Bioinspired computational heuristics to study Lane–Emden systems arising in astrophysics model
 Iftikhar Ahmad^{1}Email author,
 Muhammad Asif Zahoor Raja^{2},
 Muhammad Bilal^{3} and
 Farooq Ashraf^{1}
Received: 8 May 2016
Accepted: 11 October 2016
Published: 24 October 2016
Abstract
This study reports novel hybrid computational methods for the solutions of nonlinear singular Lane–Emden type differential equation arising in astrophysics models by exploiting the strength of unsupervised neural network models and stochastic optimization techniques. In the scheme the neural network, subpart of large field called soft computing, is exploited for modelling of the equation in an unsupervised manner. The proposed approximated solutions of higher order ordinary differential equation are calculated with the weights of neural networks trained with genetic algorithm, and pattern search hybrid with sequential quadratic programming for rapid local convergence. The results of proposed solvers for solving the nonlinear singular systems are in good agreements with the standard solutions. Accuracy and convergence the design schemes are demonstrated by the results of statistical performance measures based on the sufficient large number of independent runs.
Keywords
Background
Introduction
In astrophysics, a fluid obeys a polytropic equation of state under the assumption, and then with suitable transformation laws Eq. (1) is equivalent to equation of static equilibrium. Further in case of the gravitational potential of a selfgravitating fluids the LEE is also called Poisson’s equation. Physically, hydrostatic equilibrium provides a connection between the gradient of the potential, the pressure and the density. Numerical simulation is presented in (Shukla et al. 2015) for two dimensional Sine–Gordon equation.
Analytically, it is difficult to solve these equations, so various techniques like Adomian decomposition method (ADM), differential transformation method (DTM) and perturbation temple technique based on series solutions have been used (Wazwaz 2001, 2005, 2006; Mellin et al. 1994). Ramos (2005) solved singular IVPs of ODEs using linearization procedures. Liao (2003) presented ADM for solving LEEs (Chowdhury and Hashim 2007a). Chowdhury and Hashim (2007b) employed Homotopyperturbation method (HPM) to get the solution for singular IVPs of LEEs. Dehghan and Shakeri (2008) provides the solution of ODEs models arising in the astrophysics field using the variational iteration procedure. Further, the solution of Emden–Fowler equation (EFE) is also reported by incorporating the method of Lie and Painleve analysis by Govinder and Leach (2007). Kusano provide solutions for nonlinear ODEs based on EFEs (Kusano and Manojlovic 2011). Muatjetjeja and Khalique (2001) given the exact solution for the generalized LEEs of two kinds. Modified Homotopy analysis method (HAM) is used by Singh et al. (2009) and Mellin et al. (1994) to get the numerical solution of LEEs. Demir and Sungu (2009) gives the numerical solutions of nonlinear singular IVP of EFEs using DTM. Shukla et al. (2015) provides the studies using the cubic Bspline differential quadrature method. Moreover, neural networks applications in astronomy, Astrophysics and Space Science can be seen in Bora et al. (2008, 2009), Bazarghan and Gupta (2008), Singh et al. (1998, 2006), Gupta et al. (2004), Gulati et al. (994).
Recently, a lot of effects has been made by the researcher in the field of artificial neural networks (ANNs) to investigate the solution of the IVPs and boundary value problems (BVP) (Ahmad and Bilal 2014; Rudd and Ferrari 2015; Raja 2014a; Raja et al. 2015b). Wellestablished strength of neural networks as a universal function approximation optimized with local and global search methodologies has been exploited to solve the linear and nonlinear differential equations such as problems arising in nanotechnology (Raja et al. 2016b, e), fluid dynamics problems based on thin film flow (Raja et al. 2015a, 2016d), electromagnetic theory (Khan et al. 2015), fuel ignition model of combustion theory (Raja 2014b), plasma physics problems based on nonlinear Troesch’s system (Raja 2014c), electrical conducting solids (Raja et al. 2016c), magnetohydrodynamic problems (Raja and Samar 2014e) JafferyHamel flow in the presence of high magnetic fields (Raja et al. 2015c), nonlinear Pantograph systems (Ahmad and Mukhtar 2015; Raja 2014d; Raja et al. 2016a) and many others. These are motivating factors for authors to develop a new ANNs based solution of differential equations, which has numerous advantages over its counterpart traditional deterministic numerical solvers. First of all, ANN methodologies provide the continuous solution for the entire domain of integration, generalized method which can be applied for the solution of other similar linear and nonlinear singular IVPs and BVPs. Aim of the present research is to develop the accurate, alternate, robust and reliable stochastic numerical solvers to solve the LaneEnden equation arising in astrophysics models.
Organization of the paper is as follows: “Methods” section gives the proposed mathematical modelling of the system. In “Learning methodologies” section, learning methodologies are presented. Numerical experimentation based on three problems and cases is presented in “Results and discussion” section. In “Comparison through statistics” section comparative studies and statistical analysis are presented. In last section conclusions is drawn with future research directions.
Methods
Mathematical modelling
In Eq. (1), the generic form of nonlinear singular Lane–Emden equation is given. While in Eqs. (3–5) continuous mapping of neural networks models for approximate solution \(\hat{y}\left( x \right)\) and its derivatives are presented in term of single input, hidden and output layers. Additionally, in the hidden layers logsigmoid activation function and its derivatives are used for y(t) and its derivatives, respectively.
Fitness function formulation
Learning methodologies
Pattern search (PS) optimization technique belongs to a class of direct search methods, i.e., derivative free techniques, which are suitable to solve effectively the variety of constrained and unconstrained optimization problems. Hooke et al., are the first to introduce the name of PS technique (Hooke and Jeeves 1996), while the convergence of the algorithm was first proven by Yu (1979). In standard operation of PS technique, a sequence of points, which are adapted to desire solution, is calculated. In each cycle, the scheme finds a set of points, i.e., meshes, around the desired solution of the previous cycle (Dolan et al. 2003). The mesh is formulated by including the current point multiplied by a set of vectors named as a pattern (Lewis et al. 1999). PS technique is very helpful to get the solution of optimization problem such as minimization subjected to bound constrained, linearly constrained and augmented convergent Lagrangian algorithm (Lewis et al. 2000).
Genetic algorithms (GAs) belong to a class of bioinspired heuristics develop on the basis of mathematical model of genetic processes (Man et al. 1996). GAs works through its reproduction operators based on selection operation, crossover techniques and mutation mechanism to find appropriate solution of the problem by manipulating candidate solutions from entire search space (CantuPaz 2000). The candidate solutions are generally given as strings which is known as chromosomes. The entries of chromosome are represented by genes and the values of these genes represents the design variables of the optimization problem. A set of chromosome in GAs is called a population which used thoroughly in the search process. A population of few chromosomes may suffer a premature convergence where as large population required extensive computing efforts. Further details, applications and recent trends can be seen in (Hercock 2003; Zhang et al. 2014; Xu et al. 2013).
Sequential quadratic programming (SQP) belong to a class of nonlinear programming techniques. Supremacy of SQP method is well known on the basis of its efficiency and accuracy over a large number of benchmark constrained and unconstrained optimization problems. The detailed overview and necessary mathematical background are given by Nocedal and Wright (1999). The SQP technique has been implemented in numerous applications and a few recently reported articles can be seen in Sivasubramani et al. (2011), Aleem and Eldeen (2012).

