The purpose of this research is to construct an integrated neuro swarming scheme using the procedures of the artificial neural networks (ANNs) with the use of global search particle swarm optimization (PSO) along with the competent local search interior-point programming (IPP) called as ANN-PSOIPP. The proposed computational scheme is implemented for the numerical simulations of the third order nonlinear delay differential Emden-Fowler model (TON-DD-EFM). The TON-DD-EFM is based on two types along with the particulars of shape factor, delayed terms, and singular points. A merit function is performed using the optimization of PSOIPP to find the solutions to the TON-DD-EFM. The effectiveness of the ANN-PSOIPP is certified through the comparison with the exact results for solving four examples of the TON-DD-EFM. The scheme’s efficiency is observed by performing the absolute error in suitable measures found around 10^{−04} to 10^{−07}. Furthermore, the statistical-based assessments for 100 trials are provided to compute the accuracy, stability, and constancy of the ANN-PSOIPP for solving the TON-DD-EFM.
The delayed form of the differential system is considered one of the noteworthy, historical, and significant equation, which has attracted the research community because of its massive applications. A few of them are biological models, dynamical-based population models, communication models, engineering/economical models, propagation, and transport systems [1–5]. To solve the delay differential models, many researchers suggested a numerical and analytical schemes to tackle the difficulty of the delay terms. Brunner et al. [6] proposed a numerical discontinuous Galerkin approach, and Hsiao et al. [7] applied the Haar wavelet scheme to handle the delay factor. Wang [8] proposed the Legendre wavelet approach to solving the delay differential scheme. Rach and Adomian [9] proposed the Adomian decomposition scheme to solve the delay form of the differential system. Shakeri et al. [10] solved the delay differential model with the homotopy perturbation approach. Erdogan et al. [11] applied the finite difference numerical scheme to solve the perturbed singular delay differential system. The generic form of the delay differential is given as [12,13]:
{d3vdε3=g(ε,v(ε−t),ddεv(ε−t),d2dε2v(ε−t)),v(0)=a,dv(0)dε=b,d2v(0)dε2=c,where g indicates the linear or nonlinear-based function and t represents the delayed factor. The singular investigations have achieved huge significance because of numerous applications in engineering, and physical and biological studies. It is not easy to solve the singularity-based systems because of their hard, difficult, challengeable, and grim nature. One significant, famous, singular, and historical form is the Emden-Fowler, which has many applications, like population growth, relativistic mechanics, pattern formation, fluid dynamics, and chemical reactors modeling. The Emden-Fowler system is mathematically given as [14–18]:{d2vdε2+uεdudε+g(ε)h(v)=0,v(0)=a1,dv(0)dε=a2,where u≥1 indicates the shape vector and the Emden-Fowler model given in Eq. (2) can become the Lane-Emden for g(ε)=1. The Lane-Emden is one of the singular systems that derived a few centuries ago by the astrophysicists J. H. Lane and the R. Emden in their pioneer work. This prominent model designates the inner polytropic of structure stars, cluster galaxies, radiative cooling and model based on gas cloud. The Lane-Emden singular system has various applications in the isotropic based continuous media [19], physical scientific fields [20], density field of gaseous stars [21], dusty fluid systems [22], morphogenesis [23], stellar arrangement models [24], oscillating magnetic systems [25], catalytic diffusion reactions [26], isothermal gas sphere systems [27], mathematical sciences [28], electromagnetic theory [29] and quantum as well as classical mechanics [30]. The Lane-Emden model is given as:{d2vdε2+uεdudε+h(v)=0,v(0)=a1,dv(0)dε=a2.
The research community presented the solutions of the above model by applying different techniques. A few methods for presenting the solutions of the Lane-Emden system are the Adomian decomposition scheme suggested by Wazwaz and Shawagfeh [31,32]. Adel et al. [33] solved the pantograph Lane-Emden model using the Bernoulli collocation method. Abdelkawy et al. [34] solved the singular coupled functional Lane–Emden system using the famous spectral collocation scheme. Parand et al. [35] introduced a numerical approach for the singular equation of the Lane-Emden type. Using stochastic procedures, Sabir et al. [36] presented a nonlinear singular functional differential system.
