A neuro-swarming intelligent heuristic for second-order nonlinear Lane–Emden multi-pantograph delay differential system

The current study is related to present a novel neuro-swarming intelligent heuristic for nonlinear second-order Lane–Emden multi-pantograph delay differential (NSO-LE-MPDD) model by applying the approximation proficiency of artificial neural networks (ANNs) and local/global search capabilities of particle swarm optimization (PSO) together with efficient/quick interior-point (IP) approach, i.e., ANN-PSOIP scheme. In the designed ANN-PSOIP scheme, a merit function is proposed by using the mean square error sense along with continuous mapping of ANNs for the NSO-LE-MPDD model. The training of these nets is capable of using the integrated competence of PSO and IP scheme. The inspiration of the ANN-PSOIP approach instigates to present a reliable, steadfast, and consistent arrangement relates the ANNs strength for the soft computing optimization to handle with such inspiring classifications. Furthermore, the statistical soundings using the different operators certify the convergence, accurateness, and precision of the ANN-PSOIP scheme.


Introduction
The delay differential model is one of the historical and prominent differential model discovered four centuries ago and has numerous applications in scientific areas like economical states, population dynamics, communication networks, transport, and engineering models [1][2][3][4]. Beretta et al. [5] used the delay-dependent factors to function the geometric consistency based on the delay differential model. To solve the delay differential systems, Frazier [6] used the wavelet Galerkin scheme along with the Taylor series, Rangkuti et al. [7] applied the coupled variation iteration scheme and Chapra [8] implemented the Runge-Kutta method. The current study is about the pantograph differential model that is a form of delay differential system and has a variety of applications in medicine, chemical kinetics, light absorption, ships controlling, biology, chemistry, engineering, physics, electrodynamics, quantum mechanics, infectious diseases, electronic models, physiological kinetics and control problems [9,10]. Due to the paramount significance of these models, a variety of analytical/numerical schemes has been proposed.
To mention a few of them are the Dirichlet series that is applied to get the analytical solutions of the multi-pantograph (MP) delay differential system [11]. For higher order MP nonlinear delay differential system, one-dimensional approach based on differential transforms has been proposed [12]. The Taylor polynomials technique and numerical differential transform scheme are implemented to the 1 3 solutions of the MP delay differential system [13]. Some more potential recent studies of delay differential system arising in different fields solved with numerical or analytical schemes can be seen in [14][15][16][17].
The study of the historical Lane-Emden model that involves singularity at the origin is considered very important for the researchers due to the numerous applications in the cooling of the radiators, system of the gas cloud, cluster galaxies, and polytrophic star models. The Lane-Emden systems used to model the dusty fluids [18], physical forms of the science systems [19], density state of gasiform star [20], catalytic diffusion reactions [21], stellar configuration [22], the electromagnetic theory [23], mathematical physics [24], quantum/classical mechanics [25], oscillating magnetic areas [26], isotropic continuous media [27] and stellar structure systems [28]. It is always difficult to solve the Lane-Emden model due to a singular point, hard and grim nature. Some existing techniques that have been used to solve the singular problems shown in the ref [29][30][31]. The standard notation of the Lane-Emden system is shown as [32]: where Ω ≥ 1 is the value of shape factor and = 0 represents the singular point at the origin. The motive of the current work is to solve numerically nonlinear second-order Lane-Emden multi-pantograph delay differential (NSO-LE-MPDD) model along with a grander system understanding by applying the stochastic methods through the artificial neural networks (ANNs), optimized with both global and local search competences, particle swarm optimization (PSO) and interior-point (IP) approach, i.e., ANN-PSOIP algorithm. Some well-known submissions are HIV infection model [33], nonlinear Bratu's systems [34], heat conduction dynamics based on singular nonlinear human head model [35], singular three-point model [36], Thomas-Fermi model [37], multi-singular nonlinear models [38], model of heartbeat dynamics [39], singular periodic model [40], prey-predator models [41] and nonlinear singular functional differential model [42,43]. The generic form of the NSO-LE-MPDD model is shown as [44]: where a is the constant and the pantographs appear twice in the first and second derivative terms of the Eq. (2). The system model represented in Eq. (2) is a type of functional differential equations with multi-pantograph delays, i.e., a kind of proportional delay that exists in more than one terms. This Lane-Emden pantograph model has not solved before using the stochastic ANN-PSOIP algorithm. Some main topographies of the proposed integrated computational heuristic of ANN-PSOIP method are concisely provided as: • A mathematical NSO-LE-MPDD system is numerically solved by applying the integrated heuristic of neuroswarming computing intelligent ANN-PSOIP algorithm. • The matching/overlapping of the numerical results from the designed approach with the reference solutions of the NSO-LE-MPDD system established the worth/value of the ANN-PSOIP algorithm. • Certification of the performance is ratified via statistical explorations to find the solutions for multiple execution of the ANN-PSOIP algorithm in terms of Theil's inequality coefficient (TIC), variance account for (VAF), semiinterquartile (SI) range and Nash Sutcliffe efficiency (NSE) performance operators. • Beside the precise/practical outcomes for the nonlinear Lane-Emden pantograph second-order delay differential system, easy understanding, extensive applicability, consistency, smooth operations and robustness are other significant advantages.
The remaining portions of this work are planned as: In the next section, the suggested framework is presented using the ANN-PSOIP algorithm. In the folowing section, the mathematical formulation of the performance measures is described. In the next section, the detailed discussions of the numerical results are provided. In the last section, the conclusions together with future research guidance are listed.

