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 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]:

$$ \left\{ \begin{gathered} \frac{{{\text{d}}^{2} U}}{{{\text{d}}\chi^{2} }} + \frac{\Omega }{\chi }\frac{{{\text{d}}U}}{{{\text{d}}\chi }} + G(U) = F(\chi ), \hfill \\ U(0) = \alpha ,\,\,\frac{{{\text{d}}U(0)}}{{{\text{d}}\chi }} = 0, \hfill \\ \end{gathered} \right. $$
(1)

where \(\Omega \ge 1\) is the value of shape factor and \(\chi = 0\) 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]:

$$ \left\{ \begin{gathered} a\frac{{{\text{d}}^{2} }}{{{\text{d}}\chi^{2} }}U(a\chi ) + \frac{\Omega }{\chi }\frac{{\text{d}}}{{{\text{d}}\chi }}U(a\chi ) + G(U) = F(\chi ), \hfill \\ U(0) = \alpha ,\,\,\frac{{{\text{d}}U(0)}}{{{\text{d}}\chi }} = 0, \hfill \\ \end{gathered} \right. $$
(2)

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 neuro-swarming 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), semi-interquartile (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 \(\hat{U}(\chi )\) and its derivatives up to second order based on the log-sigmoid \(H(\chi ) = \left( {1 + \exp ( - \chi )} \right)^{ - 1}\) activation functions given respectively as follows:

$$ \begin{gathered} \hat{U}(\chi ) = \sum\limits_{i = 1}^{k} {b_{i} H(c_{i} \chi + a_{i} )} = \sum\limits_{i = 1}^{k} {\frac{{b_{i} }}{{\left( {1 + {\text{e}}^{{ - (c_{i} \chi + a_{i} )}} } \right)}}} , \hfill \\ \frac{{{\text{d}}\hat{U}}}{{{\text{d}}\chi }} = \sum\limits_{i = 1}^{k} {b_{i} \frac{{\text{d}}}{{{\text{d}}\tau }}H(c_{i} \chi + a_{i} )} = \sum\limits_{i = 1}^{k} {\frac{{b_{i} c_{i} {\text{e}}^{{ - (c_{i} \chi + a_{i} )}} }}{{\left( {1 + {\text{e}}^{{ - (c_{i} \chi + a_{i} )}} } \right)^{2} }}} , \hfill \\ \frac{{{\text{d}}^{2} \hat{U}}}{{{\text{d}}\chi^{2} }} = \sum\limits_{i = 1}^{k} {b_{i} \frac{{{\text{d}}^{2} }}{{{\text{d}}\tau^{2} }}H(c_{i} \chi + a_{i} )} = \sum\limits_{i = 1}^{k} {b_{i} c_{i}^{2} \left( {\frac{{2{\text{e}}^{{ - 2(c_{i} \chi + a_{i} )}} }}{{\left( {1 + {\text{e}}^{{ - (c_{i} \chi + a_{i} )}} } \right)^{3} }} - \frac{{{\text{e}}^{{ - (c_{i} \chi + a_{i} )}} }}{{\left( {1 + {\text{e}}^{{ - (c_{i} \chi + a_{i} )}} } \right)^{2} }}} \right)} , \hfill \\ \end{gathered} $$
(3)

where \({\varvec{b}} = [b_{1} ,b_{2} ,b_{3} ,...,b_{m} ],\,\,{\varvec{c}} = [c_{1} ,c_{2} ,c_{3} ,...,c_{m} ]\) and \({\varvec{a}} = [a_{1} ,a_{2} ,a_{3} ,...,a_{m} ]\) 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:

$$ \xi_{{{\text{Fit}}}} = \xi_{{\text{Fit - 1}}} + \xi_{{\text{Fit - 2}}} , $$
(4)

where \(\xi_{{\text{Fit - 1}}}\) is the merit functions related to the differential model and \(\xi_{{\text{Fit - 2}}}\) represents the initial conditions, respectively, shown as:

$$ \xi_{{\text{Fit - 1}}} = \frac{1}{N}\sum\limits_{m = 1}^{N} {\left( {a\frac{{{\text{d}}^{2} \hat{U}(a\chi_{m} )}}{{{\text{d}}\chi_{m}^{2} }} + \frac{\Omega }{{\chi_{m} }}\frac{{{\text{d}}\hat{U}(a\chi_{m} )}}{{{\text{d}}\chi_{m} }} + G(\hat{U}_{m} ) - F_{m} } \right)} ,\,\, $$
(5)
$$ \xi_{{\text{Fit - 2}}} = \frac{1}{2}\left( {\hat{U}_{0} - \alpha } \right)^{2} + \frac{1}{2}\left( {\frac{{d\hat{U}_{0} }}{{{\text{d}}\chi_{m} }}} \right)^{2} , $$
(6)