In this study, the TON-DD-EFM is numerically discussed through the artificial neural networks (ANNs) by using the optimization procedure based on the global particle swarm optimization (PSO) aided with the local search-based interior-point programming (IPP), i.e., ANN-PSOIPP. The singular models are assumed to be tough by using the traditional and conventional schemes, like Runge-Kutta, Adams numerical method, Milne-Predictor-Corrector scheme, and many others. However, the researcher’s alternative and best choice are to solve the singular-based models using procedures based on the ANNs. There are several applications where ANNs have been exploited to solve many models in recent years; a few of them are the multi-singular higher-order Emden–Fowler system [37–43], nonlinear SIR dengue fever model [44], HIV infection system [45,46], third-order singular Emden–Fowler equation [47], SITR system [48], second kind of singular model [49], mosquito dispersal model [50] and many more [51–54]. By keeping the worth of these models, authors are interested in exploiting the singular TON-DD-EFM, which has never been solved before by using the stochastic ANN-PSOIPP. The general forms of the singular TON-DD-EFM are based on the two types given as [55]:
{d3dε3v(ε−t)+2χεd2dε2v(ε−t)+χ(χ−1)ε2ddεv(ε−t)+g(ε)h(v)=f(ε),v(0)=α,dv(0)dε=0,d2v(0)dε2=0,{d3dε3v(ε−t)+2χεd2dε2v(ε−t)+χ(χ−1)ε2ddεv(ε−t)+g(ε)h(v)=f(ε),v(0)=α,dv(0)dε=0,d2v(0)dε2=0,where χ is a real number always taken positive and f(ε) represents the forcing factor. The terms g(ε) and h(v) represent the ε and v functions. The singularity appears twice at ε = 0 in the first form of the model, represented in Eq. (4), while a single singularity occurs in the second type derived in Eq. (5). The shape factors in the 1^{st} type are 2χ and χ(χ−1), whereas a single shape factor χ is observed in the 2^{nd} type. Likewise, the delayed factors in the 1^{st} type are noticed thrice than are observed in the first, second, and third factors. In the 2^{nd} case, the delayed factors appeared twice in the first and second factors. The novel features of the proposed ANN-PSOIPP are concisely briefed as follows:
A novel design of ANN-PSOIPP is proposed to solve the singular TON-DD-EFM numerically along with its two types.
The detail about the delay factors, singular point, and shape factor is provided for solving the singular TON-DD-EFM.
The intersection of the exact/proposed solutions through ANN-PSOIPP proves the worth in the form of convergence to solve both cases of the singular TON-DD-EFM.
The correctness of the ANN-PSOIPP is observed through the good performance of the absolute error (AE) for solving the singular system.
The statistical performance is provided for the dependability of the stochastic ANN-PSOIPP by using the “Theil’s inequality coefficient (T.I.C)”, “root mean square error (R.MSE)”, and “Nash Sutcliffe efficiency (NSE)” for solving both the cases of the TON-DD-EFM.
Alongside the reasonable precise solutions of the cases of the singular TON-DD-EFM, stability, ease of understanding, robustness, specific applicability, and smooth operation are other valued advantages.
The other paper parts are provided as follows: Section 2 describes the procedures of the stochastic ANN-PSOIPP. Section 3 represents the performance operators. The results detail for solving the singular TON-DD-EFM are provided in Section 4. Finally, the conclusions are listed in the final section.
Designed Methodology
The designed ANN-PSOIPP approach is separated into two steps to demonstrate the performance of singular TON-DD-EFM. First, to introduce a fitness function (FF) for solving the model and the hybrid of the designed ANN-PSOIPP.
Modeling Based on ANNs
Several researchers implemented the modeling based on ANNs in various investigations to understand the nonlinear models in various fields. v(ε) indicates the results based on continuous mapping by implementing the FF, i.e., log-sigmoid s(ε)=(1+e−ε)−1 is written as:
v^(ε)=∑i=1kais(wiε+ci)=∑i=1kai(1+e−(wiε+ci))−1,dv^dε=∑i=1kaiddεs(wiε+ci)=∑i=1kaiwie−(wiε+ci)(1+e−(wiε+ci))−2,d2v^dε2=∑i=1kaid2dε2s(wiε+ci)=∑i=1kaiwi2(2e−2(wiε+ci)(1+e−(wiε+ci))−3−e−(wiε+ci)(1+e−(wiε+ci))−2),d3v^dε3=∑i=1kaid3dε3s(wiε+ci)=∑i=1kaiwi3(6e−3(wiε+ci)(1+e−(wiε+ci))−4−6e−2(wiε+ci)(1+e−(wiε+ci))−3+e−(wiε+ci)(1+e−(wiε+ci))−2),where a=[a1,a2,…,ak],w=[w1,w2,…,wk] and c=[c1,c2,…,ck] show the weight vectors. To solve the singular TON-DD-EFM given in Eqs. (4) and (5), the FF in the mean square error form is given as:
e=efit−1+efit−2,efit−1=efit−a+efit−b,efit−2=efit−c+efit−d,where efit−a and efit−b are the error based FFs in the form of differential model given in the Eqs. (4) and (5), while efit−c and efit−d represent the corresponding initial conditions (ICs), written as:
efit−a=1N∑k=1N(d3v^(εk−t)dεk3+2χεkd2v^(εk−t)dεk2+χ(χ−1)εk2dv^(εk−t)dεk+gkh(v^k)−fk)2,efit−b=1N∑k=1N(d3v^(εk−t)dεk3+χεkd2v^(εk−t)dεk2+gkh(v^k)−fk)2,efit−c=13((v^0−α)2+(dv^0dεk)2+(d2v^0dεk2)2),efit−d=13((v^0−α)2+(dv^0dεk−β)2+(d2v^0dεk2)2),where, Nh=1,v^k=v^(εk),gk=g(εk),fk=f(εk) and xk=kh.
