Neural computational characterization of stokesian micropolar casson fluid flow through and around a darcy porous sphere using feed forward neural networks

The Artificial Neural Networks provide a visually intuitive, computationally efficient framework for approximating arbitrarily nonlinear physical systems, circumventing the high computational costs commonly associated with all conventional numerical solvers. In this study, we investigated the flow kinematics of a Newtonian fluid flowing over/around a porous sphere, which is embedded with a Micropolar Casson fluid through the inner permeable region. Using a stream-function formulation, we reframe governing nonlinear flow equations and obtain analytic solutions. The rationale behind this approach is that those exact solutions then serve as the baseline dataset required to train and validate the neural network. The ANN architecture that is proposed in the work is subsequently utilized to provide mappings of stream-function responses for a range of physical parameters controlling the flow regime. In particular, we investigate the perturbation of local stream-function distribution arising from variations in the couple stress parameter, Darcy number, Casson fluid parameter, and azimuthal angle. For a quantitative measure of how well the ANN predicted it, we compute a variety of conventional error metrics measured in terms of mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), Pearson Correlation Coefficient (R), normalized root mean square error (NRMSE) and maximum absolute errors (MaxAE) as well as symmetric MAE before carrying out more specific evaluations through statistical comparisons using SMAPE. For a more concrete footing of these results, relative performance with respect to a variety of traditional machine learning tools is compared. The comparative baseline includes Multiple linear, Ridge, Lasso, Poly (Polynomial), Decision Tree, Random Forest, Support Vector, and Gaussian Process regressions. Results show that, among all the models considered, ANN achieves consistently minimal residual errors and the strongest correlation with analytical baselines, thereby significantly validating its predictive accuracy and generalization ability. Thus, this neural network architecture serves as an extremely powerful surrogate model for complex fluid-flow dynamics in porous media.