Artiﬁcial Neural Networking (ANN) Model for Convective Heat Transfer in Thermally Magnetized Multiple Flow Regimes with Temperature Stratiﬁcation Effects

: The convective heat transfer in non-Newtonian ﬂuid ﬂow in the presence of temperature stratiﬁcation, heat generation, and heat absorption effects is debated by using artiﬁcial neural networking. The heat transfer rate is examined for the four different thermal ﬂow regimes namely (I) thermal ﬂow ﬁeld towards a ﬂat surface along with thermal radiations, (II) thermal ﬂow ﬁeld towards a ﬂat surface without thermal radiations, (III) thermal ﬂow ﬁeld over a cylindrical surface with thermal radiations, and (IV) thermal ﬂow ﬁeld over a cylindrical surface without thermal radiations. For each regime, a Nusselt number is carried out to construct an artiﬁcial neural networking model. The model prediction performance is reported by using varied neuron numbers and input parameters, and the results are assessed. The ANN model is designed by using the Bayesian regularization training procedure, and a high-performing MLP network model is used. The data used in the creation of the MLP network was 80 percent for model training and 20 percent for testing. The graph shows the degree of agreement between the ANN model projected values and the goal values. We discovered that an artiﬁcial neural network model can provide high-efﬁciency forecasts for heat transfer rates having engineering standpoints. For both ﬂat and cylindrical surfaces, the heat transfer normal to the surface reﬂects inciting nature towards the Prandtl number and heat absorption parameter, while the opposite is the case for the temperature stratiﬁcation parameter and heat generation parameter. It is important to note that the magnitude of heat transfer is signiﬁcantly larger for Flow Regime-IV in comparison with Flow Regimes-I, -II, and -III.


Introduction
The solutions of polymeric melts, polymeric materials, dispersions, suspensions, and slurries exhibit complex non-Newtonian flow fields and hence one cannot narrate such flow fields by the use of Newton's law of viscosity. In this regard, the involvement of convection heat transfer to such non-Newtonian flow fields makes the study more interesting and important. The heat transfer by way of these fluids largely depends upon geometric configuration, flow regime, and the rheology of the non-Newtonian being used. The coatings, printing inks, detergent, petrochemical, food, and chemical products are a few microchannels. The flow was described as laminar, time-independent, incompressible, axisymmetric, and slip. The non-Newtonian fluid's behavior was described using a powerlaw model. Here, constant heat flux, temperature, and thermal boundary conditions were used. The flow equations were solved by the use of the control volume finite difference method with acceptable boundary conditions. The outcomes show that the slip coefficient increases the heat transfer rate. Using power-law fluid, Yao and Molla [10] investigated heat transfer aspects in non-Newtonian fluid moving across a flat surface. The flow equations were solved by marching downstream from the leading edge, as was common for Newtonian flows. Non-Newtonian effects in shear-thinning and shear-thickening fluids were demonstrated using temperature and velocity, shear loads, and Nusselt number. The substantial impacts arise along the leading edge, with the variance in shear stresses gradually diminishing further downstream. Sahoo [11] expanded the heat transfer aspects of a magnetized non-Newtonian electrically conducting fluid. The flow problem was substantially non-linear when momentum equations are used. Numerical solutions for the governing nonlinear equations were obtained. The impact of flow variables on temperature and velocity fields was thoroughly studied and visually depicted. Since then, the past and recent developments on heat transfer aspects in non-Newtonian fluid flows can be assessed in Refs. [12][13][14][15][16][17][18][19][20][21].
In non-Newtonian fluids, there is relatively limited research on predictive correlations relevant for engineering designs having the convective heat transfer coefficient given as the local Nusselt number. Particularly for thermally magnetized Jeffrey fluid flow, we found less investigation by use of artificial neural networking. This is due to the appearing of the complicated mathematical model. Therefore, the main contribution of the present list attempts to include the following:

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The mathematical formulation for Jeffrey fluid flow towards a flat and cylindrical surface.

