Evolutionary Integrated Heuristic with Gudermannian Neural Networks for Second Kind of Lane–Emden Nonlinear Singular Models

: 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


Introduction
The singular models have many appreciated applications in physics, physiology, engineering, and mathematics. The paramount Lane-Emden model is a historical model, which is famous due to singularity and presented by Lane and Emden [1,2] a few centuries ago by working on the performance of thermal gas and the state of thermodynamics [3]. The generic form of the Lane-Emden nonlinear singular models (LE-NSM) is written as [4]: u (t) + η t u (t) + g(t, u) = 0, η ≥ 0, 0 < t ≤ 1, u(0) = I 1 , u (0) = I 2 , • A novel GNN-GAASM computing-based stochastic solver is designed, implemented, and exploited using differential continuous mapping of GNNs together with optimization with the hybrid combined heuristics of GAs and ASM; • The presented GNN-GAASM solver is tested accurately to effectively solve the three different examples of the nonlinear singular model; • The overlapping of the results obtained by the GNN-GAASM from the exact solutions show the consistency, precision, and correctness of GNN-GAASM to approximate the solution of the second kind of the LE-NSMs; • The obtained outcomes of proposed GNN-GAASM for multiple executions via different performance measures of mean, Nash Sutcliffe efficiency (NSE), semi-interquartile range (S.I.R), median, and variance account for (VAF) further enhanced the competence of the designed GNN-GAASM.
The other paper parts are provided as: Section 2 describes the model structure, Section 3 gives the optimization model detail, Section 4 gives the information of performance indices, Section 5 relates the detail of numerical solutions together with interpretations of the outcome. The conclusion details and future research clarifications are given in the final section.

Methodology
The current section describes the GNN operators, which are designed with the necessary explanation to solve the second kind of LE-NSM. The operations of the differential system, merit function (MF), and optimization procedures through the GAASM are also discussed.

Designed Methodology: GNN
The neural networks are familiar with delivering standardized as well as reliable solutions for numerous applications arising in a variety of diverse fields. In this modeling, u(t) shows the obtained results through GNN-GAASM and its nth derivatives, written as: where, n and m indicate the derivative order and number of neurons, respectively. The MF is z, while,q,w,p are the unknown weight vectors, which are defined as W = [q, w, p], for q = [q 1 , q 2 , q 3 , . . . , q m ],w = [w 1 , w 2 , w 3 , . . . , w m ] and p = [p 1 , p 2 , p 3 , . . . , p m ]. The Gudermannian function (GF) is written as: Using the GF given in the above equation, the approximate continuous mapping of differential operations is written as: For solving the second kind of the LE-NSM, the formulation of MF using the mean squared error metric is given as: where E denotes an unsupervised error function associated to the second kind of the LE-NSM, whereas, E 1 and E 2 are the respective error functions linked to boundary conditions of the model (1) as: where Nh = 1,û k = u(t k ), g(t, u) = g(t k , u(t k )), and t k = kh.

Network Optimization
The numerical solutions of the second kind of LE-NSM are acquired to optimize the GNN by applying the hybrid computing scheme, i.e., GAASM.
GA is one of the intelligent evolutionary computing schemes that is based on natural development. In the 7th decade of the 19th century, GA is discovered with the innovator's work of Holland [39], and later it is employed as a key leading derivative to optimize the models based on constrained/un-constrained arrangements. GA works through the optimal process of mutation, selection, heuristic, and crossover. GA is widely applied in many areas such as robotics, astrophysics, optics, digital communication, bioinformatics, signal processing, financial mathematics, nuclear power system, economics, chemical industry, and materials. Some recent submissions of GAs that works as an optimization model are wind power model [40], heart disease prediction [41], intrusion detection performance model [42], energy managing systems [43], metal-organic constructions [44], heterogeneous celebration [45], heartbeat systems [46], a study of military systems [47], aquatic weed model [48]. These presented applications inspired the authors to apply the GA as an optimization process using the GNN to find the approximate outputs of the second kind of LE-NSM.
GA hybridizes with local search ASM, i.e., local search approach to use the quick convergence by assigning the best GA values as a start/initial point. Hence, ASM is suitable to regulate the parameters. ASM has been implemented in many recent submissions, such as trade and industry load dispatch models [49], short-term hydrothermal supervision [50], bipedal walking robot dynamics [51], economic multiproduct manufacture [52], LNG process [53], model of heating in the thermal blow frame cycling [54], aircraft transportation [55], wind turbine support structures [56] and quadratic convex bilevel programming models [57]. In this study, the combination of GAASM is implemented to solve the second kind of LE-NSM, and the optimization process of GAASM is provided in Table 1. The variety of the procedure is introduced for selection, i.e., stochastic uniform, remainder, roulette, and tournament, for mutation, i.e., Gaussian, uniform and adaptive feasible, as well as for crossover, scattered, single point, two points, intermediate, arithmetic and heuristic, however, we set stochastic uniform for the selection, heuristics for the crossover and adaptive feasible for mutation. These parameter settings are adopted after a lot of experiments, knowledge, experience, and performance advantages on different applications in the presented study. Table 1. Pseudocode for the optimization GNN-GAASM for solving the second kind of LE-NSM.

GA procedure
Inputs   (5)-(7). Accumulate: Regulate function counts, W GA-Best , time, iterations and Fit for the current trials of ASM.

End of ASM Data Generations
Replicate the GAASM 100 times for a larger dataset to optimize the variables for the second kind of LE-NSM by using the GNN-GAASM for functioning the statistical interpretations.

