Feed forward back propagation neural network with Levenberg–Marquardt training for MHD Casson hybrid nanofluid flow over an infinite vertical Riga plate

This study explores the time-dependent MHD flow of a Casson hybrid nanofluid over an infinite vertical Riga plate, incorporating Darcy–Forchheimer porous medium, bio-convective boundary conditions, modified Hartmann number, and Joule heating. The base fluid is blood containing gold ( $$Au$$ ) and copper ( $$Cu$$ ) nanoparticles. The governing nonlinear PDEs are transformed into dimensionless form using non-similarity transformations and solved numerically using the Crank–Nicolson implicit finite difference method, yielding velocity and temperature profiles. Results indicate that increasing the modified Hartmann number $$\left( {E \, = \, 0.0{-}2.0} \right)$$ and the permeability parameter $$(K \, = \, 1.0{-}3.0)$$ enhances velocity by approximately 14–18%, while higher thermal radiation $$\left( {Nr \, = \, 0.0{-}2.0} \right)$$ and heat source/sink $$\left( {Q \, = \, 0.0{-}1.0} \right)$$ increase temperature by 10–15%. The numerical solutions are validated against analytical perturbation methods, confirming their reliability and accuracy. To further analyze the data, the numerical dataset is employed to train an FFBPNN using the LM algorithm, which ensures rapid convergence and optimal weight adjustment. The trained ANN achieves low MSE values and a strong coefficient of determination (R = 0.999 for SFC, R = 0.979 for NN) with prediction accuracy of 97–99%, highlighting its robustness and efficiency. These results emphasize the capability of LM-trained FFBPNNs to accurately predict complex engineering quantities and evaluate parametric effects. Overall, the study provides valuable insights for potential applications in biomedical diagnostics and bioengineering systems, particularly in thermal management and fluid transport within biological environments.