Step 1: Initialization: create an initial population with bounded real numbers with each row vector represents chromosomes or individuals. Each chromosome has number of genes equal to a number of design parameters in ANN models. The parameter settings GAs are given in Table 1.Table 1
Parameters setting for SQP, PS and GA respectively
Methods
Parameter
Setting
Parameter
Setting
SQP
Initial weight vector creation
Randomly between (−1, 1)
Maximum function evaluation
10000
Number of variable
30
Fitness limit
10^{−25}
Total initial weight vectors
1000
X tolerance
10^{−25}
Number of iterations
10000
Function tolerance
10^{−30}
Derivative
By solvers
Nonlinear constraint tolerance
10^{−30}
Finite difference type
Central
Upper bound
−10
Hessian
BFGS
Lower bound
10
Algorithm
SQP
Others
Default
PS
Solver
Pattern search
Maximum size
Inf
Start point
Randn (1, 30)
Scale
On
Poll method
GPS positive 2N
Bind tolerance
10^{−03}
Complete poll
Off
Maximum iteration
2000
Polling order
Consecutive
Max function evaluation
1,000,000
Initial size
1
X tolerance
10^{−25}
Expansion factor
2
Function tolerance
10^{−30}
Tolerance
Eps
Nonlinear constraint tolerance
10^{−30}
Mesh tolerance
10^{−32}
Plot
Function value
GA
Solver
GA
Pop type
Double vector
No of variables
30
Pop size
[30, 30, 30, 30, 30, 30, 30, 30, 30]
Time limit
Default
Initial range
[0,1]
Generations
1,000,000
Scaling fun
Rank
Stall generations
2000
Selection fun
Stochastic
Function tolerance
10^{−28}
Interval
20
Nonlinear constraint tolerance
10^{−28}
Fraction
0.2
Initial penalty
10
Plot
Best function
Penalty factor
100
Elite count
2
Crossover
Forward
Time limit
Inf
Direction
Forward
Other
Defaults

