin of error by RLS Fuzzy model. As can be seen, the Parabola2D model's error is 53.73918377% from Max. error of 23.74028975% to Min. error of −29.99889402%. The ExtremeCum model's error is 6.62089424% from Max. error of 2.656876933% to Min. error of −3.964017307%. The Poly2D model's error is 17.47650685% from Max. error of 8.032125% to Min. error of −9.444381852%. Thus, ExtremeCum model has the least margin of error and can be applied as the best fit. Figure 13 shows the Error contour by RLS Fuzzy model. As can be seen, the dash lines are the empirical datasets while the solid lines are the trained datasets. The results showed that the ExtremeCum model have better smoothness.

Conclusion
Through the present work, the dynamic viscosity of novel generated CuO/ liquid paraffin nanofluids was obtained experimentally for various temperatures and concentrations. To prepare CuO/ liquid paraffin nanofluids, CuO nanoparticles were dispersed within paraffin. Two types of Feed-Forward Neural Networks (FFNNs) were examined and compared with RLS Fuzzy model to predict CuO/ liquid paraffin nanofluids. It was seen that the ANN optimization approach was examined based on the empirical data of CuO/ liquid paraffin nanofluids. To evaluate the best optimization method of nanofluid viscosity, MLF, RBF, and RLSF were compared and discussed. MLP network provided a global approximation while the RBF acted more locally. On the contrary, the RBF network had better properties from the generalization and noise rejection points of view. Also, RBF networks could be applied in an online manner. Moreover, the results of the MLP network had better smoothness, so the MLP network showed better generalization while RBF had better precision. Further, three curves of RLS Fuzzy model by Parabola2D, ExtremeCum, and Poly2D models were fitted on the empirical data and compared. The margin of error for Parabola2D was 53.73918%, for ExtremeCum was 6.62089%, and for Poly2D was 17.47651%. Thus, The ExtremeCum model showed the least margin of error and can be employed to predict the data.
Employing of the trained models in the case studies is possible by hybrid decision tree (DT). The viscosity measurement and its optimization by ANN and Fuzzy models can be used in a wide variety of applications within many different industries, including: Pharmaceuticals (suspensions, gelatins, and syrups); Construction industry materials (cement, coatings, and mortars); Chemicals (paints, inks, detergents, adhesives, and resins); and also, Foods containing seaweed, starches, soups, sauces, etc.

Disclosure statement
No potential conflict of interest was reported by the author(s).