Optimization: ANN-Psoipp
For the singular TON-DD-EFM given in Eqs. (4) and (5), the design of the ANN-PSOIPP scheme is presented.
Particle Swarm Optimization (PSO): It is an optimization process known as replacing a genetic algorithm [56]. PSO was introduced by Eberhart and Kennedy a few decades ago, which required short memory and was applied as an easy implementation process. PSO has been extensively applied as an optimization technique, like optical stuff based on multilayer thin films [57], electric daily peak-load forecasting [58], high-dimensional clustering statistics [59], prediction differential models [60], parameter approximation of chaotic plots [61], optimization of nonlinear benchmark model [62] and parameter estimate models in electromagnetic waves of the plane [63].
A particular candidate result for the optimization process is authenticated as a particle in space study. The network is revealed in the PSO scheme to make a swarm. For the ideal presentation of the approach, the primary swarms escalate larger. To adjust the parameters of the PSO, PLBn−1 and PGBn−1 indicates the swarm’s position as well as velocity. The mathematical notations are written as:
Xin=Xin−1+Vin−1,Vin=ωVin−1−n1r1(Xin−1−PLBn−1)−a2r2(Xin−1−PGBn−1).
In the above equations, the particle and velocity components are X_{i} and V_{i} for the i^{th} vector, ω is an inertia weight vector. The random vectors are r1 and r2, whereas the acceleration constants are n1 and n2. The velocity element vector lies in the interval [-v_{max}, v_{max}], (v_{max} shows the maximum velocity).
Interior-point programming (IPP): It adjusts the PSO parameters to converge more promptly by integrating the best global weights. These best global PSO weights are applied as an initial input. In recent years, IPP has been applied in numerous applications, e.g., riveting simulation in aircraft parts [64], complementarity monotone systems [65], viscoplastic fluidics system [66], dispatch system of the financial load [67], identification of the nonlinear stable system [68], non-smooth interaction dynamics [69], reactive optimal power flow problem with discrete control variables [70] and flow constraints in a pressure-dependent water distribution system [71]. This study is related presenting the hybrid form of the PSOIPP, which is pragmatic to compute the variables for the TON-DD-EFM. The pseudocode details using the ANN-PSOIPP are given in Tab. 1.
Optimization procedure of the ANN-PSOIPP scheme
PSO procedure starts
1: Initialization: Produce the prime swarms and adjust the optimizations.
2: Fitness Assessment: Examine the e for each particle in Eq. (7).
3: Ranking: Define the Rank of each particle using the minimum e values
4: Stopping criteria: Stop if
• Fit level proficient
• Cycles performed
If the terminating criteria obtain, move to Step-5
5: Regenerate: Check the velocity and position, using Eqs. (14) and (15).
6: Improvement: Replicate till the total flights are attained.
7: Storage: Store e with the best-accomplished values and indicate the global best particle, i.e., W_{PSO}.
PSO process ends
PSOIPP process starts
Inputs: W_{PSO}
Output: Best PSOIPP vectors are signified as W_{PSOIPP}
Initialize Use W_{PSO} as an initial point.
Terminate: The method terminates, when (e = 10^{−20}), (TolX = TolCon = 10^{−20}), (Iterations = 850), (TolFun = 10^{−21}), (MaxEvals = 262000).
While (Terminate)
Fitness Evaluations: For e, use the model (6)
Modifications: Invoke the routine of fmincon.
Move to the “fitness step” by taking the enhanced weight vector.
