Modeling and prediction of Carreau fluid flow with generalized thermal conductivity using hybrid artificial neural network and symbolic regression

This study introduces a novel hybrid framework integrating artificial neural networks and symbolic regression to analyze boundary layer flow and heat transfer in Carreau fluids over a shrinking sheet in a parallel free stream, incorporating the effects of generalized thermal conductivity. This investigation is motivated by the limitations of constant-conductivity models, which fail to describe realistic thermal transport in non-Newtonian systems encountered in polymer processing, biomedical flows, and energy devices. The governing nonlinear partial differential equations are reduced to a system of coupled ordinary differential equations using the Lie symmetry method and solved numerically to identify dual similarity solutions. A linear stability analysis based on the smallest eigenvalue confirms that only the upper solution is physically realizable. To enable fast and generalized prediction of flow and thermal characteristics, an artificial neural network (ANN) trained via the Levenberg–Marquardt algorithm achieves mean absolute errors below 1% for the skin-friction coefficient and Nusselt number, demonstrating excellent agreement with numerical simulations while significantly reducing computational cost compared to numerical solvers. To further enhance interpretability, symbolic regression (SR) is employed as a post-processing step to extract compact analytical expressions from the ANN predictions, effectively bridging the gap between data-driven modeling and physical insight. The hybrid ANN–SR framework accurately reproduces both solution branches and reflects the influence of the power-law index on flow and thermal responses. This approach offers a fast, interpretable, and generalizable modeling strategy for non-Newtonian boundary-layer dynamics with improved predictive precision and transparency, possessing strong potential for real-time optimization in advanced engineering applications.