Neuro-swarm computational heuristic for solving a nonlinear second-order coupled Emden–Fowler model

The aim of the current study is to present the numerical solutions of a nonlinear second-order coupled Emden–Fowler equation by developing a neuro-swarming-based computing intelligent solver. The feedforward artificial neural networks (ANNs) are used for modelling, and optimization is carried out by the local/global search competences of particle swarm optimization (PSO) aided with capability of interior-point method (IPM), i.e., ANNs-PSO-IPM. In ANNs-PSO-IPM, a mean square error-based objective function is designed for nonlinear second-order coupled Emden–Fowler (EF) equations and then optimized using the combination of PSO-IPM. The inspiration to present the ANNs-PSO-IPM comes with a motive to depict a viable, detailed and consistent framework to tackle with such stiff/nonlinear second-order coupled EF system. The ANNs-PSO-IP scheme is verified for different examples of the second-order nonlinear-coupled EF equations. The achieved numerical outcomes for single as well as multiple trials of ANNs-PSO-IPM are incorporated to validate the reliability, viability and accuracy.


Introduction
The historical Emden-Fowler (EF) system is considered very important for the research community because of singularity at the origin and has various applications in wide-ranging fields of applied science and engineering. Some well-known applications are catalytic diffusion reactions using the error estimate models (Burdyny and Smith 2019), stellar configuration (Abbas et al. 2019), density profile of gaseous star (Bacchini et al. 2019), spherical annulus (Soliman 2019), isotropic continuous media (Adel and Sabir 2020), extrinsic thermionic maps (Barilla et al. 2020), the theory of electromagnetic (Guirao et al. 2020) and morphogenesis (Dridi and Trabelsi 2022). Due to the specialty of the singular point and extensive applications, the researcher has always shown keen interest to solve these models all the time. These models are not easy to solve due of this singular model, nonlinearity and stiff nature, and only a few techniques are available in the literature to solve these models. Few of them are Legendre spectral wavelets scheme (Dizicheh et al. 2020), Adomian decomposition scheme (Abdullah Alderremy et al. 2019), Haar quasilinearization wavelet scheme (Singh et al. 2020;Verma and Kumar 2019), an analytic algorithm approach (Arqub et al. 2020), rational Legendre approximation scheme (Dizicheh et al. 2020), modified variational iteration scheme (Verma et al. 2021), differential transformation scheme (Xie et al. 2019), fourth-order B-spline collocation scheme (Roul and Thula 2019), Chebyshev operational matrix scheme (Sharma et al. 2019) and variation of parameters scheme with an auxiliary parameter (Khalifa and Hassan 2019). Beside these, the numerical methodologies introduced in (Abdelrahman and Alharbi 2021; Alharbi et al. 2020;Almatrafi et al. 2021;Lotfy 2019;Sabir 2022a, Sabir et al. 2022c) can be exploited for EF equations-based systems.
All these mentioned schemes have their specific merits/ advantages and demerits/imperfections, whereas soft computing stochastic solver is used to manipulate the artificial neural networks (ANNs) strength optimized by global/local search proficiencies of particle swarm optimization (PSO) and interior-point method (IPM), i.e., ANNs-PSO-IPM, have not been implemented for the nonlinear coupled EF model of second kind. The researchers have been generally practiced the numerical computing meta-heuristic schemes along with the neural network strengths for solving the various mathematical linear/nonlinear models (Guerrero-Sánchez et al. 2021;Guirao et al. 2022;Lu et al. 2019Lu et al. , 2021Mehmood et al. 2020;Sabir et al. 2020a, e). Few recent applications of the stochastic solvers are financial market forecasting (Bukhari et al. 2020), food chain model (Sabir 2022b), nonlinear smoking models (Saeed et al. 2022), nonlinear fractional Lane-Emden systems (Sabir et al. 2022d), nonlinear second-order Lane-Emden pantograph delay differential systems (Nisar et al. 2021), peristaltic motion of a third-grade fluid involving planar channel (Mahmood et al. 2022), nonlinear predator-prey system , elliptic partial differential model (Fateh et al. 2019), mathematical form of the Leptospirosis system (Botmart et al. 2022), HIV mathematical models (Sabir et al. 2021c, nonlinear multiple singularity-based systems (Raja et al. 2019), singular Thomas-Fermi equation , heartbeat dynamics (Malešević et al. 2020), a corneal model for eye surgery Wang et al. 2022) and heat conduction model of the human head (Raja et al. 2018). These proposed stochastic solvers verified the values of the exactness, convergence, and accurateness of the ANNs-PSO-IPM.
Keeping in view all the consequences of above proposals, authors are interested to exploit the numerical stochastic solvers for consistent, stable, and efficient scheme for nonlinear second-order coupled EF system. The literature form of the coupled EF model of second kind is written as (Sabir et al. 2020b): where G 1 and G 2 are the nonlinear functions, a and b are the constants, while F 1 and F 2 are designated as a source functions. The aim of this current study is to solve the model given in Eq.
(1) through intelligent computing schemes based on ANN-PSO-IP scheme. Some inventive inspiration of the current study is presented as: • A neuro-swarm novel intelligent computing ANNs-PSO-IPM is designed and presented to solve secondorder nonlinear coupled EF model. The remaining forms of the present work are shown as; Sec 2 presents the detailed methodology of the neural networks using the optimization process ANNs-PSO-IP scheme. Sec 3 presents the performance measures. Sec 4 indicates the numerical measures of the ANNs-PSO-IPM together with the statistical measures. Finally, some concluding remarks along with future work plans are described.