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Examination of the Nusselt number at both flat and cylindrical surfaces. • For both surfaces, prediction of the Nusselt number by using an artificial neural networking model.
We believe the findings on convective heat transfer by use of artificial neural networking will be helpful for researchers having an affiliation with thermal engineering.

Flow Formulation and Data Set
For the present analysis, we considered the dataset offered as an open source by Rehman et al. [22]. They considered Jeffery fluid flow as a non-Newtonian fluid. Two unlike stretched inclined surfaces, namely plane surface and cylindrical, are carried. The flow regimes are further established with the succeeding physical effects, namely, thermal radiations, mixed convection, stagnation point flow, heat generation, applied magnetic field, temperature stratification, and heat absorption. The domain of interest is geometrically illustrated in Figure 1 while the endpoint conditions are summarized as  In light of Equations (1)-(3), the ultimate flow narrating differential equations for the flow field of Jeffrey fluid are summarized as follows: while the endpoint conditions are summarized as Here, Equation (4) is the equation of continuity, and it guaranties the balance of mass entering a domain to the rate at which mass leaves the domain. Equation (5) is the component form of the momentum equation. The fluid flow is two-dimensional, and flows normal to the surfaces are negligible, so we have a U-component form of momentum equation. The last three terms own mixed convection, stagnation point, and an externally applied magnetic field assumptions. Equation (6) is the energy equation. The last two terms own non-linear thermal radiations and heat generation/absorptions effects. Equation (7) is the boundary conditions representation subject to the temperature stratification effect and stagnation point flow. One can note from boundary conditions that the fluid flow is above the cylindrical surface. The vertical velocity is zero and temperature at the surface is supposed to be greater in comparison with temperatures far away from the surface. Rehman et al. [22] introduce: while the reduced endpoint conditions are Equations (9) and (10) own the following fluid effects parameters: It is important to note that H ± represents the heat generation/absorption parameter. To be more specific, the positive values (H + ) identify the heat generation parameter, while negative values (H − ) show the heat absorption parameter.
In this reference study, Rehman et al. [22] considered heat transfer as a surface quantity; they examine the heat transfer rate normal to the surface in various flow regimes. The mathematical relation in this regard is summarized as follows: and the corresponding dimensionless form is as follows: To examine the Nusselt number in detail, we have considered four various flow regimes, namely Flow Regime-I, Flow Regime-II, Flow Regime-III, and Flow Regime-IV. Equations (4)-(7) are non-linear and difficult to solve in terms of the exact solution. Equation (8) is utilized to attain the coupled differential equations; see Equations (9) and (10). Therefore, we seek a numerical solution. For the numerical solution of Equations (9) and (10) along with boundary conditions (11), we utilized the shooting method. In this regard, firstly, the boundary value problem given by Equations (9)-(11) is transformed into six first-order ordinary differential equations as an Initial Value Problem (IVP). Such an IVP is solved by using missing conditions as an initial guess. The Runge-Kutta scheme solves the set of first-order equations, and initial estimates are improved by the use of Newton's method. If boundary residuals are less than the tolerance error 10 −6 , the computed solution converges; otherwise, the initial guesses are updated by use of Newton's method, and the process is repeated until the solution reaches the desired convergence threshold. The complete flow chart of the solution methodology for obtaining the numerical data of the Nusselt number is given in Figure 2.
firstly, the boundary value problem given by Equations (9)-(11) is transformed into six first-order ordinary differential equations as an Initial Value Problem (IVP). Such an IVP is solved by using missing conditions as an initial guess. The Runge-Kutta scheme solves the set of first-order equations, and initial estimates are improved by the use of Newton's method. If boundary residuals are less than the tolerance error 10 −6 , the computed solution converges; otherwise, the initial guesses are updated by use of Newton's method, and the process is repeated until the solution reaches the desired convergence threshold. The complete flow chart of the solution methodology for obtaining the numerical data of the Nusselt number is given in Figure 2.

