A novel design of Gudermannian function as a neural network for the singular nonlinear delayed, prediction and pantograph differential models

: The present work is to solve the nonlinear singular models using the framework of the stochastic computing approaches. The purpose of these investigations is not only focused to solve the singular models, but the solution of these models will be presented to the extended form of the delayed, prediction and pantograph differential models. The Gudermannian function is designed using the neural networks optimized through the global scheme “genetic algorithms (GA)”, local method “sequential quadratic programming (SQP)” and the hybridization of GA-SQP. The comparison of the singular equations will be presented with the exact solutions along with the extended form of delayed, prediction and pantograph based on these singular models. Moreover, the neuron analysis will be provided to authenticate the efficiency and complexity of the designed approach. For the correctness and effectiveness of the proposed approach, the plots of absolute error will be drawn for the singular delayed, prediction and pantograph differential models. For the reliability and stability of the proposed method, the statistical performances “Theil inequality coefficient”, “variance account for” and “mean absolute deviation’’ are observed for multiple executions to solve singular delayed, prediction and pantograph differential models.

analysis; complexity analysis; statistical performances

Introduction
The models based on Lane-Emden (LE) have a variety of valuable applications in physiological, physical models, mathematics science and engineering studies. The celebrated form of the LE has a great significance due to the singular point and this model is explored by famous scientists Lane and Emden [1,2] working on the thermal gas performance and the thermodynamics state [3]. The literature/generic form of the LE model is given as [4][5][6][7]: where  is the shape factor, ( , ) g   is the continuous real valued based function, 1 u and 2 u are the initial conditions (ICs).
The LE based singular nonlinear systems define an area of physical sciences [8], gaseous density stars [9], stellar models [10], morphogenesis [11], oscillating fields [12], mathematical systems [13], dusty fluidics [14] and an isotropic source [15]. To present the solutions of the singular systems is not easy and considered tough due to the involvement of singularity at the origin. Only a few schemes in literature are available to solve the singular based LE types of models [16,17].
A variety of physical phenomena are investigated using the sense of differential systems, among them the delay differential (DD) form is the most prominent due to the vast applications in technology as well as engineering. The DD systems were introduced few centuries ago and has number of applications in the variety of scientific areas including a communication system, engineering models, population dynamics, transport systems and economic circumstances [18][19][20][21]. Few associated investigations of DD systems are the geometric functions reliability through DD systems to exploit the factors of delay-dependent studied by Beretta et al. [22]. The biological based mathematical systems based on the DD models were studied by the Forde [23]. The implementations based on the Galerkin wavelet scheme together with the Taylor series investigations to obtain the numerical outcomes of the DD system defined by the Frazier [24]. The coupled variation iteration approach to calculate the analytical outcomes of the DD system introduced by Rangkuti et al. [25]. The Runge-Kutta approach to calculate the numerical procedures of the DD systems is implemented by the Chapra [26]. Few more schemes have been implemented to solve the DD models are reported in the literature [27][28][29]. The PD system is considered using the ideas of the DD system. The prediction differential system is applied in weather forecasting, transport, stock markets, technology, engineering, astrophysics and biological networks [30]. The literature form of the DD and PD systems are provided in Eqs (2) and (3) as [31]: where ( )   shows the ICs and 1 ( ) r    represents the delayed form, while 1 r is delayed term, l and m are constants. The term w is the value derivative of  at l and  represents a small positive constant. As l     , then the necessary condition for the delay term bound should be 0 ≤ r1 ≤ | l − ε | such that system in Eq (2), remained DD equation.
When any of the value is added in  , it becomes prediction, i.e., 1 ( ) r    , where r is used as a prediction term and PD system is given as: The pantograph differential systems (PDSs) have a variety of submission due to its enormous significance in the engineering fields, biological models and science areas. Some well-known applications are light absorption in the solid, dynamical population networks, communication networks, control systems, infectious viruses, propagation systems, electronic frameworks, transports and quantum mechanism [32][33][34][35][36]. There are various techniques that have been implemented to treat the PDSs, e.g., one-dimensional transformation approach, Taylor polynomial method and Direchlet series scheme [37][38][39][40][41]. The generic form of the PDSs is given as: For solving the LE based singular systems, every technique has specific sensitivity, correctness, potential and efficiency, over and above, flaws, demerits and weaknesses. The wide-ranging computing potential schemes are used for the singular LE systems, DD models, PD systems and PDSs. To solve these singular systems, Gudermannian function is designed as a neural network optimized with the global scheme "genetic algorithms (GA)", local method "sequential quadratic programming (SQP)" and the hybridization of GA-SQP. The stochastic numerical schemes have been explored to solve various applications like HIV infection nonlinear system based infected latently cells [42][43][44][45], higher order singular nonlinear systems [46][47][48], mosquito dispersal nonlinear system [49], heat conduction system based human head [50], doubly singular systems [51,52] and SIR nonlinear dengue fever system [53].These well-known submissions authenticated the implication of the stochastic computing solvers in terms of stability, exactitude and convergence. Hence, the design of Gudermannian function that work as a neural network is never been applied to solve the LE systems, DD models, PD systems and PDSs by using the optimization through the GA-SQP procedures. Some novel topographies of the proposed Gudermannian neural network along with GA-SQP are summarized as:  A novel Gudermannian function is designed as a neural network is presented for solving the singular LE systems, DD models, PD systems and PDSs by using the optimization through the GA-SQP procedures.  The numerical solutions of the stochastic procedures for solving four different examples of the singular systems are found precise and accurate.  The analysis based on small and large neurons are effectively provided to authenticate the efficiency and complexity of the designed approach.  The matching of the best and mean outcomes obtained by the stochastic procedures authenticate the consistency, accuracy and perfection of the singular LE systems, DD models, PD systems and PDSs.  The reliability of the outcomes obtained by the proposed stochastic procedures through single/multiple executions via performance operators based on mean, Theil inequality coefficient (TIC), median (Med), variance account for (VAF), semi-interquartile range (SIR), maximum (Max) and mean absolute deviation (MAD) improve the capability of the scheme. The remaining parts of the paper are given as: Section 2 labels the designed structure along with the statistical performance, Section 3 indicates the details of numerical results together with clarifications of the results. The neuron analysis will be presented in Section 4. The concluding remarks are provided in the last Section.