Design methodology
The proposed ANN-PSOIP algorithm for numerical results of the NSO-LE-MPDD model is categorized into two steps.
• Introduced an objective function using the differential system and associated boundary/initial conditions. • The optimal combination of PSO and IP algorithm, i.e., PSOIP algorithm is provided in the form of preliminary material along with pseudocode.

ANN modeling
Mathematical models of the NSO-LE-MPDD system are accumulated with the strength of feed-forward ANNs, which designate the continuous mapping for an approximate solution Û ( ) and its derivatives up to second order based on the log-sigmoid H( ) = (1 + exp(− )) −1 activation functions given respectively as follows: are the weight vectors. The log-sigmoid activation function is normally used exhaustively for the hidden layers due to established strength of stability, efficiency, and accuracy in the majority of the applications in diversified fields. For solving the NSO-LE-MPDD system, an error-based merit function is written as: where Fit -1 is the merit functions related to the differential model and Fit -2 represents the initial conditions, respectively, shown as: where hN = 1, F m = F( m ),Û m =Û( m ) and m = mh. , while h be the step size.

Optimization procedure: PSO-IP algorithm
A kind of memetic computing paradigm through hybrid computational heuristics of global search efficacy of particle swarm optimization (PSO) aided with rapid local refinements with efficient interior-point (IP) algorithm, i.e., PSOIP, is ratified for the parameter optimization for NSO-LE-MPDD system due to their established strength of accuracy, convergence, and stability over the standalone techniques based on global and local search methodologies [45].
PSO is a global search optimization process in which the process of search space, a candidate single result relating the procedure of optimization is represented as a particle. For the PSO optimization, preliminary swarms spread into the larger. To modify PSO parameters, the scheme delivers optimal outcomes iteratively P Φ−1 LB and P Φ−1 GB , that is the position and velocity of swarm, mathematically shown as: where X i and V i are ith position and velocity of the particle, respectively, Φ be the flight index or cycle of the algorithm, Ψ ∈ [0, 1][0, 1] is the inertia weight, Φ 1 and Φ 2 are cognitive and social acceleration constants, respectively, while r 1 and r 2 are the random positive real number between 0 and 1.
PSO is a global search optimization process represented in Eqs. (7)(8) used as an operational alternate of the genetic algorithms [46] for NSO-LE-MPDD system. Kennedy and Eberhart suggested PSO, i.e., a famous global search easy implementation algorithm introduced at the last of the previous century and required short requirements of the memory [47]. Few recent applications of the PSO are fuel ignition model [48], balancing stochastic U-lines problems [49], nonlinear physical systems [50], feature classification [51] and operation scheduling of microgrids [52].
The PSO rapidly converges to hybrid with an appropriate local search approach by using the PSO values as a primary weight. Therefore, a rapid and operative local search approach based on interior-point (IP) algorithm is implemented to adjust the solutions attained by the designed optimization algorithm. Few recent submissions of the IP algorithm are active noise control systems [53], mixed complementarity monotone systems [54], simulation of aircraft parts riveting [55], nonlinear system identification [56] and economic load dispatch model [57].
The pseudo code for the combination of PSOIP algorithm trains the ANN along with the essential parameter settings of PSO and IP algorithm are given in Table 1. The scheme based on optimization becomes impulsive by a slight change to set the parameters, thus, it needs numerous experiences, repetitions, and information on necessary optimization imitations of suitable settings using the hybrid of PSO-IP algorithm.