Flow Regime-I
In this regime, Jeffrey fluid flow over an inclined stretching plane surface is considered. Mixed convection, heat generation, magnetic field, stagnation point flow, temperature stratification, and heat absorption physical effects are considered in this case. The concluding mathematical formulation for the present case can be achieved by setting K 0 = and R 0, = in Equations (9) and (10). The outcome will be

Flow Regime-I
In this regime, Jeffrey fluid flow over an inclined stretching plane surface is considered. Mixed convection, heat generation, magnetic field, stagnation point flow, temperature stratification, and heat absorption physical effects are considered in this case. The concluding mathematical formulation for the present case can be achieved by setting K = 0 and R = 0, in Equations (9) and (10). The outcome will be While the quantity of interest reduces to For Regime-I, the impacts of Prandtl number, temperature stratification, heat generation, and absorption on the Nusselt number are examined and concluded in Table 1.

Flow Regime-II
In this regime, the flow of Jeffrey fluid is considered over a flat surface in the presence of non-linear thermal radiations. The other physical effects include magnetic field, heat generation, mixed convection, stagnation point flow, temperature stratification, and heat absorption. The concluding mathematical formulation for Regime-II can be achieved by setting K = 0 and R = 0.3, in Equations (9) and (10). The reduced equations are For Regime-II, the impacts of Prandtl number, temperature stratification, heat generation, and absorption on the Nusselt number are examined and concluded in Table 2.

Flow Regime-III
In this regime, the flow of Jeffrey fluid is considered over a cylindrical surface in the absence of non-linear thermal radiations. The other physical effects include heat generation, mixed convection, stagnation point flow, applied magnetic field, temperature stratification, and heat absorption. The concluding mathematical formulation for Regime-III can be achieved by setting R = 0 in Equations (9) and (10). The reduced equations are while the quantity of interest, namely the Nusselt number, reduces to For Regime-III, the impacts of Prandtl number, temperature stratification, heat generation, and absorption on the Nusselt number are examined and concluded as Table 3.

Flow Regime-IV
In this regime, the flow of Jeffrey fluid is considered over a cylindrical surface in the presence of non-linear thermal radiations. The other physical effects include heat generation, mixed convection, stagnation point flow, magnetic field, temperature stratification, and heat absorption. The concluding mathematical formulation for Regime-IV is reported as Equations (9) and (10). The mathematical relation for the Nusselt number subject to Flow Regime-IV is concluded as Equation (14). By using Equation (14), the variations in Nusselt number subject to Prandtl number, temperature stratification, heat generation, and heat absorption are summarized in Table 4. The Nusselt number numerical values represent the heat transfer rate normal to the surfaces. We found that as heat generation and the Prandtl number rise, the rate of heat transfer normal to the surface also rises. The heat transfer rate is found decreasing the function of the heat generation parameter and temperature stratification parameter. Such outcomes hold for both the flat and cylindrical surfaces.

