Intelligent computing through neural networks for numerical treatment of non-Newtonian wire coating analysis model

In the current study, a modern implementation of intelligent numerical computational solver introduced using the Levenberg Marquardt algorithm based trained neural networks (LMA-TNN) to analyze the wire coating system (WCS) for the elastic-viscous non-Newtonian Eyring–Powell fluid (EPF) with the impacts of Joule heating, magnetic parameter and heat transfer scenarios in the permeable medium. The nonlinear PDEs describing the WCS-EPF are converted into dimensionless nonlinear ODEs containing the heat and viscosity parameters. The reference data for the designed LMA-TNN is produced for various scenarios of WCS-EPF representing with porosity parameter, non-Newtonian parameter, heat transfer parameter and magnetic parameter for the proposed analysis using the state of the art explicit Runge–Kutta technique. The training, validation, and testing operations of LMA-TNN are carried out to obtain the numerical solution of WCS-EPF for various cases and their comparison with the approximate outcomes certifying the reasonable accuracy and precision of LMA-TNN approach. The outcomes of LMA-TNN solver in terms of state transition (ST) index, error-histograms (EH) illustration, mean square error, and regression (R) studies further established the worth for stochastic numerical solution of the WCS-EPF. The strong correlation between the suggested and the reference outcomes indicates the structure’s validity, for all four cases of WCS-EPF, fitting of the precision \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-5}$$\end{document}10-5 to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-9}$$\end{document}10-9 is also accomplished.

Wire coating covers an electrical conductor with a layer of dielectric material through a process called the Extrusion Process. This process is one of the most important and accurate production processes manufacturing the insulated wires and cables usually used in polymer melt industries. The system model operations for WCS-EPF are shown below in Fig. 1 1 . In this process, the coating material, which is in the shape of granules, is melted that introducing this material into the Extruder, where the high temperature and pressure are appropriate to meet the material and deliver it to the required state. Then the material reaches so-called Extruder Head, which guides the liquid material through the tip and dies to give the desired tube shape and the required thickness. Then the material comes out from the Head's core to stick to the wire directly and form according to the wire's shape. Immediately after that, the water-cooling stage comes inside a stream of water that the coated wire passes through for a certain distance and then dried with direct air. Then it is rolled onto a drum. The wire coated, regardless of its shape or content, by two types of separate dies, one in the form of a tube and the other pressure die like a ring. The flow through this mold is identical to the flux by the guttural area consisting of two molds, one is fixed externally, and the other internal is dynamic moving in the flow path. Depending on the die geometry, dynamic velocity, and heat of the wire and melt polymer, various kinds of liquids are used for wire coating.
The study of WCS in Newtonian fluids systems has attracted the research community with their broad interest. Fluidic systems are classified into two types depending on the viscosity: Newtonian fluids as water, honey, oil, and alcohol, and non-Newtonian fluids as butter, ketchup, mayonnaise, milk, and blood. The non-Newtonian fluidic systems with the variable viscosity havind wide range of applications in industry and fluid mechanics. Many researchers [1][2][3][4] have used different non-Newtonian fluid models for materials of wire coating along with the affects of joule heating and hall current.A magnetic field provides an influencing source in the magnetic hydrodynamic (MHD) process and significantly impacts fluid movement in the wire coating processes. WCS analyses involving MHD have been extensively exploited by scholars in the latest years due to its widespread implementations in the industrial system, such as glazier processing and attractive-materials. Many researchers [5][6][7] studied the MHD process with the impact of an applied magnetic field for the dynamics of WCS.
Owing to the broad reach in engineering science, fluid flow across a porous medium has unparalleled interest for researchers. Some common permeable media are timber, mineral foams, and crags of the carbonate, etc. Over time, the study about the application of heat transfer for WCS subjected to non-Newtonian fluids has acquired popularity owing to its application for different manufacturers. Rehman and Nadeem 8 examined transfer of heating impact on the processing for the motion of multi-directional stagnation flow. MHD along with heating impacts on the WCS for various types of fluids are investigated by several researchers [9][10][11][12][13][14][15][16] . All these researches have so far introduced numerical/analytical of deterministic techniques for the solution of WCS subjected to different types of Newtonian and non-Newtonian fluids. In contrast, artificial intelligence-based numerical soft computing solver is relatively less discovered/exploited in the field of computing fluid mechanics problems, especially for WCS in different flow dynamics.
The approximate numerical solutions based on stochastic techniques are obtained primarily by modeling artificial neural networks (ANN) and optimizing them with both the mixture of global and local search approaches for solving a range of problems based on ordinary and partial differential systems. Recent applications of stochastic numerical computing solvers include nonlinear systems emerging in astrophysics 17,18 , nanotechnologies 19 , fluid mechanics [20][21][22][23][24] , plasma physics 25,26 , fuel catching fire model 27 , magneto-hydrodynamics 28,29 , electrical conduction solids 30 , energy 31 , rotating electrical devices 32 , thermodynamics 33 , heat conduction 34 , electromagnetic 35 , bioinformatics 36 , and COVID-19 virus spread model [37][38][39] are few important examples of such solutions. These facts motivate the researchers to explore and use soft-computing stochastic methods to create an effective, alternative, and feasible computing model for solving the fluid dynamics systems associated with the wire coating operation.
Throughout this research study, the innovative ideas about the proposed problem and soft computational model are illustrated as follows: • A new implementation of intelligent computational system of the artificial intelligence is introduced by incorporating the solver LMA-TNN for interpreting the fluidic system WCS-EPF along with the impacts of Joule heating, applied magnetic field and transfer of heat in the permeable medium for different scenarios. • The mathematical formulation is introduced with nonlinear differential equations systems for WCS-EPF, which are converted into dimensionless nonlinear ODEs representing the mathematical modeling of heatbased changing viscosity. • A set of data for suggested LMA-TNN is produced for WCS-EPF on the in terms of physical quantities such as the porosity parameter, non-Newtonian parameter, magnetic parameter, heat transfer parameter utilizing the capability of explicit Runge-Kutta technique. Figure 2 illustrates the structure of the study. Let R d , be the radius, L is the length of die and the saturated temperature θ d since the viscoelastic Eyring-Powell material is not compressible, therefore the wire temperature exceeds θ w , the radius is equivalent to R w and the velocity U w in the porous medium. After that, the wire is dragged across the center length in the fixed stress mold. The outflow liquid is concurrently dominated by the unified differential stress dp dz across the axial direction with the attractive strength B o . The magnetic force is vertical along the path of the incompressible non-Newtonian Eyring-Powell fluid flow. We used the concept of low Reynolds number in our study to minimize or ignore the disturbance in the magnetic field.