Step 2: Fitness evaluations: Determined the value of the fitness row vector of the population using the Eq. (7).

Step 3: Stoppage: Terminate GAs on the basis of following criteria.

The predefined fitness value \(\in\) is achieved by the algorithm, i.e., \(\in < 10^{  15} .\)

Total number of generations/iterations are executed.

Any of the termination conditions given in Table 1 for GAs are achieved.

Step 4: Ranking: Rank each chromosome on the basis minimum fitness \(\in\) values. The chromosome ranked high has small values of the fitness and vice versa.

Step 5: Reproduction: Update population for each cycle using reproduction operators based on crossover, mutation selection and elitism operations. Necessary settings of these fundamental operators are given in Table 1.

Step 6: Refinement: The Local search algorithm based on SQP is used for refinement of design parameters of ANN model. The values of global best chromosome of GA are given to SQP technique as an initial start weight. In order to execute the SQP algorithm, we incorporated MATLAB build in function ‘fmincon’ with algorithm SQP. Necessary parameter settings for SQP algorithm is given in Table 1.

Step 7: Data Generation and analysis: Store the global best individual of GA and GASQP algorithm for the present run. Repeat the steps 2–6 for multiple independent runs to generate a large data set so that reliable and effective statistical analysis can be performed.
Results and discussion
The results of simulation studies are presented here to solve LEEs by the proposed ANN solver. Proposed results are compared with reported analytical as well as numerical methods.
Problem I: (Case I: n = 0)
Best weights trained for neural network modelling by SQP, PS and PSOSQP algorithms
Methods  I  \(\delta_{i}\)  β _{ i }  _{ ω i }  i  δ i _{ i }  β _{ i }  _{ ω i } 

