In this paper, we propose a group Lasso regularization term as a hidden layer regularization method for feedforward neural networks. Adding a group Lasso regularization term into the standard error function as a hidden layer regularization term is a fruitful approach to eliminate the redundant or unnecessary hidden layer neurons from the feedforward neural network structure. As a comparison, a popular Lasso regularization method is introduced into standard error function of the network. Our novel hidden layer regularization method can force a group of outgoing weights to become smaller during the training process and can eventually be removed after the training process. This means it can simplify the neural network structure and it minimizes the computational cost. Numerical simulations are provided by using K-fold cross-validation method with $K=5$ to avoid overtraining and to select the best learning parameters. The numerical results show that our proposed hidden layer regularization method prunes more redundant hidden layer neurons consistently for each benchmark dataset without loss of accuracy. In contrast, the existing Lasso regularization method prunes only the redundant weights of the network, but it cannot prune any redundant hidden layer neurons.

A group L _1/2 regularization term is defined and introduced into the conventional error function for pruning the hidden layer nodes of feedforward neural networks. This group L _1/2 regularization method (GL _1/2 ) can prune not only the redundant hidden nodes but also the redundant weights of the surviving hidden nodes of the neural networks. As a comparison, the popular group lasso regularization (GL ₂ ) can prune the redundant hidden nodes, but cannot prune any redundant weights of the surviving hidden nodes, of the neural networks. A disadvantage of the GL _1/2 is that it involves a non-smooth absolute value function, which causes oscillation in the numerical computation and difficulty in the convergence analysis. As a remedy, the absolute value function is approximated by a smooth function, resulting in a smooth group L _1/2 regularization method (SGL _1/2 ). Numerical simulations on a few benchmark data sets show that, compared with GL ₂ , SGL _1/2 can achieve better accuracy and remove more redundant nodes and weights of the surviving hidden nodes. A convergence theorem is also proved for SGL _1/2 .

This study investigates the steady, incompressible magnetohydrodynamic (MHD) flow of a tangent hyperbolic nanofluid over an exponentially stretching porous sheet considering variable heat and mass fluxes, thermal radiation, Brownian motion, thermophoresis, porous‐medium resistance, and non‐Newtonian effects. Similarity transformations are applied to convert the governing partial differential equations into a nonlinear system of ordinary differential equations, which is solved numerically using MATLAB's bvp5c solver. The influences of suction/blowing, exponential stretching, magnetic field strength, Prandtl number, Schmidt number, Brownian motion, thermophoresis, thermal radiation, and tangent hyperbolic parameters on the velocity, temperature, and concentration fields are systematically examined. The results indicate that wall blowing increases the surface temperature and nanoparticle concentration, and , by 130.8% and 98.3%, respectively, compared with suction conditions. Increasing the exponential stretching parameter reduces and by 73.3% and 62.0%, respectively. Thermophoresis enhances and by 12.4% and 30.0%, whereas higher Prandtl and Schmidt numbers reduce them by 41.3% and 42.9%, respectively. The positive skin‐friction magnitude is further optimized using response surface methodology (RSM) with a face‐centered central composite design (CCD). The developed quadratic RSM model exhibits excellent predictive capability with and adjusted . The normalized skin‐friction magnitude ranges from 0.754773 to 1.260247, and ANOVA and sensitivity analysis identify the tangent hyperbolic parameter as the dominant factor, followed by suction/injection and Weissenberg number, with a significant interaction between the latter two parameters. The findings provide a numerical‐statistical framework for optimizing tangent hyperbolic nanofluid transport in thermal management, coating processes, and enhanced oil recovery applications.