Application of neural network analysis to study thermally stratified hybrid nanofluid over a variable thickness sheet

Abstract Enhancing the efficiency of thermal systems requires advanced working fluids with superior heat transfer characteristics. In this study, we investigate the thermally stratified magnetohydrodynamic (MHD) flow of a TiO $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>2</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> –Fe $$_3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>3</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> O $$_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>4</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> /H $$_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mn>2</mml:mn> <mml:mrow/> </mml:mmultiscripts> </mml:math> O hybrid nanofluid over a nonlinear stretching sheet with variable thickness. The model incorporates temperature-dependent viscosity and thermal conductivity, thermal radiation, internal heat generation, and a non-Fourier heat flux based on the Cattaneo–Christov theory. The governing partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations using appropriate similarity transformations. These equations are solved numerically using a shooting method combined with a Runge–Kutta scheme. Moreover, an artificial neural network (ANN) model is created based on the Levenberg–Marquardt algorithm to estimate velocity and temperature profiles as a surrogate model. ANN performance is measured by statistical measures of error such as mean squared error (MSE), root-mean-square error (RMSE) and regression coefficient ( R ), and it is found that they are in excellent agreement with the numerical results. Parametric study indicates that an increase in the magnetic parameter ( M ) decreases the velocity of the object considerably because of the Lorentz force and increases the temperature field. Temperature decreases as Prandtl number (Pr) and thermal stratification parameter (St) increase, and this means that the thermal diffusion is lower. The heat transfer rate increases with radiation parameter ( R ) and convective parameter ( $$B_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ). The results demonstrate that the hybrid nanofluid enhances heat transfer performance compared to single nanofluids, and the proposed ANN model provides an efficient predictive tool for complex nonlinear thermal systems.