Formulation of ANN
An artificial intelligence approach, which was developed as an alternative to traditional mathematical models, was used to analyze the Nusselt number in four different flow regimes. The first flow regime is considered as a plane surface, with K = 0 and R = 0 (without thermal radiation). In the second flow regime, R = 0.3 (with thermal radiation) is considered. The third flow regime was evaluated under conditions of a cylindrical surface, with K = 0.3 and R = 0 (without thermal radiations). The fourth flow regime was investigated for a cylindrical surface with K = 0.3 and R = 0.3 (with thermal radiations) conditions. These models are named Flow Regime-I, Flow Regime-II, Flow Regime-III, and Flow Regime-IV, respectively. The ANN model, which is one of the important artificial intelligence algorithms, was used as the analysis methodology. ANN models are an engineering tool with high performance even in the simulations of complex structures where there is no functional relationship between them due to their advanced systematics [23,24]. In the ANN model, which is designed to estimate the Nu number, the multi-layer perceptron (MLP) network model, which has a reliable estimation capability with its strong architecture, is used [25][26][27][28]. Each layer in the topology of MLP networks is directly coupled to the layer above it [29]. Data is input into the system at the input layer, which is the first layer of an MLP network. Each MLP network has at least one hidden layer, which is the layer connected to the input layer after that. The output layer, the final layer of the MLP neural network and the layer following the hidden layer, is where simulation results are obtained. The symbolic architecture in this regard is offered in Figure 3.
In the training of MLP neural networks, previously obtained data sets were used. The MLP network model built for calculating the Nu number utilized a total of 80 numerically acquired data sets. One of the most significant phases in the design process was optimizing and organizing the data set utilized in training ANN models [30]. The data set required to create the network model was broken down into three sections: training, validation, and testing. In determining the data set used in each section, the methodology frequently preferred by the researchers and with an ideal accuracy was used [31][32][33]. Of the dataset, 56 were reserved for training the model, 12 for validation, and 12 for testing. A neuron is a fundamental computational element found in the buried layers of MLP network models [34]. One of the challenges in the design of MLP neural network models is optimizing this computational element in the hidden layer. The challenge stems from the fact that there is no standard approach for calculating the number of neurons [35]. In order to overcome this difficulty, the outputs of ANN models developed with different neuron numbers were analyzed, and the number of neurons was optimized by choosing the model with the highest accuracy. The same method was used to optimize the number of neurons to be used in the developed MLP neural network, and the model with nine neurons in the hidden layer was preferred. The diagram showing the basic structure of the ANN model developed for the optimization of the Nu number is given in Figure 4. gime-IV, respectively. The ANN model, which is one of the important artificial intelli gence algorithms, was used as the analysis methodology. ANN models are an engineering tool with high performance even in the simulations of complex structures where there is no functional relationship between them due to their advanced systematics [23,24]. In the ANN model, which is designed to estimate the Nu number, the multi-layer perceptron (MLP) network model, which has a reliable estimation capability with its strong architec ture, is used [25][26][27][28]. Each layer in the topology of MLP networks is directly coupled to the layer above it [29]. Data is input into the system at the input layer, which is the firs layer of an MLP network. Each MLP network has at least one hidden layer, which is the layer connected to the input layer after that. The output layer, the final layer of the MLP neural network and the layer following the hidden layer, is where simulation results are obtained. The symbolic architecture in this regard is offered in Figure 3. In the training of MLP neural networks, previously obtained data sets were used. The MLP network model built for calculating the Nu number utilized a total of 80 numerically acquired data sets. One of the most significant phases in the design process was optimiz ing and organizing the data set utilized in training ANN models [30]. The data set re quired to create the network model was broken down into three sections: training, vali dation, and testing. In determining the data set used in each section, the methodology frequently preferred by the researchers and with an ideal accuracy was used [31][32][33]. O the dataset, 56 were reserved for training the model, 12 for validation, and 12 for testing A neuron is a fundamental computational element found in the buried layers of MLP net work models [34]. One of the challenges in the design of MLP neural network models i optimizing this computational element in the hidden layer. The challenge stems from th fact that there is no standard approach for calculating the number of neurons [35]. In orde to overcome this difficulty, the outputs of ANN models developed with different neuron numbers were analyzed, and the number of neurons was optimized by choosing th model with the highest accuracy. The same method was used to optimize the number o neurons to be used in the developed MLP neural network, and the model with nine neu rons in the hidden layer was preferred. The diagram showing the basic structure of th ANN model developed for the optimization of the Nu number is given in Figure 4. In the MLP network model, the Levenberg-Marquardt training algorithm was pre ferred as the training algorithm. The Levenberg-Marquardt training algorithm is one o the widely used training algorithms in MLP neural networks due to its advanced learning ability [36,37]. Tan-Sig and Purelin transfer functions are preferred as transfer function in the hidden and output layers of the MLP network, respectively. The models of th transfer functions used are given below [38,39]: After completing the design phase of the MLP network model, it is time to analyz the training, learning, and prediction performance of the model. First of all, the training and learning accuracy of the MLP neural network was examined, and it was ensured tha the training phase was completed in an ideal way. In order to ensure that the network In the MLP network model, the Levenberg-Marquardt training algorithm was preferred as the training algorithm. The Levenberg-Marquardt training algorithm is one of the widely used training algorithms in MLP neural networks due to its advanced learning ability [36,37]. Tan-Sig and Purelin transfer functions are preferred as transfer functions in the hidden and output layers of the MLP network, respectively. The models of the transfer functions used are given below [38,39]: Pureline(x) = x.
After completing the design phase of the MLP network model, it is time to analyze the training, learning, and prediction performance of the model. First of all, the training and learning accuracy of the MLP neural network was examined, and it was ensured that the training phase was completed in an ideal way. In order to ensure that the network model can accurately predict the Nu number, the compatibility of the outputs obtained from the network model with the target data was examined. Following that, the coefficient of determination (R) and Mean Squared Error (MSE) parameters, which are commonly utilized in the literature, were calculated and thoroughly studied. The formulae used to calculate the performance parameters are as follows: [40,41]: The margin of deviation (MoD) values, which express the percentage divergence between the neural network outputs and the target data, were also calculated and evaluated to further the performance study of the ANN model. The equation used to calculate the MoD values is given below [42][43][44][45]:

Results and Discussion
The process of making ANN models ready to make predictions by learning the relationship between the data is called the "training phase". The training phase of an ANN model must be completed in order for the model's prediction accuracy to be accurate. One of the data used in the analysis of the training and learning processes of ANN models is the examination of the training performance graphics of the model. In Figure 5, the training performance graph of the developed MLP neural network is given. When looking at the graph depicting the estimated MSE values for each data set divided into three parts, it is clear that the MSE values are high at the start of the ANN model's training phase. After each epoch, it is seen that the MSE values are decreasing, and the ideal MSE value is reached at the 35th epoch. With the MSE values reaching the most ideal value, the most ideal training performance point, which is expressed with a dotted line, is reached for the three data groups.
The results obtained from the training performance graph show that the training phase of the developed MLP network model is ideally completed with the most ideal performance values.
Another methodology used in the analysis of training performance is the analysis of error histograms, which show the errors obtained from the training phase. In Figure 6, the error histogram showing the errors obtained from the training phase of the MLP neural network is presented. When the error values on the x-axis of the error histogram are considered, it can be seen that the values are very low. Looking at the data histogram in which the error values are expressed for three separate data sets, it is seen that the errors are concentrated very close to the zero error line, which is symbolized by the yellow line. These results, obtained from the analysis of the error histogram, confirm that the training phase of the MLP neural network model was completed with very low errors. In Figure 7 ing performance graph of the developed MLP neural network is given. When looking at the graph depicting the estimated MSE values for each data set divided into three parts, it is clear that the MSE values are high at the start of the ANN model's training phase. After each epoch, it is seen that the MSE values are decreasing, and the ideal MSE value is reached at the 35th epoch. With the MSE values reaching the most ideal value, the most ideal training performance point, which is expressed with a dotted line, is reached for the three data groups.  Another methodology used in the analysis of training performance is the analysis of error histograms, which show the errors obtained from the training phase. In Figure 6, the error histogram showing the errors obtained from the training phase of the MLP neural network is presented. When the error values on the x-axis of the error histogram are considered, it can be seen that the values are very low. Looking at the data histogram in which the error values are expressed for three separate data sets, it is seen that the errors are concentrated very close to the zero error line, which is symbolized by the yellow line. These results, obtained from the analysis of the error histogram, confirm that the training phase of the MLP neural network model was completed with very low errors. In Figure  7    The fact that the MSE values are as low as possible also shows the closeness of the outputs to be obtained from the developed ANN model to the truth. When the MSE values calculated from the data points are reviewed, the MSE values calculated for each of the four flow regimes are very near to zero. The closeness of the line denoting the MSE values to zero clearly confirms that the MLP network model has been trained to reach values very close to the truth. Analyzing the compatibility of the outputs obtained from the developed MLP neural network with the target data plays a vital role in the evaluation of the prediction performance of the neural network. In Figure 8, both values are shown on the same graph so that the harmony of output and target data can be clearly seen. When the states of the data points expressing the Nu values estimated for four different flow regimes and the Nu values calculated numerically are studied, we see that the ANN model outputs and the target data are in perfect harmony. The congruence of the output and target data for each data point shows that the designed neural network model can predict Nu values for all four flow regimes with very high accuracy.  The fact that the MSE values are as low as possible also shows the closeness of the outputs to be obtained from the developed ANN model to the truth. When the MSE values calculated from the data points are reviewed, the MSE values calculated for each of the four flow regimes are very near to zero. The closeness of the line denoting the MSE values to zero clearly confirms that the MLP network model has been trained to reach values very close to the truth. Analyzing the compatibility of the outputs obtained from the developed MLP neural network with the target data plays a vital role in the evaluation of the prediction performance of the neural network. In Figure 8, both values are shown on the same graph so that the harmony of output and target data can be clearly seen. When the states of the data points expressing the Nu values estimated for four different flow regimes and the Nu values calculated numerically are studied, we see that the ANN model outputs and the target data are in perfect harmony. The congruence of the output and target data for each data point shows that the designed neural network model can predict Nu values for all four flow regimes with very high accuracy.