where \(hN = 1,\,\,F_{m} = F(\chi_{m} ),\,\,\hat{U}_{m} = \hat{U}(\chi_{m} )\,\,{\text{and}}\,\,\,\chi_{m} = mh.\), 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 \({\varvec{P}}_{LB}^{\Phi - 1}\) and \({\varvec{P}}_{GB}^{\Phi - 1}\), that is the position and velocity of swarm, mathematically shown as:

$$ X_{i}^{\Phi } = X_{i}^{\Phi - 1} + V_{i}^{\Phi - 1} , $$
(7)
$$ V_{i}^{\Phi } = \Psi V_{i}^{\Phi - 1} + \Phi_{1} (P_{{{\text{LB}}}}^{\Phi - 1} - X_{i}^{\Phi - 1} )r_{1} + \Phi_{2} (P_{{{\text{GB}}}}^{\Phi - 1} - X_{i}^{\Phi - 1} )r_{2} , $$
(8)

where Xi and Vi are ith position and velocity of the particle, respectively, \(\Phi\) be the flight index or cycle of the algorithm, \(\Psi \in [0,1][0,1]\) is the inertia weight, \(\Phi_{1}\) and \(\Phi_{2}\) are cognitive and social acceleration constants, respectively, while r1 and r2 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.

Table 1 Pseudo code based on the optimization operator PSOIP algorithm to achieve the ANNs weights
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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.

The mathematical formulations of TIC is presented as:

$$ {\text{TIC}} = \frac{{\sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {U_{i} - \hat{U}_{i} } \right)}^{2} } }}{{\left( {\sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {U_{i}^{2} } } + \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {\hat{U}_{i}^{2} } } } \right)}}, $$
(9)

where \(U_{i}\) and \(\hat{U}_{i}\) be the reference and estimation solutions for the ith input of nonlinear Lane–Emden pantograph second-order 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:

$$ \left\{ \begin{gathered} {\text{VAF}} = - \left( {\frac{{{\text{var}} \left( {U_{i} (\chi ) - \hat{U}_{i} (\chi )} \right)}}{{{\text{var}} \left( {U_{i} (\chi )} \right)}} - 1} \right) \times 100, \hfill \\ {\text{EVAF}} = \left| {\frac{1}{0.01} - {\text{VAF}}} \right|. \hfill \\ \end{gathered} \right. $$
(10)

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:

$$ \left\{ \begin{gathered} {\text{SIR}} = 0.5 \times \left( {Q_{3} - Q_{1} } \right), \hfill \\ Q_{3} = \,3{\text{rd}}\,{\text{quartile,}}\,\,\,Q_{1} = \,1{\text{st}}\,{\text{quartile}}{.}\, \hfill \\ \end{gathered} \right. $$
(11)

The mathematical based definition of NSE and error in NSE (ENSE) are given respectively as follows

$$ \left\{ \begin{gathered} {\text{NSE = }}1 - \frac{{\sum\nolimits_{i = 1}^{n} {\left( {U_{i} (\chi ) - \hat{U}_{i} (\chi )} \right)}^{2} }}{{\sum\nolimits_{i = 1}^{n} {\left( {\hat{U}_{i} (\chi ) - \overline{U}_{i} (\chi )} \right)}^{2} }},\;\quad \overline{U}_{i} (\chi ) = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {U_{i} (\chi )} \hfill \\ {\text{ENSE = }} - \left( {{\text{NSE}} - 1} \right){.} \hfill \\ \end{gathered} \right.. $$
(12)

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:

$$ \left\{ \begin{gathered} \frac{1}{2}\frac{{{\text{d}}^{2} }}{{{\text{d}}\chi^{2} }}U\left( {\frac{1}{2}\chi } \right) + \frac{3}{\chi }\frac{{\text{d}}}{{{\text{d}}\chi }}U\left( {\frac{1}{2}\chi } \right) + U^{2} = \chi^{8} + 2\chi^{4} + 3\chi^{2} + 1, \hfill \\ U(0) = 1,\,\,\frac{{{\text{d}}U(0)}}{{{\text{d}}\chi }} = 0. \hfill \\ \end{gathered} \right. $$
(13)