Methodology
This section presents the design of ANNs-PSO-IPM for second-order nonlinear coupled EF model in two stages as given below: Stage 1: A mean square error-based objective/fitness function is constructed for nonlinear coupled EF model Stage 2: The training/learning of the networks is presented with the help of hybrid PSO-IPM.

ANNs modeling
The neural networks are extensively applied to solve the diverse applications arising in sundry domains of engineering and applied sciences (Nasirzadehroshenin et al. 2020;Sabir et al. 2021bUmar et al. 2020). The proposed results are indicated asÛðWÞ andVðWÞ, while d nÛ dW n and d nV dW n are the derivatives of n th order, respectively, and are given as follows: The objective functions E FitÀ1 and E FitÀ2 are linked with coupled differential systems, and E FitÀ3 is used for the initial conditions.

Optimization: PSO-IPM
The optimization to solve the second-order nonlinearcoupled EF system is ratified by the hybrid-computing of PSO-IPM.
PSO is a well-organized search algorithm used as a global search methodology like genetic algorithms (GAs). The PSO algorithm introduced by Eberhart and Kennedy (Hussain and Ismail 2020; Sibalija 2019) and works as an easy procedure that needs minor memory. In search space, an applicant single solution of decision variables by applying optimization is known as a particle and these particles set formulate a swarm. The PSO operates via local P qÀ1 LB and global P qÀ1 GB best particle positions in a swarm. The position X i and velocity V i are mathematical expressed as follows:  (Kuntoji et al. 2020), and optimization of permanent magnets synchronous motor (Mesloub et al. 2020). The convergence performance of PSO quickly achieved by using the combination with local search procedure by taking the global best particle of PSO as an initial weight. Consequently, an operative and quick local search approach named as interior-point method (IPM) is oppressed for rapid refinement of the outcomes obtained via PSO scheme. The integrated heuristics of PSO-IPM is exploited to train the networks, while the essential parameter settings of importance elements for PSO-IPM are given in Table 1. Few recently IP scheme applications are power flow security constraint optimization (Casacio et al. 2019), image processing (Chouzenoux et al. 2020), multistage nonlinear nonconvex problems (Zanelli et al. 2020) and nonlinear benchmark models (Wambacq et al. 2021). The PSO-IP scheme is used to train the networks as per process and parameter settings provided in Table 1.

Performance indices/metrics
The performances is measured using RMSE, VAF, TIC indices along their globals, i.e., mean values. The mathematical forms of these statistical operatives are given as:

Results and discussions
The detail for presenting the solving the four examples of second-order coupled EF model is presented in this section. ; ð11Þ Problem I Consider the second-order nonlinear-coupled EF model is given as: The exact solutions of Eq. (13) are [e W 2 ; e ÀW 2 ], whereas the fitness function becomes as: here N =20, 25 and 30 for input span [0, 1], [0, 1.25] and [0, 1.5], respectively.
Problem II Consider the second-order nonlinear-coupled EF system is written as: The exact solutions of Eq. Problem III Consider the second-order nonlinear-coupled EF model is given as: The exact solutions of Eq. (17) are 1 þ W 2 ; 1 À W 2 Â Ã , and the fitness/objective function is given as follows: Problem IV Consider the second-order nonlinear-coupled EF model is given as: The exact solutions of Eq. (17) are ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 þ W 2 p ; 1 ffiffiffiffiffiffiffiffi ffi 1þW 2 p ! , and the fitness/objective function is given as follows: here N =20, 25 and 30 for input span [0, 1], [0, 1.25] and [0, 1.5], respectively.
To calculate/determined the proposed numerical outcomes for the Problems I to IV based on the second-order nonlinear-coupled EF model using the proposed PSO-IPM executed for 50 multiple runs to attain the adjustable weights. The numerical values of the weights are presented in Fig. 1 forÛ andV. These parameters are applied to get the estimated results for all four variants based on the second-order nonlinear-coupled EF model and the mathematical representations becomes as: The optimization is performed for all the problems of the nonlinear-coupled EF system with ANNs-PSO-IPM for 50 independent runs. A set of the best weights along with proposed and exact outcomes are shown in Fig. 1. It is stated that all the problems of the nonlinear-coupled EF system of second kind, the exact/reference solution and ANNs-PSO-IPM results overlapped consistently forÛðWÞ andVðWÞ. This overlapping of the outcomes depicts the One may see that results are consistently overlapping for small as well as large interval. The AE plots forÛðWÞ andVðWÞ are drawn in Fig. 2a and b for N = 20, while the performance measures forÛðWÞ andVðWÞ are provided in Fig. 2c and d for N = 20. It is observed that the AE values ofÛðWÞ lie around 10 -05 -10 -06 , 10 -04 -10 -05 , 10 -06 -10 -08 and 10 -06 -10 -07 for Problem I, II, III and IV in case of N = 20, 25 and 30. While the AE values ofVðWÞ lie around 10 -05 -10 -06 , 10 -04 -10 -05 , 10 -06 -10 -09 and 10 -06 -10 -07 for Problems I-IV for N = 20. The performance measures ofÛðvÞ andVðWÞ based on FIT, RMSE, TIIC and EVAF are plotted in Fig. 2c and d. It is seen that the FIT forÛðWÞ andVðWÞ lie close to 10 -08 -10 -10 , for problems I, III and IV, and similarly the FIT for Problem II lies around 10 -06 -10 -08 . The RMSE and TIC forÛðWÞ andVðWÞ lie around to 10 -04 -10 -06 , for all the problems. The TIC values lie around 10 -06 -10 -08 for both indexes of all the Problems. The values of the EVAF for both indices of all the problems lie around 10 -10 -10 -12 . The convergence measures for the Problems I-IV based on the second-order nonlinear-coupled EF model using the fitness values, boxplots and histograms with 10 neurons are plotted in For more satisfaction, accuracy and precision examination of the ANNs-PSO-IP scheme, statistical measures are made based on minimum (MIN), mean, standard deviation (SD), median and semi interquartile range (S-IR). S-IR range is 0.5 times of the difference of the third quartile, i.e., Q 3 = 75% data and first quartile, i.e., Q 1 = 25% data, is calculated for 50 runs of ANNs-PSO-IP scheme to solve four different examples of the nonlinear-coupled EF system of second kind. These statistical results for Problems I-IV are provided in Tables 2 as well as 3

Conclusion
In this investigation, a reliable, stable, consistent and precise numerical ANNs-PSO-IPM is presented for solving the nonlinear-coupled EF system by using the ANNs strength. The objective function is optimized of these networks using the global as well as local search competences of PSO-IPM. The suggested ANNs-PSO-IPM is viably executed to solve four different examples of the nonlinear-coupled EF system. The detailed, precise and particular presentation is obtained for ANNs-PSO-IPM in terms of AE with steadfast precision that is measured around 4-7 decimals of accurateness of the present reference solutions for all four problems of the nonlinear-coupled EF system of second kind. Furthermore, the statistical clarifications achieved good measures using the Min, standard deviation, Mean, S-IR and Median to check the convergence, robustness and accuracy of the ANNs-PSO-IPM for solving the second-order nonlinear-coupled EF model-based problems I-IV.

Future research directions
In the future, one can exploit/explore the knacks of ANNs-PSO-IPM to solve the singular higher order models (Sabir et al. 2020c, d;2021a), fractional order models ( Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. The authors have not disclosed any funding.
Data availability Enquiries about data availability should be directed to the authors.

Declarations
Conflict of interest The authors have not disclosed any competing interests.