Neuro-swarms intelligent computing using Gudermannian kernel for solving a class of second order Lane-Emden singular nonlinear model

The present work is to design a novel Neuro swarm computing standards using artificial intelligence scheme to exploit the Gudermannian neural networks (GNN)accomplished with global and local search ability of particle swarm optimization (PSO) and sequential quadratic programming scheme (SQPS), called as GNN-PSO-SQPS to solve a class of the second order Lane-Emden singular nonlinear model (SO-LES-NM). The suggested intelligent computing solver GNN-PSO-SQPS using the Gudermannian kernel are unified with the configuration of the hidden layers of GNN of differential operators for solving the SO-LES-NM. An error based fitness function (FF) applying the differential form of the differential model and corresponding boundary conditions. The FF is optimized together with the combined heuristics of PSO-SQPS. Three problems of the SO-LES-NM are solved to validate the correctness, effectiveness and competence of the designed GNN-PSO-SQPS. The performance of the GNN-PSO-SQPS through statistical operators is tested to check the constancy, convergence and precision.


Introduction
The singular based models have gotten much importance due to its valued applications in physiology, physics and mathematics submissions. The Lane-Emden based model is one the historic paramount singular model introduced by famous astrophysics Lane and the further explored by Emden [1,2]. The general form of the second order Lane-Emden singular nonlinear model (SO-LES-NM) is given as [3]: 1 2 ( ) ( ) ( , ) 0, , 0, 0 1, where 1 i and 2 i are constants,  is the shape factor, while ( , ) g v  is known as the real valued continuous function. The singular Lane-Emden nonlinear systems are widely implemented to solve an assortment of phenomena in the fields of physical science [4], gaseous star density [5], electromagnetic theory [6], morphogenesis study [7], stellar structure model [8], study of mathematical physics [9], oscillating magnetic fields [10], dusty fluid system [11] and an isotropic standard [12]. The singular models due to singularity are found to be hard and grim at the origin. A small number of numerical and analytical approaches are available to tackle such nonlinear singular systems are given in these references [13][14][15][16].
These prospective and potential submissions demonstrated the importance, value and significance of the numerical stochastic based computing solvers in terms of exactitude, stability and convergence. Consequently, the novel features of the Gudermannian neural network (GNN) are designed together with particle swarm optimization (PSO) and the sequential quadratic programming scheme, i.e., GNN-PSO-SQPS to solve the SO-LES-NM.
The motive of this research work is to solve SO-LES-NM by integrating the intelligent computing approach based on the GNN-PSO-SQPS. The innovative features of the GNN-PSO-SQPS are given as follows: • A novel computingGNN-PSO-SQPSintelligentsolveris exploited and explored using the GNN along with the hybrid-combination of PSO-SQPS.
• The designed GNN-PSO-SQPS are tested precisely and accurately for solving three different problems of the SO-LES-NM. • The coinciding of the outcomes achieved by the proposed GNN-PSO-SQPS and the exact  results demonstrations the correctness of the scheme to solve SO-LES-NM.  • The results through proposed GNN-PSO-SQPS for single/multiple runsvia performance investigations of mean, root mean square error (RMSE), semi inter quartile range(S-I-R), Theil's inequality coefficient (TIC), median and standard deviation certified the competence, consistency, precision, accuracy and correctness of the designed GNN-PSO-SQPS. The rest parts of this research paper are given as: The methodology is given in Sec 2, the performance indices information is given in Sec 3, the detail of numerical results together with future research clarifications is given in Section 4.

Materials and method
In this section, the design of the differential operator GNN is presented to solve the SO-LES-NM. The detail of the differential model, fitness function (FF) and optimization using the suggested PSO-SQPS are provided.

Proposed procedure: Gudermannian function
The models based on neural network are familiar to provide the reliable, standardized and consistent solutions for a number of applications indifferent fields. In the below modeling, ˆ( ) v  represents the obtained outcomes from the GNN-PSO-SQPS together with its n th derivatives are given as: To solve the SO-LES-NM, the formulation of FF using the mean squared error metric is written as: While the networks presented in (6,7), used Gudermannian function (3) and its derivatives as a activation function. Now after learning of the weights of the networks, one can optimize the fitness/cost function in (6,7) and accordingly the solution of system (1) is approximately by proposed methodology.