SQP  1  0.78636  0.11707  −1.72866  6  2.24414  −1.07067  3.55435 
2  −2.80683  0.84866  −1.47352  7  0.72638  0.80491  −1.05825  
3  −0.66561  0.11462  −1.56349  8  −0.35824  −0.30248  −0.81978  
4  −2.77591  −0.10056  −0.56814  9  −0.93043  −0.67351  0.00426  
5  0.35878  −1.54018  −0.39116  10  0.77053  1.44174  1.15517  
PS  1  3.03789  −0.35385  0.02289  6  −1.38575  4.00223  −8.80831 
2  0.82521  −0.82359  1.73409  7  1.10965  −0.74481  −1.14468  
3  −8.99993  −3.18106  −4.91916  8  −0.27787  −3.01028  0.61488  
4  −1.05818  0.54020  1.74266  9  −5.93029  0.35017  −2.00264  
5  7.54212  0.28198  −0.83137  10  −2.05182  0.45484  2.87243  
PSSQP  1  −0.72218  0.55533  3.06589  6  −1.26715  1.30549  −1.92557 
2  0.70591  3.76635  4.43837  7  −1.18046  −1.15109  −0.18455  
3  0.09391  3.18733  1.15390  8  0.00790  1.08650  0.35001  
4  7.63559  −0.07725  −1.55487  9  0.06668  −4.96006  7.78676  
5  1.38422  −0.76144  −1.69561  10  0.00121  −6.38647  6.67553  
GA  1  1.29680  −0.70586  0.94321  6  0.18766  2.70748  −1.62678 
2  −1.08867  0.43791  1.16502  7  0.31812  2.33138  −0.61683  
3  0.95837  −0.89643  1.47983  8  −0.06982  0.73937  −0.09366  
4  0.415759  3.43301  1.48746  9  −0.55372  0.98114  1.17121  
5  −0.01698  5.29995  −2.90769  10  0.14876  1.78333  0.46163  
GASQP  1  −0.17494  1.78344  0.98665  6  −0.19778  0.02527  0.14497 
2  −0.91882  −0.05487  0.84314  7  −0.18451  −0.15034  0.50728  
3  −0.07822  0.34068  0.50753  8  0.32558  −1.24882  2.01797  
4  0.75215  1.77652  1.46356  9  2.07449  −1.21908  3.30059  
5  −0.08314  1.50680  −1.73595  10  −1.08201  0.038511  1.28033 
Results of learning plots shows that rate of convergence (reduction in the fitness values) and accuracy of GA is relatively better than that of PS techniques, while, the results of the hybrid approaches i.e., PSSQP and GASQP, are found better than that of GA and PS techniques. Additionally, very small difference is observed between the results of the hybrid approaches, however GAPS results are a bit superior.
Comparative studies of the results of proposed methodologies for Problem I
X  Exact  Reported solution  Proposed approximated solution \(\hat{y}({\text{x}})\)  

y(x)  Analytical result  ChNN (Mall and Chakraverty 2014)  SQP  PS  GA  PSSQP  GASQP  
0  1.000000  1.0000  1.0000  1.000000  1.000001  1.000037  1.000000  1.000000 
0.1  0.998333  0.9983  0.9993  0.998349  0.998322  0.998451  0.998224  0.998339 
0.2  0.993333  0.9933  0.9901  0.993376  0.993513  0.993512  0.993197  0.993341 
0.3  0.985000  0.9850  0.9822  0.985053  0.985395  0.985151  0.984859  0.985006 
0.4  0.973333  0.9733  0.9766  0.973380  0.973868  0.973433  0.973186  0.973341 
0.5  0.958333  0.9583  0.9602  0.958369  0.958901  0.958420  0.958184  0.958341 
0.6  0.940000  0.9400  0.9454  0.940032  0.940514  0.940122  0.939851  0.940007 
0.7  0.918333  0.9183  0.9134  0.918369  0.918761  0.918511  0.918183  0.918340 
0.8  0.893333  0.8933  0.8892  0.893375  0.893700  0.893543  0.893182  0.893340 
0.9  0.865000  0.8650  0.8633  0.865044  0.865370  0.865192  0.864848  0.865007 
1  0.833333  0.8333  0.8322  0.833373  0.833750  0.833480  0.833181  0.833340 
Comparative studies based on values of absolute errors (AE) for Problem I
X  Proposed methods (AE)  Reported (AE)  