Performance procedures
In the current study, the statistical forms of the Nash Sutcliffe efficiency (NSE), Theil's inequality coefficient (TIC), semi-interquartile (SI) range, and variance account for (VAF) are presented to nonlinear Lane-Emden pantograph second-order delay differential system. Table 1 Pseudo code based on the optimization operator PSOIP algorithm to achieve the ANNs weights 1 3 The mathematical formulations of TIC is presented as: where U i and Û i be the reference and estimation solutions for the ith input of nonlinear Lane-Emden pantograph secondorder delay differential system. The desire/optimal value of TIC is 0 for the perfect scenarios.
The mathematical formulations of VAF and error in VAF (EVAR) are presented as: here 'var' stands for variance operator, and the desire/optimal values of VAF and EVAF are 100 and 0 for the perfect modelling scenarios.
Mathematical expression for SIR metric is given as follows: The mathematical based definition of NSE and error in NSE (ENSE) are given respectively as follows The desire/optimal values of NSE and ENSE are o and 1 for the perfect scenario, respectively.

Results and discussions
In this section, the details for solving three problems of the NSO-LE-MPDD system are provided.

Problem-I
Consider a NSO-LE-MPDD model based equation is written as: The true result of the Eq. (13) is 1 + 4 and the merit function is given as:

Consider a NSO-LE-MPDD model based equation involving trigonometric values in its forcing factor is given as:
The exact solution of the Eq. (15) is cos( ) and the fitness function becomes as:

Consider a NSO-LE-MPDD model based equation is written as:
The exact form of the solution of the Eq. (17) is 1 + 3 and the merit function is given as: , The optimization of the NSO-LE-MPDD model-based problems I-III is accomplished by the hybrid of PSOIP algorithm for 60 independent executions to get the ANN parameters for 10 neurons. The values of the ANN-PSOIP algorithm using the best values of the weight vector are plotted in Fig. 1   The statistical soundings for the designed ANN-PSOIP  algorithm through the Fitness, TIC, EVAF and ENSE values  using the boxplots and histogram values for the NSO-LE-MPDD model-based problems I-III are presented in Figs. 3, 4, 5 and 6. It is indicated that maximum fitness, TIC, EVAF and ENSE values are found to be around 10 −06 to 10 −08 , 10 −04 to 10 −08 , 10 −02 to 10 −08 and 10 −04 -10 −06 , respectively. One may realize from these solutions that most of the independent executions got specific and reasonable accuracy for the statistical values of TIC, EVAF and ENSE.
To find the statistical measures, minimum (Min), median (Med), Mean, and SI operatives are performed for 60 independent runs using the ANN-PSOIP algorithm for solving the NSO-LE-MPDD model-based problems I-III. The statistic values using these gages are presented in Table 2 for solving all the problems. These numerical values are calculated adequate and adequate, which designates the precision and accuracy of the designed ANN-PSOIP scheme. Further analysis of performance is conducted by the implementation of the proposed integrated heuristic ANN-PSOIP for optimization problem based on multiple neurons in the hidden layers, i.e., 5, 10, and 15 neuron-based network models. It is shown that with the increase of the number of neurons the accuracy and stability of the ANN-PSOIP increase but at the cost of more computations. Therefore, there is always a trade-off between the complexity and accuracy, so increasing the neurons more than 10 in the networks, the complexity of the algorithm increases rather more rapidly while very little advantage or gain in the accuracy and convergence.

Conclusion
In this study, a novel submission of stochastic numerical computing solvers is proposed to solve the NSO-LE-MPDD model-based equations using 10 numbers of neurons optimized with the global/local search proficiencies of particle swarm optimization enhanced with the rapid In the future, the proposed ANN-PSOIP algorithm can be used as an accurate/efficient stochastic numerical approach for singular higher order models [58][59][60], biological models [61,62], prediction differential model [63], dynamical investigations of computational fluid models [64][65][66][67][68] and stiff nonlinear systems [69][70][71][72][73][74][75]. Moreover, the polynomial, radial, wavelet, support vector machine-based    neural networks looks promising to be exploited in future for the improved performance [76].
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