The true result of the Eq. (13) is \(1 + \chi^{4}\) and the merit function is given as:

$$ \zeta_{{{\text{Fit}}}} = \frac{1}{N}\sum\limits_{m = 1}^{N} {\left( {\frac{1}{2}\frac{{{\text{d}}^{2} }}{{{\text{d}}\chi_{m}^{2} }}\hat{U}\left( {\frac{1}{2}\chi_{m} } \right) + \frac{3}{{\chi_{m} }}\frac{{\text{d}}}{{{\text{d}}\chi }}\hat{U}\left( {\frac{1}{2}\chi_{m} } \right) + \hat{U}_{m}^{2} - \chi_{m}^{8} - 2\chi_{m}^{4} - 3\chi_{m}^{2} - 1} \right)}^{2} + \frac{1}{2}\left( {(\hat{U}_{0} - 1)^{2} + \left( {\frac{{{\text{d}}\hat{U}_{0} }}{{{\text{d}}\chi_{m} }}} \right)^{2} } \right). $$
(14)

Problem-II

Consider a NSO-LE-MPDD model based equation involving trigonometric values in its forcing factor is given as:

$$ \left\{ \begin{gathered} \frac{1}{2}\frac{{{\text{d}}^{2} }}{{{\text{d}}\chi^{2} }}U\left( {\frac{1}{2}\chi } \right) + \frac{3}{\chi }\frac{{\text{d}}}{{{\text{d}}\chi }}U\left( {\frac{1}{2}\chi } \right) + U^{ - 2} \quad= - \frac{1}{2}\cos \left( {\frac{1}{2}\chi } \right) + \sec^{2} (\chi ) - \frac{3}{\chi }\sin \left( {\frac{1}{2}\chi } \right), \hfill \\ U(0) = 1,\,\,\frac{{{\text{d}}U(0)}}{{{\text{d}}\chi }} = 0. \hfill \\ \end{gathered} \right. $$
(15)

The exact solution of the Eq. (15) is \(\cos (\chi )\) and the fitness function becomes as:

$$ \zeta_{{{\text{Fit}}}} = \frac{1}{N}\sum\limits_{m = 1}^{N} {\left( \begin{gathered} \frac{1}{2}\frac{{{\text{d}}^{2} }}{{{\text{d}}\chi_{m}^{2} }}\hat{U}\left( {\frac{1}{2}\chi_{m} } \right) + \frac{3}{{\chi_{m} }}\frac{{\text{d}}}{{{\text{d}}\chi }}\hat{U}\left( {\frac{1}{2}\chi_{m} } \right) + \hat{U}_{m}^{ - 2} \hfill \\ \quad + \frac{1}{2}\cos \left( {\frac{1}{2}\chi_{m} } \right) - \sec^{2} (\chi_{m} ) + \frac{3}{\tau }\sin \left( {\frac{1}{2}\chi_{m} } \right) \hfill \\ \end{gathered} \right)}^{2} \, + \frac{1}{2}\left( {(\hat{U}_{0} - 1)^{2} + \,\left( {\frac{{{\text{d}}\hat{U}_{0} }}{{{\text{d}}\chi_{m} }}} \right)^{2} } \right). $$
(16)

Problem-III

Consider a NSO-LE-MPDD model based equation is written as:

$$ \left\{ \begin{gathered} \frac{1}{2}\frac{{{\text{d}}^{2} }}{{{\text{d}}\chi^{2} }}U\left( {\frac{1}{2}\chi } \right) + \frac{3}{\chi }\frac{{\text{d}}}{{{\text{d}}\chi }}U\left( {\frac{1}{2}\chi } \right) + {\text{e}}^{U} = {\text{e}}^{{1 + \chi^{3} }} + \frac{15}{4}\chi , \hfill \\ U(0) = 1,\,\,\frac{{{\text{d}}U(0)}}{{{\text{d}}\chi }} = 0. \hfill \\ \end{gathered} \right. $$
(17)

The exact form of the solution of the Eq. (17) is \(1 + \chi^{3}\) and the merit function is given as:

$$ \zeta_{{{\text{Fit}}}} = \frac{1}{N}\sum\limits_{m = 1}^{N} {\left( {\frac{1}{2}\frac{{{\text{d}}^{2} }}{{{\text{d}}\chi_{m}^{2} }}\hat{U}\left( {\frac{1}{2}\chi_{m} } \right) + \frac{3}{{\chi_{m} }}\frac{{\text{d}}}{{{\text{d}}\chi }}\hat{U}\left( {\frac{1}{2}\chi_{m} } \right) + {\text{e}}^{{\hat{U}_{m} }} - {\text{e}}^{{1 + \chi_{m}^{3} }} - \frac{15}{4}\chi_{m} } \right)}^{2} \,\, + \frac{1}{2}\left( {(\hat{U}_{0} - 1)^{2} + \,\left( {\frac{{{\text{d}}\hat{U}_{0} }}{{{\text{d}}\chi_{m} }}} \right)^{2} } \right). $$
(18)

