Levenberg–Marquardt Backpropagation for Numerical Treatment of Micropolar Flow in a Porous Channel with Mass Injection

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, KP 23200, Pakistan Department of Computer Science, College of Computer and Information Sciences, King Saud University, Riyadh 11543, Saudi Arabia Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, Douliou, Yunlin 64002, Taiwan Department of Mathematics, COMSATS University Islamabad, Attock Campus, Attock 43600, Pakistan Department of Computer Science Faculty of Applied Sciences, Taiz University, Taiz 6803, Yemen Information and Communication Engineering Technology, School of Engineering Technology and Applied Science, Centennial College, Toronto, Canada


Introduction
A few years ago, Eringen [1,2] firstly presented the idea of micropolar fluids. eories of non-Newtonian fluid are developed to describe the behavior of the fluid that does not obey Newton's law, such as micropolar fluids.
is fluid summarizes specific non-Newtonian behaviors, such as liquid with polymer additives, liquid crystals, animal blood particles, suspensions, and topographic features. e governing equations of many physical problems are nonlinear in nature and cannot be solved analytically; therefore, the scientist developed some approximate and numerical techniques, such as perturbation-based methods [3,4], homotopy perturbation-based methods [5][6][7], homotopy analysis-based methods [8][9][10], collocation-based method [11][12][13][14][15][16][17][18], and Adomian decomposition-based methods [19,20]. Among these methods, artificial intelligence (AI)based numerical methods have been broadly designed for solving differential equations in several diverse applications [21][22][23]. A few latest research works to solve the problems of nonlinear systems include the study introduced for local fractional partial differential equations [24], fourth-order nonlinear differential equations [25], Riccati equation control of nonlinear uncertain systems [26], analytic solution of micropolar flow using the homotopy analysis method [27], and pantograph delay differential equation [28]. However, these numerical-based methods need discretization and improved linearization techniques, which only allow computing the solution for certain standards variables and required huge computer memory and time. For optimizing the results, convergence and stability should be considered to avoid divergence. e perturbative methods required the assumption of small parameters, which is itself an issue. Besides, there is no study yet has been applied a fast backpropagation method for finding an accurate series solution to micropolar flow in a porous channel with mass injection (MPFPCMI

Problem Formulations
Consider the steady, incompressible, and laminar flow of micropolar fluid along two-dimensional channel with porous walls. e mass fluid is introduced with speed q. e walls of the channel are adjusted at y � ±h, where 2h is the width of the channel [29,30]. e fundamental equations governing are as follows [31,32]: e suitable conditions for physical boundaries are as follows [26,27]: e symmetric flow (SF) is as follows [33]: Here, q greater than zero relates to suctions, q less than zero corresponds to injections, and "s" is a finite-parameters worn to model the degree to which microelements in the region of the channel walls are free to rotate, e.g., as "s" is equal to zero is the case where microelements near the boundary cannot turn around when s � 0.5, the situation of microrotation is identical to the velocity of the fluid at the end. Kelson et al. [29] developed the following equations: where η � y h , e Navier-Stokes equations (1)-(4) decrease via applying equations (7) and (8): Dimensionless parameters are established as follows: 2 Complexity When viscosity parameter (Re) is more significant than zero used for suction and less than for injection, the BCs are e SF is as follows: By applying Kelson et al. [29], we put s � 0, N 1 � 1, N 2 � 1, and N 3 � 0.1: e BCs are as follows:

Numerical Experimental Results with Discussion
A short overview of the scheme proposed for finding the proposed LMA-BANN numerical experimentation continuity and momentum equations, i.e., 2-4, based on MPFPCMI is accessible in this section. e proposed structure of stepwise flow is presented in Figure 1 by using "nftool" of the NN tool-box existed in MATLAB. LMA-BANN is implemented for two-layer structures that include single input hidden and output of feed-forward network by LMA-based backpropagation process. Figure 2 demonstrates the structural design of ANN based on ten neuron numbers with a data-sigmoid activation function.
A reference dataset for LMA-BANN for equations (14) and (15) is created between intervals [0, 1] for 201 input grids. Now, 80 percent of data are used for training as 10 percent is for testing and 10 percent is for validation in the event of a 2-layer feed-forward ANN structure fitting tool with LMA backpropagation to solve all problems of MPFPCMI. Training data are used to establish the estimated solution on the source of the MSE, validation data are used to LMA-BANN, and even as test data are used to assess the truthful input performance. Figures 3 and 4 show the effects of LMA-BANN performance represented by error histograms and fitting of solutions for two cases of the MPFPCMI scenario, while the regression tests are shown in Figure 5 for two cases of the MPFPCMI scenario. Furthermore, the MSE, number epochs, and other convergence parameters for training, validation, and testing data are tabulated in Tables 1 and 2  Increasing the Reynolds number with a minimum rotation occurred does not make it move away from the origin of the channel.
According to the results in Table 1, we can see that the model achieves the values 2.07665E − 12, 4.16509E − 9, and 2.3595E − 10 of the MSE on the test data for cases 1, 2, and 3, when the number of epochs are 121, 214, and 223, respectively. We notice that all these values of the MSE are too low, confirming the effectiveness of the model. However, the lowest value is obtained for case 1 at the smallest value of     Table 2, it obvious that the model gets the values 1.08132E − 9, 4.7336E − 10, and 8.78423E − 11 for the test data at epoch's number 92, 329, and 177 and through the settings of case 1, case 2, and case 3. e lowest value is obtained for case 3 at epoch's number 177, which is also an acceptable number of epochs. is also confirms the efficiency of the LMA-BANN model. Finally, from Figures 6(b)-8(b), we can see that lowest values of error plot are for Re � 10 for all scenarios, which means that this value is suitable for Re parameter.

Conclusion
e LMA-BANN is used as an artificial intelligence-based integrated method to find an accurate series solution for the MPFPCMI. e partial differential equations (PDEs) system of the MPFPCMI is converted to the order differential equations (ODEs) system by using the ability of similarity variables. e OHA method is used for producing the dataset of the MPFPCMI. Different measurable quantity of a set of metrics is utilized for evaluating the developed model. For training the model, a percentage equals to 80% of the data used, and a percentage of 10% from the remaining is used for testing, as well as the last 10% of the reference data is applied for validating the LMA-BANN model. e near values of both planned and reference outcomes matching of level between 10 −05 to 10 −07 confirm the rightness of solution, and supposed feature is additional authentic via numerical and graphical design of for the convergence of MSE, AE, error histogram plot, and regression plot measures. After this assurance, the results are demonstrated for the rotating and velocity profile when there are different values of Reynolds number and viscosity parameter (Re). From the numerical results of the problem, we get that increasing the value of Re decreases the velocity in x-direction and mass injection. Moreover, the experimental results confirmed the effectiveness of the LMA-BANN model for accurate analysis of MPFPCMI. In future work, we plan to apply the designed model for finding a solution to another nonlinear system with a large dataset and with different percentages of testing for more analyzing and investigating.
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Data Availability
All the relevant data are included within the manuscript.

Conflicts of Interest
e authors declare that there are no conflicts of interest about the publication of the research article.