SQP  PS  GA  PSSQP  GASQP  ChNN (Mall and Chakraverty 2014)  
0  1.90E−09  4.97E−07  6.45E−07  5.60E−10  9.38E−09  0.00E+00 
0.1  1.47E−06  3.47E−05  2.72E−05  1.03E−07  2.07E−06  1.00E−03 
0.2  4.35E−06  8.80E−05  1.41E−05  3.60E−07  3.51E−06  3.20E−03 
0.3  5.61E−06  1.03E−04  8.75E−05  4.93E−07  2.58E−06  2.80E−03 
0.4  5.16E−06  8.88E−05  1.33E−04  4.67E−07  2.19E−06  3.30E−03 
0.5  4.09E−06  7.04E−05  1.26E−04  3.73E−07  2.95E−06  1.90E−03 
0.6  3.47E−06  6.62E−05  8.39E−05  3.12E−07  3.24E−06  5.40E−03 
0.7  3.71E−06  7.61E−05  3.88E−05  3.12E−07  2.62E−06  4.40E−03 
0.8  4.39E−06  8.77E−05  2.09E−05  3.50E−07  2.42E−06  4.10E−03 
0.9  4.69E−06  8.86E−05  4.08E−05  3.78E−07  2.99E−06  1.70E−03 
1  4.27E−06  8.01E−05  7.66E−05  3.46E−07  2.69E−06  1.10E−03 
Results of statistics based on 100 independent runs of each algorithm
Methods  x  Min  Max  Mean  Median  STD  Variance  MSE 

SQP  0.1  2.06E−09  1.58E−05  3.35E−06  2.71E−06  2.81E−06  7.92E−12  3.35E−06 
0.3  3.54E−07  7.33E−05  1.39E−05  1.10E−05  1.19E−05  1.41E−10  1.39E−05  
0.5  3.19E−08  1.03E−04  1.23E−05  8.93E−06  1.32E−05  1.74E−10  1.23E−05  
0.7  3.07E−07  8.17E−05  1.06E−05  7.53E−06  1.07E−05  1.15E−10  1.06E−05  
0.9  4.25E−07  7.35E−05  1.19E−05  9.49E−06  1.08E−05  1.16E−10  1.19E−05  
PS  0.1  8.94E−07  1.24E−03  2.73E−04  2.08E−04  4.82E−05  2.32E−09  2.95E−05 
0.3  1.71E−06  1.52E−03  4.66E−04  3.66E−04  1.09E−04  1.20E−08  4.73E−05  
0.5  2.67E−06  1.39E−03  4.76E−04  3.61E−04  1.06E−04  1.12E−08  4.74E−05  
0.7  3.82E−07  1.56E−03  4.52E−04  3.65E−04  8.49E−05  7.20E−09  4.54E−05  
0.9  1.56E−06  1.52E−03  4.65E−04  3.42E−04  1.01E−04  1.01E−08  4.74E−05  
PSSQP  0.1  7.20E−08  3.51E−04  2.95E−05  1.14E−05  2.65E−04  7.02E−08  2.73E−04 
0.3  3.38E−07  9.57E−04  4.73E−05  1.72E−05  3.78E−04  1.43E−07  4.65E−04  
0.5  3.73E−07  8.88E−04  4.74E−05  1.66E−05  3.87E−04  1.50E−07  4.76E−04  
0.7  2.18E−07  6.01E−04  4.54E−05  1.81E−05  3.72E−04  1.39E−07  4.52E−04  
0.9  1.37E−07  8.15E−04  4.74E−05  1.70E−05  3.86E−04  1.49E−07  4.65E−04  
GA  0.1  1.34E−07  1.55E−03  1.65E−04  7.79E−05  2.54E−04  6.45E−08  1.65E−04 
0.3  1.75E−07  2.86E−03  3.20E−04  2.13E−04  3.83E−04  1.47E−07  3.20E−04  
0.5  1.79E−07  5.57E−03  4.23E−04  1.66E−04  7.39E−04  5.46E−07  4.23E−04  
0.7  1.07E−05  5.82E−03  4.59E−04  1.52E−04  8.02E−04  6.43E−07  4.59E−04  
0.9  1.39E−06  4.39E−03  3.85E−04  1.67E−04  6.13E−04  3.76E−07  3.85E−04  
GASQP  0.1  2.42E−07  3.68E−05  5.58E−06  4.46E−06  2.54E−04  6.45E−08  5.58E−06 
0.3  4.71E−08  5.00E−05  1.19E−05  7.77E−06  3.83E−04  1.47E−07  1.18E−05  
0.5  1.01E−07  5.00E−05  9.80E−06  7.51E−06  7.39E−04  5.46E−07  9.80E−06  
0.7  7.70E−07  5.10E−05  9.94E−06  7.26E−06  8.02E−04  6.43E−07  9.94E−06  
0.9  1.20E−07  5.21E−05  1.07E−05  7.77E−06  6.13E−04  3.76E−07  1.07E−05 
Problem II: (Case II: n = 1)
Comparative studies of the results of proposed methodologies for Problem II
X  Exact  Reported solution  Proposed approximate solution \(\hat{y}({\text{x}})\)  

