###### Abstract  
In this work, a new heuristic computing design is presented with an artificial intelligence approach to exploit the models with feed-forward (FF) Gudermannian neural networks (GNN) accomplished with global search capability of genetic algorithms (GA) combined with local convergence aptitude of active-set method (ASM), i.e., FF-GNN-GAASM to solve the second kind of Lane-Emden nonlinear singular models (LE-NSM). The proposed method based on the computing intelligent Gudermannian kernel is incorporated with the hidden layer configuration of FF-GNN models of differential operatives of the LE-NSM, which are arbitrarily associated with presenting an error-based objective function that is used to optimize by the hybrid heuristics of GAASM. Three LE-NSM-based examples are numerically solved to authenticate the effectiveness, accurateness, and efficiency of the suggested FF-GNN-GAASM. The reliability of the scheme via statistical valuations is verified in order to authenticate the stability, accuracy, and convergence. Guedermannian kernel; Lane-Emden model; Gudermannian neural networks; active-set method; numerical solutions; genetic algorithms +
Footnote †: journal: Applied Sciences  
0000-0001-9993-3300-3300-3300-3300-33000-33000-33000-33000-33000-3000  
where the shape factor is \(\eta\), \(g(t,u)\) is the real-valued continuous function, and \(I_{1}\) and \(I_{2}\) represent constants values. The LEM singular nonlinear models define a collection of phenomena in the gaseous star density [5], the physical area of science [6], the theory of electromagnetic [7], stellar construction system [8], morphogenesis [9], physics-based mathematical model [10], oscillating magnetic areas [11], an isotropic medium [12] and models of dusty fluid [13]. Solving the singular models is found to be grim and tough due to a singular point at the origin. Few analytical and numerical schemes are accessible to handle such singular nonlinear models are presented in these references [14; 15; 16]. All presented above schemes have their individual sensitivity, potential, efficiency, and correctness, as well as weaknesses, flaws, and demerits over each other. The extensive computing heuristic approach potential is to solve the singular systems applying the widespread capacity of artificial neural networks (ANNs) collectively with local and global based search approaches [17; 18; 19; 20; 21; 22; 23]. Few noteworthy illustrations contain neuro-intelligent computing approach to study the dynamics of convective heat transfer involving carbon nanotubes [24], dusty plasma nonlinear model [25], model of mosquito release in the heterogeneous atmosphere [26], Navier Stokes problems [27], singular functional differential model [28], HIV infection system of CD4+ T cells [29], plasma-based physics investigations [30], Thomas-Fermi singular system [31], prey-predator biological system [32], nanotechnology [33], killing well control system [34], biological model based on corneal shape [35], Jeffery Hamel flow problem [36], parameter estimation in biodiesel studies [37] and model of atomic physics model [38]. These potential applications proved the significance of the stochastic solvers on the basis of stability, convergence, and exactitude. Therefore, the novel design of the Gundermannian neural network (GNN) is exploited using the genetic algorithm (GA) and active-set method, i.e., GNN-GAASM for the second kind of LE-NSM. The basic aim of this study is to solve the second kind of LE-NSM by introducing a new intelligent scheme based on combined heuristics of GNN-GAASM.