Performance Procedures
The demonstration actions to solve the second kind of LE-NSM to authenticate the GAASM are constructed in terms of the Nash Sutcliffe efficiency (NSE) and variance account for (VAF), mathematically given as:

Result and Simulations
The comprehensive simulations for the numerical outcomes using GNN-GAASM to solve the second kind of LE-NSM are presented in this section.
Problem I: Consider the second kind of LE-NSM involving exponential functions is written as: For the above equation, the MF is given as follow: The exact solution is ln 2 (t 2 +1) . Problem II: Consider the second kind of LE-NSM is written as: For the above equation, the MF is given as follow: The exact solution is 3 t 2 +3 . Problem III: Consider the second kind of LE-NSM having an exponential function is written as: For the above equation, the MF is given as follow: The exact solution is 2 ln 2)t 2 . To optimize the second kind of LE-NSM based on all problems by functional the GAASM system using the activation GF for independent hundred executions to find the system parameter variables. The set of best weight validates the estimated numerical outcomes for 10 neurons. The mathematical form of the obtained results is given as: The best weights set for 10 neurons and comparison of the mean, exact, and the best results for all the problems of the second kind of LE-NSM is shown in Figure 1. The set of best weight is drawn by using Equations 16 to 18. It is seen that the overlapping of the mean, exact, and best results are performed for all the problems of the second kind of LE-NSM. These results assessment shown in Figure 1 indicates the correctness and exactness of the suggested GNN-GAASM. Figure 2 shows the performance investigations based on the TIC, ENSE, and EVAF operator together with the best, worst, and mean values of AE for all variants of the second kind of LE-NSM. In order to evaluate the performance measures for Problem I, the calculations of the best Fit, ENSE, and EVAF values lie 10 −10 -10 −12, and the mean values of the Fit, ENSE and EVAF lie 10 −4 -10 −6 . The performance of the best Fit, ENSE, and EVAF values for Problem 2 and 3 lie 10 −10 -10 −15 , 10 −6 -10 −10 , and 10 −7 -10 −10 , respectively, while the mean Fit, ENSE, and EVAF are close to 10 −5 . In order to measure the absolute error (AE), the best values have been calculated around 10 −6 to 10 −7 for Problem I, while for another two problems, the best AE lie 10 −5 -10 −6 . The mean AE values have also been noticed in suitable measures for all variants of the second kind of LE-NSM. The plots of the convergence measures for the Fit, ENSE, and EVAF together with the boxplots and histogram are given in Figures 3-5 for for all variants of the second kind of LE-NSM. It is indicated that most of the Fit, EVAF, and ENSE values for all the problems lie around 10 −5 to 10 −10 , 10 −6 to 10 −9 , and 10 −7 to 10 −9 , respectively. One can determine that the accurate, precise, and specific values of the ENSE and EVAF operators have been obtained for all variants of the second kind of LE-NSM.
Problem I Problem II Problem III   The plots of the convergence measures for the Fit, ENSE, and EVAF together with the boxplots and histogram are given in Figures 3-5 for for all variants of the second kind of LE-NSM. It is indicated that most of the Fit, EVAF, and ENSE values for all the problems lie around 10 −5 to 10 −10 , 10 −6 to 10 −9 , and 10 −7 to 10 −9 , respectively. One can determine that the accurate, precise, and specific values of the ENSE and EVAF operators have been obtained for all variants of the second kind of LE-NSM.      Table 2. The mathematical form of the S.I.R is one-half of the difference between the third and first quartiles, respectively. These statistics-based performances for all variants of the second kind of LE-  Table 2. The mathematical form of the S.I.R is one-half of the difference between the third and first quartiles, respectively. These statistics-based performances for all variants of the second kind of LE-NSM are found to be satisfactory and endorse the precision/accuracy of the proposed GNN-GAASM. The global demonstrations for all the variants of the second kind of LE-NSM using the proposed GNN-GAASM are given in Table 3 Table 4. The complexity analysis for the designed second kind of LE-NSM using GNN-GAASM using the generation, implemented time, and function calculations are given in Table 5. It is concluded that the average generations, implemented time together with the function calculations are around 39.1709, 383.0966, and 29,623.3967, respectively, for the second kind of LE-NSM using GNN-GAASM.     I  100  93  81  100  100  72  100  100  100  II  98  93  78  97  93  77  100  100  100  III  99  97  93  99  99  93  100 100 100

Conclusions
The current research work is related to design a novel Gudermannian neural network for solving the nonlinear Lane-Emden singular model of the second kind using GNN-GAASM containing the singular point at the origin using 10 neurons. The optimization is produced by the global skill of genetic algorithms and rapid modification of applicant solutions by working the local search through the active-set scheme. The solver based on Gudermannian computing intelligent neural network is designed with the layer structure neural network models for solving the second kind of nonlinear Lane-Emden singular model. The precision, convergence, and accuracy of the stochastic numerical solver are anticipated to attain the matching/overlapping results with the exact solutions having 6 to 8 decimal levels of accuracy for the second kind of nonlinear Lane-Emden singular model. Furthermore, statistical interpretations based on 100 executions for the second kind of nonlinear Lane-Emden singular model, in the form of maximum, minimum, median, mean, standard deviation, and semi-interquartile range, validate the trustworthiness, exactness, robustness, and correctness of the proposed GNN-GAASM that is specified further by the procedures of ENSE and EVAF.