Store: W_{PSOIPP}, the final weight vector, time, e, generations, and function count for the present trials.
PSOIPP process ends
Statistical Performance
Three statistical measures T.I.C, R.MSE, and ENSE are presented in this section. The mathematical form of these operators by taking the exact and proposed solutions v and v^ are written as:
R.MSE=[1n∑m=1n(vm−v^m)2],T.I.C=ab,a=1n∑m=1n(vm−v^m)2,b=(1n∑m=1nvm2+1n∑m=1nv^m2)NSE = {1−cd,c=∑m=1n(vm−v^m)2,d=∑m=1n(vm−v¯m)2,v¯m=1n∑m=1nvm,ENSE = 1−NSE
Results and Discussions
The detail for two examples of both the types of the singular TON-DD-EFM using the design ANN-PSOIPP scheme is provided in this section. The first two examples are obtained by taking the values of χ=2 and t=1 in Eq. (4), while the third and fourth examples are obtained by taking χ=1 and t=1 in Eq. (5).
Example 1: Consider the singular TON-DD-EFM having triple singular points is written as:
{d3dε3v(ε−1)+4εd2dε2v(ε−1)+2ε2ddεv(ε−1)+εv2=ε7+2ε4+ε+30−36ε+6ε2,v(0)=1,dv(0)dε=0,d2v(0)dε2=0.
The true solution of the above Eq. (20) is 1+ε3.
Example 2: Consider the singular TON-DD-EFM having triple singular points involving trigonometric ratios are given as:
{d3dε3v(ε−1)+4εd2dε2v(ε−1)+2ε2ddεv(ε−1)+εv2=ε54−2ε2+6ε+ε2−2ε2sin(ε−1)−4εcos(ε−1)+ε3cosε+εcos2ε,v(0)=1,dv(0)dε=d2v(0)dε2=0.
The true solution is cos(ε+0.5ε2).
Example 3: Consider the singular TON-DD-EFM involving exponential based function is written as:
{d3dε3v(ε−1)+1εd2dε2v(ε−1)+εev=12−6ε+εe1+ε+ε3,v(0)=1,dv(0)dε=1,d2v(0)dε2=0.
The true solution of Eq. (22) is 1+ε+ε3.
Example 4: Consider the singular TON-DD-EFM involving trigonometric based function is written as:
{d3dε3v(ε−1)+1εd2dε2v(ε−1)+εv2=εsin2ε+2εsinε+ε−cos(ε−1)−1εsin(ε−1),v(0)=dv(0)dε=1,d2v(0)dε2=0.
1+sin(ε) is the exact solution of Eq. (23).
The proposed procedure based on the ANN-PSOIPP is implemented for the singular TON-DD-EFM based Examples for 100 trials to get the system optimization of the model parameters. The best vectors are described to demonstrate the estimated forms of the TON-DD-EFM using ten neurons. The obtained numerical standards are given as follows:
v^1(ε)=−0.62811+e−(4.500ε+5.8496)+6.37471+e−(2.429ε+4.0923)−3.28901+e−(−3.4927ε−7.719)+…−5.99391+e−(1.187ε+0.0367),v^2(ε)=8.67411+e−(3.964ε−13.084)+8.32181+e−(−2.947ε−11.536)+1.44361+e−(−20.00ε+10.213)+…−9.98611+e−(5.377ε−7.163),v^3(ε)=1.52211+e−(−1.0355ε+5.657)+5.58441+e−(−0.805ε−2.2374)+0.44641+e−(1.8942ε+5.7719)+…−5.28121+e−(−2.0613ε−2.5638),v^4(ε)=−15.54171+e−(14.063ε−15.968)+6.9991+e−(−6.825ε−9.814)+7.35351+e−(0.942ε−1.5638)+…+1.07631+e−(−0.2511ε+14.2354).
Optimization is performed for solving the singular TON-DD-EFM based examples 1-4 using the combination of the PSOIPP for 100 runs. Figs. 1a–1d signifies the optimized weight vectors of ANNs to solve each example of the singular TON-DD-EFM, and these weights are given in Eqs. (24)–(27). The result comparisons for all the examples of the singular TON-DD-EFM based on the obtained and exact solutions are provided using the proposed ANN-PSOIPP scheme in Figs. 1e–1h. The results are overlapped for the TON-DD-EFM, which specifies the exactness of ANN-PSOIPP. For the level of accuracy, the absolute error (AE) is calculated in Fig. 2. The second portion of Fig. 2 specifies the performance procedures of T.I.C, ENSE, and R.MSE, for each example of the singular TON-DD-EFM. It is indicated that the RMSE lies as 10^{−04} to 10^{−06}. The TIC measures for each Example lies 10^{−08} to 10^{−10} , and the ENSE for each example lie 10^{−06} to 10^{−08}, whereas, for example four the ENSE is found 10^{−10} to 10^{−12}. These achieved results state the good tendency of routine using different measures for TON-DD-EFM.