Mathematical model
For this study, the governing system of the suggested wire coating as follows 16 :

Methodology and discussions
In artificial intelligence (AI), supervised machine learning relates to a category of algorithms and paradigms that describe a predictive model utilizing datasets with known outputs. The approach is learned via an effective teaching algorithm such as artificial neural networks that usually use optimization procedures to reduce error function.
Here, the technical solution involves two steps: the first part includes the required explanation for the design of LMA-TNN data sets; the second part describes the process for applying LMA-TNN. The complete typical procedure is shown in Fig. 3, whereas the suggested solver as a single neural paradigm is illustrated in Fig. 4. Numerical treatment with LMA-TNN is performed for the heat-based changing viscosity paradigm provided in Eqs. (4)- (12). The suggested LMA-TNN is introduced for different scenarios, i.e., S-1 to S-5 corresponding each case of Reynolds model whereas S-1 to S-4 corresponding each case of Constant Viscosity and Vogel's models, as shown in Tables 1, 2 and 3 , respectively. The reference data for LMA-TNN is obtained by employing the Runge-Kutta technique with the help of NDSolver in Mathematica. The suggested LMA-TNN is employed in the form of data sets; the output toward a single input is integrated with the assist of 'nftool' in the toolbox of the neural network through MATLAB www.nature.com/scientificreports/ software (R2020b Update 5 (9.9.0.1592791), https:// www. mathw orks. com/ acade mia/ tah-portal/ king-abdul azizunive rsity-40773 215. html). Total 1001 given data points for each variable (u(r), θ(r) ) is determined between 1 and δ = 2 by keeping the step size 0.001. Then this data is divided into three datasets: the testing, the validation, and the training, in different proportions to determine the percentage that gives a better convergence. These datasets have been generated by reference standard solutions for the system of non-linear higher-order differential equations interpreting the elastic-viscous Eyring-Powell fluidic system WCS-EPF along with the impacts of Joule heating, applied magnetic field, and transfer of heat in the permeable medium for different scenarios.
System of ODEs contains three separate models as follow: • Viscosity Constant, • Reynolds mode, • Vogel's model.