Methodology
In this section, the Gudermannian function is presented as a neural network for the singular LE systems, DD models, PD systems and PDSs by using the optimization through the GA-SQP procedures. The differential operators, objective function and optimization-based procedures using the proposed scheme are also presented.

Designed procedures through Gudermannian function as a neural network
The artificial neural networks are considered important to form the consistent and steadfast solutions for frequent submission arising in the various fields. In this modelling, ˆ( ) indicates the obtained performances via the Gudermannian function as a neural network and its derivatives are described as: where, k shows the neurons and n represents the derivative. The objective function is y, whereas, s, w, . The mathematical form of the Gudermannian function is given as: The approximate form of the continuous mapping based differential operations is shown as:

Network optimization GA-SQP
In this section, the Gudermannian function as a neural network is designed and numerical solutions for solving the singular LE systems, DD models, PD systems and PDSs through the optimization of GA-SQP are presented.
The evolutionary intelligent computing approach "GA" is designed on the basis of natural growth. GA is introduced by Holland in the previous century and then it is used as a leading factor in optimization using the constrained/unconstrained systems [54]. GA shows the optimal performances of mutation, crossover, heuristic and selection and broadly implemented in various fields of robotics, Bioinformatics, optics, astrophysics, digital communication, financial based mathematics, signal processing, nuclear based power systems, chemical industry and economics. Recently, it is implemented in the optimization of wind power systems [55], pipe networks [56], intrusion detection system [57], energy management models [58], circularity error unified evaluation [59], heterogeneous celebration [60], drying process of carrot [61], 2D industrial packing problems [62] and aquatic weed systems [63]. These reputed submissions inspired the authors to optimize through GA the singular LE systems, DD models, PD systems and PDSs for finding the best proposed outputs.

