Neural network study of gyrotactic microorganism dynamics in nanofluid through porous stretched surface

In this article, the effects of swimming gyrotactic microorganisms on magnetohydrodynamic nanofluid flow through a porous medium governed by Darcy–Forchheimer law are investigated. The nonlinear coupled mathematical model is solved using a hybrid algorithm that combines MATLAB's bvp4c solver with the Levenberg–Marquardt technique (LMT). The proposed LMT-based neural network model exhibits excellent agreement with the reference solutions, achieving extremely low mean squared error (MSE) values of 10 − 10 , confirming its robustness and efficient convergence. The reliability of the proposed hybrid methodology is confirmed through excellent agreement between neural network predictions and numerical solutions. Parametric studies indicate that the velocity profile F ′ ( η ) enhances with higher motile microorganism parameter (Nr) and bioconvection Rayleigh number (Rb), but declines with increasing Darcy permeability parameter ( β D ) and magnetic parameter M. Temperature θ ( η ) rises with thermophoresis ( N t ), Brownian motion ( N b ), Eckert number (Ec) and magnetic parameter M, while decreasing for larger Prandtl number (Pr). The nanoparticle concentration ϕ ( η ) is diminished for higher Schmidt (Sc) and Brownian motion ( N b ) parameters, whereas it is augmented under stronger thermophoretic effects ( N t ). Similarly, the motile microorganism profile χ ( η ) is suppressed by larger microorganism Schmidt number ( Sc m ), Péclet number ( P e ), and microorganism concentration difference ratio ( Ω d ), indicating reduced motile microorganism density under these conditions. These results demonstrate the effectiveness of the hybrid bvp4c–LMT framework in accurately capturing complex nonlinear transport and bioconvective dynamics in gyrotactic nanofluids. • Developed a hybrid bvp4c–Levenberg–Marquardt neural network framework. • Modeled MHD nanofluid flow with gyrotactic microorganisms in porous media. • Achieved very low mean squared error (MSE ≈ 10 −10 ) ensuring robustness. • Velocity enhanced by motile microorganism (Nr) and bioconvection Rayleigh (Rb). • Temperature rises with thermophoresis, Brownian motion, and Eckert number. • Microorganism density suppressed by higher Sc, Pe, and concentration ratio (Ωd).