Examining the MoD values, which express the percentage deviation between the ANN outputs and the target data, plays a vital role in determining the error rates of the model. In Figure 9, the calculated MoD values of each data used in the development of the ANN model are shown for the four flow regimes. When looking at the MoD values produced for four distinct flow regimes, it is evident that the data points are quite close to the zero error line. The closeness of the data points expressing the MoD values to the zero error line means that it is very low. These low MoD values confirm that the designed MLP neural network is capable of predicting Nu values for four different flow regimes with very low and acceptable deviations. the same graph so that the harmony of output and target data can be clearly seen. When the states of the data points expressing the Nu values estimated for four different flow regimes and the Nu values calculated numerically are studied, we see that the ANN model outputs and the target data are in perfect harmony. The congruence of the output and target data for each data point shows that the designed neural network model can predict Nu values for all four flow regimes with very high accuracy. Examining the MoD values, which express the percentage deviation between the ANN outputs and the target data, plays a vital role in determining the error rates of the model. In Figure 9, To investigate the ANN model error rates in greater depth, the discrepancies between the target values and the ANN model outputs were calculated for each data set utilized in the model training and displayed in Figure 10. When the difference data shown sepa- To investigate the ANN model error rates in greater depth, the discrepancies between the target values and the ANN model outputs were calculated for each data set utilized in the model training and displayed in Figure 10. When the difference data shown separately for each of the four flow regimes are examined, for each data point, the variations between the goal values and the ANN outputs are quite small. The fact that the differences are so low indicates that the outputs obtained from the developed MLP neural network are very close to the real data. These results clearly prove that the developed ANN model is designed to predict the Nu number in each flow regime with very low error. For the compatibility between the Nu values obtained from the MLP neural network and the real data more clearly and to be sure of the estimation accuracy of the model, the estimation and target data were placed on different axes of the same graph, and the obtained results were evaluated. In Figure 11, Nu values obtained from the ANN model and target values for four different flow regimes are shown. When the state of the data points is analyzed, it is seen that each data point is located on the zero error line. It should also be noted that the data points remain within the ±10% error band. The results obtained from the graphs given for the four flow regimes clearly show that the developed ANN model can predict the Nu number for each flow regime with ideal accuracy. These results clearly prove that the developed ANN model is designed to predict the Nu number in each flow regime with very low error. For the compatibility between the Nu values obtained from the MLP neural network and the real data more clearly and to be sure of the estimation accuracy of the model, the estimation and target data were placed on different axes of the same graph, and the obtained results were evaluated. In Figure 11, Nu values obtained from the ANN model and target values for four different flow regimes are shown. When the state of the data points is analyzed, it is seen that each data point is located on the zero error line. It should also be noted that the data points remain within the ±10% error band. The results obtained from the graphs given for the four flow regimes clearly show that the developed ANN model can predict the Nu number for each flow regime with ideal accuracy.
Performance parameters were calculated and evaluated for the MLP neural network model, which was developed to estimate Nu number values for four different flow regimes. The R value calculated for the training phase of the neural network was 0.99998, the R value calculated for the validation phase was 0.99322, and the R value calculated for the testing phase was 0.99938. The R values' proximity to zero shows that the ANN model prediction accuracy is quite high. The MSE and MoD values given in Table 5 also show that the prediction accuracy of the model is very high, and the developed ANN model is an ideal tool that can be used to estimate the Nu number in different flow characteristics. The values of the Nusselt number in comparison to Mahapatra and Gupta's [46] are presented in Table 6. For two different examples where the Prandtl number is Pr = 1.5 and Pr = 0.5, the numerical values of the Nusselt number are obtained by altering the velocity ratio parameter. We saw a decent match in both instances, which gives us confidence in the present numerical results.  Performance parameters were calculated and evaluated for the MLP neural network model, which was developed to estimate Nu number values for four different flow regimes. The R value calculated for the training phase of the neural network was 0.99998, the R value calculated for the validation phase was 0.99322, and the R value calculated for the testing phase was 0.99938. The R values' proximity to zero shows that the ANN model prediction accuracy is quite high. The MSE and MoD values given in Table 5 also show that the prediction accuracy of the model is very high, and the developed ANN model is an ideal tool that can be used to estimate the Nu number in different flow characteristics. The values of the Nusselt number in comparison to Mahapatra and Gupta's [46] are presented in Table 6. For two different examples where the Prandtl number is Pr = 1.5 and Pr = 0.5, the numerical values of the Nusselt number are obtained by altering the velocity ratio parameter. We saw a decent match in both instances, which gives us confidence in the present numerical results.

