magnetic hydrodynamic flow and heat transfer of williamson nanofluids in a porous medium impact of chemical reactions and melting effects

Radiation and chemical reaction effects on the steady magnetohydrodynamic (mhd) boundary layer flow of williamson nanofluid through a porous medium over a horizontally linearly stretching sheet are numerically investigated, incorporating coupled influences of melting heat transfer and nanoparticle dispersion. the governing partial differential equations are reduced to a system of nonlinear ordinary differential equations using similarity transformations and solved via the fourth-order runge-kutta (rk-4) method to generate reference datasets. a novel supervised machine learning framework, feed-forward neural network optimized with the backpropagated levenberg-marquardt algorithm (ffnn-blma), is proposed, trained on 1001 data points with 70% training, 15% validation, and 15% testing splits. the ffnn-blma yields exceptional predictive accuracy with absolute errors ranging from 10⁻⁸ to 10⁻¹⁰ across velocity temperature  and concentration  profiles, validated through 10-fold cross-validation, error histograms, regression analysis, and curve superposition. parametric studies reveal that increasing the melting parameter  enhances velocity and reduces temperature, while the chemical reaction parameter  diminishes concentration trends consistent with prior literature. skin friction, nusselt, and sherwood numbers are computed to quantify engineering performance. the ffnn-blma outperforms traditional rk-4 and analytical methods in accuracy, convergence, and computational efficiency, establishing a robust, discretization-free paradigm for solving complex non-newtonian multi-physics flow problems with potential extension to fractional-order systems.