Output:
The global best weight vectors are WGA-Best Initialization: Form a W, which is a weight vector having real elements to choose a chromosome. For the initial population, the design of 'W' is presented. To regulate the generations and assertions for the gaoptimset.
Fitness assessment: Proficient the fitness (E Fit ) in population for W using Eqs (8)

End of SQP Data Generations
The process of GASQP is replicate 50 times for a larger dataset for the singular LE systems, DD models, PD systems and PDSs using the optimization-based GA-SQP through the statistical clarifications.
The hybridization of global search with any local search method performs the rapid convergence using the hybridize with the local search scheme. The best GA values are assigned as an initial input. The local search SQP is applied to normalize the parameters. SQP has been applied in various directions, e.g., bilinear model predictive control of a HVAC system [64], optimization of the engineering models [65], optimal control of building HVAC&R systems [66], ESP-implemented wells [67], optimal coordination of automated vehicles at intersections [68], dynamic combined economic emission dispatch models [69], cost minimization of a hybrid photovoltaic, diesel generator, and battery energy storage system [70]. Figure 1 indicates the graphical illustrations based Gudermannian function as a neural network and optimization procedures of GA-SQP, while the details of the optimization steps based on GA-SQP are given in Table 1 for singular LE systems, DD models, PD systems and PDSs. The settings of the parameter are adopted based on the experiments, experience, performance advantages and knowledge using different applications in the current work. The Problem

Stopping Standards
Obtained.

Fitness Valuation
Stopping Standards attained.

Performance procedures
In this section, the performances through different statistical gages T.I.C, VAF, S.I.R and MAD is provided to validate the constancy and reliability of the proposed Gudermannian function as a neural network using the optimizations of GA-SQP for solving the nonlinear singular LE systems, DD models, PD systems and PDSs. The mathematical formulations of these measures are provided as:

Simulations of the results and discussions
In this section, the simulation of the results and comprehensive discussions is performed to solve the singular LE systems, DD models, PD systems and PDSs by exploiting the Gudermannian function as a neural network along with the optimal performances of GA-SQP. One example of each nonlinear singular model based on the singular LE systems, DD models, PD systems and PDSs along with the statistical procedures is numerically discussed through the designed scheme.
Problem I: Consider a nonlinear singular differential model having an exponential function The fitness function of the above function is given as: The exact form of the solution is cos (1 ) sin (1 ), The fitness function of the above function is given as: The exact form of the solution is cos . Problem 3: Consider a nonlinear singular PD form of the equations with MTFs in its forcing function The fitness function of the above function is given as: The exact form of the solution is sin . Problem 4: Consider a nonlinear singular PDS having MTFs in its forcing function     The fitness function of the above function is given as: The exact form of the solution is cos .
To solve the nonlinear singular LE systems, DD models, PD systems and PDSs, the optimization procedures through GA-SQP based on the Gudermannian function as a neural network for hundred independent trials are applied. The mathematical for of the best weight sets that authenticate the estimated outcomes for 30 variables. The convergence/learning curves along with the updated iterations through the merit functions, are drawn in Figures 2-5 for problems 1-4, respectively. It can be found that the performance of GA initially performed fast using the optimization procedures, but after a few generations the convergence ability decreased, enhanced further with the SQP hybridization process. Hence, the GA-SQP scheme provided reliable convergent results for all four problems. In addition, it can be authenticated that the stochastic scheme is reliable for solving the singular LE systems, DD models, PD systems and PDSs.

Conclusions
In this study, the design of the Gudermannian neural network is presented along with the optimization procedures of global and local search schemes. The nonlinear models based on the singular delay, prediction and the pantograph forms have been numerically investigated using the proposed solver. These kinds of nonlinear models are considered tough to solve due to the singularity at the origin, so it becomes more complicated when the delayed, prediction and pantograph terms are involved in the singular equations. Therefore, the designed scheme is considered as an impressive solver for these different types of the singular systems. The overlapping of mean and best solutions with the exact results indicates the correctness of the proposed scheme. The plots of AE are found in good measures, which are calculated around 10 -5 to 10 -7 in the singular delay, prediction and the pantograph forms using 10 numbers of neurons. The best calculated statistical performances based on EVAF, TIC and MAD shows the reliability of the proposed stochastic numerical scheme. The statistical explanations using 50 trials are also provided for the nonlinear singular LE systems, DD models, PD systems and PDSs using the Min, SIR and Med operators, which validates the exactness, robustness and perfection of the scheme. Moreover, neuron analysis by taking small and large neurons is provided to solve these nonlinear singular systems using the optimization procedures-based GA-SQP.