Conclusions
We developed artificial neural networking model for the convective heat transfer in stagnation point non-Newtonian fluid flow by using two inclined stretching surfaces, namely flat and cylindrical surfaces. The heat transfer flow field is manifested with heat generation, heat absorption, thermal radiations, magnetic field, and temperature stratification. Four different thermal flow regimes are considered and the corresponding Nusselt number is evaluated by varying thermophyiscal flow variables, namely Prandtl number, temperature stratification parameter, heat generation parameter, and heat absorption parameter. Different input parameters were used to create the feed forward back propagation multilayer perceptron network model. The training phase consumed 80% of the data set, whereas the testing phase consumed 20%. The Bayesian regularization training procedure was used to develop the network estimation performance, and the compatibility between the estimate values and the target data was investigated. The outcomes by the use of MoD and MSE admitted the accuracy of the artificial neural networking model for the estimation of convective heat transfer in various thermal flow regimes. Besides this, we have noticed R = 0.99998 for training phase, 0.99322 for validation phase, and 0.99938 for the testing phase. Such proximity to zero subject to R for the ANN model guarantees that the accuracy is quite high. Data Availability Statement: The adopted methodology can be offered upon request by readers.

Acknowledgments:
The authors would like to thank Prince Sultan University, Saudi Arabia, for the technical support through the TAS research lab.

Conflicts of Interest:
The authors declare no conflict of interest.