Optimization of the network: PSO-SQPS
The capability of the ANNs based optimization models to solve the SO-LES-NM using the designed GNN-PSO-SQPS.
PSO is used as an alteration of the genetic algorithm and work as a global search method [38][39]. PSO is an easy implementation, needs less memory and global search optimization process introduced in the previous century [40]. Recently, PSO is applied in many applications like as Some recent PSO applications are traveling salesman problem [41], SFO-DTC induction motor drive [42], to evaluate the parameters of the reaction kinetic parameters [43], prediction of asphalting precipitation [44], reducing cost and increasing reliability [45] and object detection in autonomous driving [46].
To modify the PSO parameters, the scheme provides optimal iterative solutions, where Vi and Xi denote the velocity and position, 1  and 2  are the constant values of the accelerations, while [0,1]   shows the weight inertia vector. In order to perform the rapid convergence, the global PSO approach is hybridized with an appropriate local search scheme taking the PSO results. Therefore, an operative local search scheme named as SQS is executed to regulate the obtained outcomes through the GNN-PSO-SQPS. Some latest applications of SQS are transient heat conduction model [47], geometric optimization of radioactive enclosures [48], nonlinear predictive control model [49], cognitive radio system [50] and optimal management of automated vehicles at intersections [51]. The detail of the optimization procedure using the hybrid of PSO-SQPS are tabulated in the pseudo code Table 1. Table 1. Pseudo code for the GNN-PSO-SQPS to solve the SO-LES-NM.

PSO process starts
Step-1: Start: Generate the arbitrarily initial swarms and adjust the PSO parameters using the optimoptions.
Step-2:Fitness Scheme: Examine the fit values to each particle in the Eq. (5).
Step-3: Ranking: Rank all those element that have minimum standards of the FF.
Step-4: Stopping Measures: Stop, when one of the below condition meets.
Step-6: Elevation: Repeat the steps 02-06 until the whole flights are completed.
Step-7: Storage: store the best fit and designated as WPSO.

End of PSO-SQPS Data Generations:
The process PSO-SQPS repeats 100 times to find a comprehensive data-set of the optimization process for the SO-LES-NM

Performance form
The presentation of two different measures for solving the SO-LES-NM are constructed in terms of the RMSE and TIC that are executed to verify the proposed PSO-SQPS, the mathematical notations of these procedures are given as: AIMS Mathematics Volume 6, Issue 3, 2468-2485.

Result simulations
The detailed simulation based results of the numerical results through the GNN-PSO-SQPS for the SO-LES-NM are described in this section.
For the Eq (12), the FF becomes as: The true solution of the Eq (12) is For the Eq (14), the FF becomes as: The true solution takes the form as For the Eq (16), the FF becomes as: The true solution of the Eq (16) is The optimization for all the examples of the SO-LES-NM optimized by the hybrid of PSO-SQPS using the Gudermannian activation function for hundred independent runs to get the system variables of the parameter. A set of the best weights authenticate the numerical results by taking 10 neurons and the mathematical notations of these proposed outcomes is written as:

Conclusions
The present research investigations are associated to present a novel Gudermannian neural network to solve the nonlinear second order singular Lane-Emden system using the hybrid computing GNN-PSO-SQPS framework involving singularities at the origin. An error based objective function is optimized by using the global capability of particle swarm optimization and fast/rapid sequential quadratic scheme using 10 numbers of neurons throughout the present study. The design of the Gudermannian neural network is successfully exploited to solve the nonlinear second order singular Lane-Emden system. The obtained results using the process of optimization through the Gurmannian kernels have been compared with the true solutions for three different examples to check the precision and correctness of the suggested GNN-PSO-SQPS. One can find that the obtained and exact results overlapped over one another and examined the accuracy of order 5 to 7 decimal places. Furthermore, the performance of the scheme is investigated through the RMSE and TIC operators by taking the mean, worst and best AE values for all examples of the nonlinear second order singular Lane-Emden system. It is observed that the best values lie around 10 -06 -10 -08 , while the mean results and even the worst results lie in the good ranges for all the measures. The statistical clarifications for the 100 independent trials are implemented to the nonlinear second order singular Lane-Emden system in terms of the gages minimum, median, standard deviation, maximum, semi inter quartile range, mean authenticates the trustworthiness, robustness, accurateness and exactness of the proposed GNN-PSO-SQPS that is identified further by the performance measures of RMSE and TIC.