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 and the mathematical illustrations of the proposed numerical solutions are shown as:

$$ \hat{U}_{1} (\chi ) = \frac{1.5160}{{1 + {\text{e}}^{ - (1.867\chi - 4.688)} }} + \frac{1.1578}{{1 + {\text{e}}^{ - ( - 3.586\chi + 6.613)} }} - \frac{2.5214}{{1 + {\text{e}}^{ - ( - 6.906\chi + 9.005)} }} + \cdots + \frac{1.4091}{{1 + {\text{e}}^{ - (1.608\chi + 0.007)} }}, $$
(19)
$$ \hat{U}_{2} {(}\chi ) = \frac{2.6919}{{1 + {\text{e}}^{ - ( - 1.452\chi + 2.039)} }} - \frac{1.8104}{{1 + {\text{e}}^{ - ( - 1.864\chi - 1.655)} }} - \frac{{{0}{\text{.4569}}}}{{1 + {\text{e}}^{ - ( - 0.900\chi - 5.096)} }} + \cdots + \frac{0.8616}{{1 + {\text{e}}^{ - ( - 0.269\chi - 1.412)} }}, $$
(20)
$$ \hat{U}_{3} (\chi ) = \frac{1.3590}{{1 + {\text{e}}^{ - ( - 3.903\chi - 7.407)} }} + \frac{4.7087}{{1 + {\text{e}}^{ - ( - 0.215\chi + 1.035)} }} - \frac{0.1978}{{1 + {\text{e}}^{ - ( - 3.214\chi - 5.136)} }} + \cdots + \frac{11.2505}{{1 + {\text{e}}^{ - (2.331\chi - 4.023)} }}. $$
(21)
Fig. 1
figure 1

Best weights set and comparison of the reference, mean and exact form of the results based on NSO-LE-MPDD model for problems I–III

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Optimization is performed for solving the NSO-LE-MPDD model-based problems I-III using the ANN-PSOIP algorithm for 60 independent executions. In Fig. 1, a set of best weight vectors and comparison of the reference, mean and obtained results of the NSO-LE-MPDD model-based problems I-III using 10 neurons is presented. It is indicated that the reference, mean and proposed results overlapped over one another for all the examples of the NSO-LE-MPDD model. This overlapping of the outcomes designates the accomplishment and excellence of the ANN-PSOIP algorithm. Figure 2 represents the absolute error (AE) values and performance investigations of the ANN-PSOIP approach for the NSO-LE-MPDD model-based problems I–III. The Fig. 2a shows the AE plots for all the problems of the model. It is resulted that the AE values exist around 10−04 to 10−07, 10−04 to 10−05 and 10−05 to 10−07 for the problems I-III, respectively. The Fig. 2b represents the performance measures for all the problems in terms of the fitness, TIC, EVAF and ENSE. It is observed that the fitness values exist around 10−10 to 10−11 for Problem-I, while the fitness lie around 10−09 to 10−10 for Problems II and III. The values of the TIC gages for all the Problems lie 10−08 to 10−09 ranges. Furthermore, the EVAF and ENSE gages values for Problem I lie 10−07 to 10−08 range, while the EVAF and ENSE values for Problems II and III lie around 10−08–10−09.

Fig. 2
figure 2

AE and performance measures of the ANN-PSOIP scheme for NSO-LE-MPDD model-based problems I–III

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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.

Fig. 3
figure 3

Statistical measures for the designed ANN-PSOIP algorithm through Fitness using the values of boxplots/histograms for NSO-LE-MPDD model-based problems I–III

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Fig. 4
figure 4

Statistical measures for the designed ANN-PSOIP algorithm through TIC using the values of boxplots/histograms for NSO-LE-MPDD model-based problems I–III

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Fig. 5
figure 5

Statistical measures for the designed ANN-PSOIP algorithm through EVAF using the values of boxplots/histograms for NSO-LE-MPDD model-based problems I–III

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Fig. 6
figure 6

Statistical measures for the designed ANN-PSOIP algorithm through ENSE using the values of boxplots/histograms for NSO-LE-MPDD model-based problems I–III

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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.

Table 2 Statistics solutions for NSO-LE-MPDD model-based problems I–III
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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 refinement of decision variables by manipulating the local search strength via interior-point algorithm. An objective function is designed using the differential system/initial conditions and then optimization is performed by the hybrid of local/global competencies of particle swarm optimization and interior-point algorithm, respectively. The accuracy and exactness of the designed scheme are certified by finding identical solutions with the exact/reference results having 5–7 decimal places of precision for solving all the problems of the NSO-LE-MPDD model. Statistical interpretation through performance measures of TIC, ENSE and EVAF based on 60 trials/executions for obtaining the solution of NSO-LE-MPDD model-based equations in terms of semi-interquartile range, mean and median authenticate the robustness, accurateness and trustworthiness of the proposed scheme.

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].