A qualitative analysis of the artificial neural network model and numerical solution for the nanofluid flow through an exponentially stretched surface

This article aims to analyze the two-dimensional (2D) nanofluid (Ag/C 2 H 6 O 2 ) flow past an exponentially stretched sheet. The magnetic field impact, heat source/sink, and convection in the thermal profile are taken into account. The complexity of the problem is reduced by introducing a dimensionless group of functions. The reduced model is transformed into a system of first-order ordinary differential equations (ODEs). This system is further analyzed with the artificial neural network (ANN), which is trained using the Levenberg–Marquardt algorithm. The whole dataset is sub divided into three parts: training ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mn>70</mml:mn><mml:mi>%</mml:mi></mml:math> ), validation ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mn>15</mml:mn><mml:mi>%</mml:mi></mml:math> ), and testing ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mn>15</mml:mn><mml:mi>%</mml:mi></mml:math> ). The impact of nonlinear heat source/sink parameter, magnetic parameter, volume fraction of nanoparticles, and Prandtl number is displayed through graphs. The heat source, volume fraction, and the Prandtl number cause an increase in the thermal profile with its larger values. The magnetic parameter causes a decline in both the thermal and momentum boundary layers with its higher values. The analysis shows that the thermal energy profile is enhanced with the larger values of the volume fraction of silver nanoparticles and heat source. For each case study, the residual error (RE), regression line, and validation of the results are presented. The performance of the proposed methodology is numerically tabulated for the nanoparticle volume fraction shown in Table 3 , where the minimum absolute error (AE) is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mn>5.3373</mml:mn><mml:mi>e</mml:mi><mml:mo>−</mml:mo><mml:mn>11</mml:mn></mml:math> at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.05</mml:mn></mml:math> . Based on this, we recommend <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"><mml:mi>ϕ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.05</mml:mn></mml:math> for better performance. The AEs for the ANN and bvp4c are computed for the state variables in Tables for the magnetic parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mn>5,10</mml:mn></mml:math> , and 15. These tables show the overall performance of the ANN and further validate the present study. We have also validated the results of the ANN through the mean squared error graphically, where the accuracy of the proposed methodology is proven.