Prandtl–Eyring Nanofluid Transport for Gyrotactic Microorganisms With Heat Source/Sink Over a Permeable Stretching Sheet Using Machine‐Learning Computing

This study investigates the complex transport behavior of Prandtl–Eyring nanofluid (PENF) flow over a permeable stretching sheet, incorporating Brownian motion, thermophoresis, convective boundary conditions, heat source/sink, magnetic field, and gyrotactic microorganisms. Such flow models are essential for various engineering and bioengineering applications, including polymer sheet extrusion, metal forming, targeted drug delivery, and tissue engineering. The governing equations are solved numerically using MATLAB’s BVP5C solver and verified through artificial neural network (ANN) modeling, showing excellent agreement and validating the robustness of the computational approach. The ANN surrogate model demonstrates excellent agreement with BVP solutions for unseen parameters, achieving high accuracy with R ² ≈ 0.99 and very low error metrics. The ANN surrogate was evaluated on unseen values of different flow indicators and achieved values of approximately 0.994–0.999 with low MSE, RMSE, and MAE. The reported cases demonstrated prediction for unseen parameter values, with computational speedups of approximately 2.19–8.80 × over BVP5C. The analysis reveals that injection thickens the boundary layer and reduces flow velocity, while suction enhances heat and mass transfer. Increasing the Hartmann number strengthens Lorentz forces, compressing the boundary layer and elevating thermal gradients. Radiation and viscous dissipation further raise fluid temperature, enhancing energy transport but slightly increasing species diffusion. The Biot number and slip effects critically influence temperature, concentration, and microorganism distribution. Larger Prandtl and Biot numbers promote heat transfer, while the Eckert number suppresses it due to enhanced viscous heating. The Eyring–Powell parameter modulates non‐Newtonian shear behavior, and higher Schmidt numbers and stronger chemical reactions limit mass diffusion, affecting bioconvective stability.