Intelligent Computational
Neuron paradigms are embedded to build the suggested network.
Neural networks with Levenberg-Marquardt based Backpropagation.

Intelligent Computational
Reference data creation for NN modeling using the Runge-Kutta technique.
NN model.

Analysis of Evaluations as follows:
• Mean square error (MSE) with fitness function, • Analysis of state transition, • Error Histograms views, • Regression views, • Analysis of Absolute Error (A-E). www.nature.com/scientificreports/ The training, validation, and testing processes for Levenberg-Marquardt backpropagation neural networks are divided as follow:

V. Comparative Survey IV. Outcomes
• 80% of the dataset are assigned for the training.
• 10% of the dataset are assigned for the validation.
• 10% of the dataset are assigned for the testing.
After performing several tests to obtain optimum measurement accuracy using some hidden neurons, the best is 50 neurons, whereas training the weights of neural networks requires the Levenberg Marquardt based backpropagation. The design of the suggested network can be seen in Fig. 5. As shown in the above figure, the best artificial neural network structure for the data under study is (1 50 2), i.e., one input layer and a hidden layer containing 50 processing elements with two outputs. Using the neural network under supervision, the set of data consisting of 1001 points for each output treated according to different scenarios of all four cases, as shown by Tables 1, 2 and 3. The efficiency and precision investigation of the LMA-TNN process for All the scenarios of cases 2, 2, and 4 in Constant Viscosity, Vogel's, and Reynolds models, respectively, is achieved graphically in Figs. 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20. The comparison of all numerical and random data that contains performance, Gradient, Mu, epochs, time, and mean squared error for testing, validation, and training for all four cases of every scenario as shown in Tables 4, 5 and 6. Figures 6, 11 and 16 describe the mean squared error (MSE) based on training, testing, and validation operations for all scenarios in Constant Viscosity, Vogel's and Reynolds models, respectively, for measuring neural network performance for predicting and relying on results while ensuring predictive accuracy. As in Fig. 6a-d, it clear that have the best accuracy and performance along with MSE about ( 10 −12 , 10 −13 , 10 −11 to 10 −12 , and 10 −11      Table 3. Variants of WCS-EPF for Reynolds model.

Conclusion
In this work, soft computing artificial intelligence is introduced using the LMA-TNN for solving the mathematical model describing the WCS-EPF with a transfer of heat and non-Newtonian Eyring-Powell fluid flow past a porous medium under the impacts of Joule heating and the magnetic field for different scenarios. Findings in brief are listed below: • The nonlinear PDEs for the WCS-EPF are converted into nonlinear ODEs containing the heat-based changing viscosity framework.    www.nature.com/scientificreports/ • Runge-Kutta numerical solver is used to produce reference data for the proposed WCS-EPF in the fluid dynamics with the impacts of various physical amounts of significance such as the porous parameter, non-Newtonian parameter, magnetic parameter, thermal transfer. • The 80 % , 10 % , and 10 % of the data set is chosen as validation, testing, and training for LMA-TNN. The dataset is further validated by the graphical and numerical representations in terms of convergence of the outcomes by the mean square error, the dynamics of state transition, error histograms, and regression analysis plots. • The strong correlation between the suggested and the reference outcomes indicates the structure's validity for all four cases of WCS-EPF, fitting of the precision 10 −5 to 10 −9 is also accomplished. • We observed that the dimensionless temperature profile increases caused by the rise in the values of parameters B r , Q, n, α , and while decreasing by the rise of the parameter D. Besides, the variation in K p does not have any observable contribution to the temperature distribution.