Best weights and results comparison using the ANN-PSOIPP scheme for the TON-DD-EFMAE and performance measures using the ANN-PSOIPP scheme for solving all examples of the singular TON-DD-EFM
Statistics presentations using 100 executions for the proposed ANN-PSOIPP scheme using the analysis of fitness, RMSE, TIC, and ENSE together with the histogram (Hist) plots are provided in Figs. 3–6 for solving the singular TON-DD-EFM. It is evident in the figures that the maximum values of these statistical operators lie in suitable ranges for solving all examples of the singular TON-DD-EFM.
Statistics measures of ANN-PSOIPP over Fit values together with the histogram for the TON-DD-EFMStatistics measures based on ANN-PSOIPP through R.MSE values for the plots of the histogram for the TON-DD-EFMStatistics measures for ANN-PSOIPP over TIC values using the plots of the histogram for the TON-DD-EFMStatistics measures for the ANN-PSOIPP over ENSE values for the TON-DD-EFM
The convergence inquiries of the proposed ANN-PSOIPP scheme are shown further for global minimum and median performances of ‘G-FIT’, ‘G-TIC’, and ‘G-ENSE’ in Tab. 2. The Min G-FIT, G-TIC and G-ENSE lie 10^{−09}-10^{−10}, 10^{−09}-10^{−11}, 10^{−07}-10^{−11}, while the Med G-FIT, G-TIC, and G-ENSE were found as 10^{−06} to 10^{−08}, 10^{−05} to 10^{−08}, 10^{−02} to 10^{−06} for solving all examples of the singular TON-DD-EFM using the proposed ANN-PSOIPP scheme. The relative optimal performances enhance the accuracy of the ANN-PSOIPP scheme.
Global measures for the TON-DD-EFM
Problem
G-FIT
G-TIC
G-ENSE
Min
Med
Min
Med
Min
Med
1
3.6451E-09
2.7317E-06
6.0194E-09
5.0914E-05
4.9722E-08
8.0258E-03
2
1.4088E-10
2.5963E-06
2.2477E-09
6.9910E-07
1.0648E-07
1.2287E-02
3
5.2131E-10
2.4361E-07
7.2487E-11
2.7574E-07
1.1618E-08
1.3322E-05
4
1.0169E-10
1.0808E-08
2.9898E-10
6.0008E-08
6.8193E-11
1.6666E-06
The complexity of the ANN-PSOIPP scheme is observed over the generations, execution time and count of functions. Complexity investigations for each example of the singular TON-DD-EFM are provided. Tab. 3 shows the average generations, implementation time, and function counts are 348.52201, 5925.55750, and 113290.64250, for each example of the singular TON-DD-EFM using the proposed ANN-PSOIPP scheme.
Complexity performance for the TON-DD-EFM
Example
Generations
Implementation time
Function counts
Mean
SD
Mean
SD
Mean
SD
1
308.41584
101.38361
5850.68000
1903.06119
100782.91000
20986.77951
2
520.50647
1896.54312
6021.64000
1726.61116
103962.76000
22631.48651
3
284.61720
66.05547
5946.71000
1892.85971
124857.60000
26124.96281
4
280.54852
65.36187
5883.20000
1974.41039
123559.30000
26577.49751
Conclusion
The present study shows that a precise, stable, accurate, and reliable ANN-PSOIPP scheme is accessible for the third-order delay differential Emden-Fowler model by applying the continuous mapping and approximation capability of ANNs. The optimization of the fitness/merit of these networks is obtained by applying the global and local search capabilities of PSO and the IPP approach. The ANN-PSOIPP scheme is viably executed to solve four examples of the third kind of singular delay differential singular system. The precise performances are examined using the numerical ANN-PSOIPP scheme for singular delay differential Emden-Fowler system based on AE with reliable precision of about 5-7 decimals of correctness from the true solutions. The statistical explanations are also obtainable in the form of Min, Mean and Median actions to authenticate the robustness of the numerical ANN-PSOIPP scheme for the singular model.
Funding Statement: This project is funded by National Research Council of Thailand (NRCT) and Khon Kaen University: N42A650291.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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