<p>This article investigates the effect of the radiation parameter on MHD tangent hyperbolic nanofluid flow over a nonlinear stretching sheet. The governing PDEs are reduced to coupled nonlinear ODEs via similarity transformations, followed by the bvp4c solver in MATLAB. The results of earlier studies have shown that velocity boundary layer thickness decreases with increasing magnetic and Weissenberg numbers. In contrast, with increasing radiation and Brownian motion, the thickness of the thermal boundary layer increases. Using these numerical results, an Artificial Neural Network (ANN) is trained to improve computing performance and generate generalised predictions. The Levenberg–Marquardt (trainlm) technique is used in the ANN’s feed-forward backpropagation model to predict velocity, temperature, and concentration profiles based on eight input parameters. The main measures used to assess training success are mean squared error (MSE), gradient, validation failure (val fail), and adaptive learning rate (mu). Convergence is confirmed when the gradient drops dramatically during training from high starting values to as low as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1062_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(10^{-10}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. The learning rate is controlled by the mu value, which begins at about <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1062_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(10^{-5}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and gradually decreases to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1062_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(10^{-10}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>10</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, signifying stable optimization. Consistent model behaviour and the lack of overfitting are indicated by the validation failure being low, eventually approaching zero. The ANN achieves extremely low MSEs (as low as <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1062_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">\(2.341\times 10^{-9}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2.341</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>9</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for velocity, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1062_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">\(2.5744\times 10^{-11}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2.5744</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>11</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for temperature, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2025_1062_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">\(5.3561 \times 10^{-11}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>5.3561</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>11</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for concentration) throughout 270, 397 and 282 epochs respectively, and regression coefficients R=1. Error histograms are centred at zero, and regression plots confirm excellent agreement with numerical solutions. This work demonstrates that a precise, efficient, and scalable framework for studying complex radiative MHD non-Newtonian flows can be obtained by combining validated numerical modeling with ANN-based predictive modeling. Applications for this framework can be found in thermal engineering, biomedical transport, polymer processing, and advanced energy systems.</p>

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Machine learning-based numerical study of radiative MHD hyperbolic tangent nanofluid flow over a stretching sheet

  • Bishnu Charan Rout,
  • U. K. Saha,
  • O. D. Makinde

摘要

This article investigates the effect of the radiation parameter on MHD tangent hyperbolic nanofluid flow over a nonlinear stretching sheet. The governing PDEs are reduced to coupled nonlinear ODEs via similarity transformations, followed by the bvp4c solver in MATLAB. The results of earlier studies have shown that velocity boundary layer thickness decreases with increasing magnetic and Weissenberg numbers. In contrast, with increasing radiation and Brownian motion, the thickness of the thermal boundary layer increases. Using these numerical results, an Artificial Neural Network (ANN) is trained to improve computing performance and generate generalised predictions. The Levenberg–Marquardt (trainlm) technique is used in the ANN’s feed-forward backpropagation model to predict velocity, temperature, and concentration profiles based on eight input parameters. The main measures used to assess training success are mean squared error (MSE), gradient, validation failure (val fail), and adaptive learning rate (mu). Convergence is confirmed when the gradient drops dramatically during training from high starting values to as low as \(10^{-10}\) 10 - 10 . The learning rate is controlled by the mu value, which begins at about \(10^{-5}\) 10 - 5 and gradually decreases to \(10^{-10}\) 10 - 10 , signifying stable optimization. Consistent model behaviour and the lack of overfitting are indicated by the validation failure being low, eventually approaching zero. The ANN achieves extremely low MSEs (as low as \(2.341\times 10^{-9}\) 2.341 × 10 - 9 for velocity, \(2.5744\times 10^{-11}\) 2.5744 × 10 - 11 for temperature, and \(5.3561 \times 10^{-11}\) 5.3561 × 10 - 11 for concentration) throughout 270, 397 and 282 epochs respectively, and regression coefficients R=1. Error histograms are centred at zero, and regression plots confirm excellent agreement with numerical solutions. This work demonstrates that a precise, efficient, and scalable framework for studying complex radiative MHD non-Newtonian flows can be obtained by combining validated numerical modeling with ANN-based predictive modeling. Applications for this framework can be found in thermal engineering, biomedical transport, polymer processing, and advanced energy systems.