y(x)  Analytical results  ChNN (Mall and Chakraverty 2014)  SQP  PS  GA  PSSQP  GASQP  
0  1.00000  1.00000  1.00000  1.00000  0.99998  0.99999  1.00000  0.99999 
0.1  0.99800  0.99830  1.00180  0.99800  0.99805  0.99828  0.99833  0.99833 
0.2  0.99300  0.99330  0.99050  0.99300  0.99278  0.99346  0.99335  0.99336 
0.3  0.98500  0.98510  0.98390  0.98500  0.98470  0.98542  0.98513  0.98513 
0.4  0.97300  0.97350  0.97340  0.97400  0.97343  0.97418  0.97375  0.97375 
0.5  0.95800  0.95890  0.95980  0.95900  0.95891  0.95983  0.95935  0.95935 
0.6  0.94100  0.94110  0.94170  0.94200  0.94169  0.94255  0.94209  0.94209 
0.7  0.91800  0.92030  0.92100  0.92200  0.92187  0.92256  0.92217  0.92217 
0.8  0.89300  0.89670  0.89250  0.91200  0.89943  0.90019  0.89979  0.89979 
0.9  0.86500  0.87040  0.87000  0.87500  0.87489  0.87544  0.87521  0.87521 
1  0.83300  0.8415  0.8431  0.84900  0.84832  0.84868  0.84865  0.84865 
Comparative studies based on values of absolute errors (AE) for Problem II
X  Proposed methods (AE)  Proposed (AE)  

SQP  PS  GA  PSSQP  GASQP  ChNN (Mall and Chakraverty 2014)  
0  5.56E−08  1.84E−07  9.06E−07  7.06E−09  4.98E−09  0.00E+00 
0.1  3.28E−06  6.20E−06  2.49E−06  1.40E−06  5.89E−06  3.50E−03 
0.2  3.08E−05  8.52E−06  2.87E−05  2.08E−05  3.41E−05  2.80E−03 
0.3  1.39E−04  6.54E−06  1.36E−04  1.29E−04  1.41E−04  1.20E−03 
0.4  4.25E−04  7.20E−06  4.21E−04  4.17E−04  4.26E−04  1.00E−04 
0.5  1.02E−03  8.28E−06  1.02E−03  1.02E−03  1.02E−03  9.00E−04 
0.6  2.09E−03  7.49E−06  2.09E−03  2.08E−03  2.10E−03  6.00E−04 
0.7  3.84E−03  7.19E−06  3.84E−03  3.83E−03  3.84E−03  7.00E−04 
0.8  6.47E−03  7.98E−06  6.47E−03  6.46E−03  6.46E−03  4.20E−03 
0.9  1.02E−02  7.58E−06  1.02E−02  1.02E−02  1.02E−02  4.00E−04 
1  1.53E−02  7.95E−06  1.53E−02  1.53E−02  1.53E−02  1.60E−03 
Results of statistical operators based on 100 independent runs of each algorithm for Problem II
Methods  X  Min  Max  Mean  Median  STD  Variance  MSE 

SQP  0.1  0.00E+00  1.00E−05  0.00E+00  0.00E+00  2.43E−06  5.92E−12  1.05E−11 
0.3  0.12E−04  0.19E−02  0.14E−03  0.14E−03  9.59E−06  9.19E−11  1.92E−08  
0.5  0.99E−03  0.10E−02  0.10E−02  0.10E−02  8.71E−06  7.59E−11  1.05E−06  
0.7  0.31E−02  0.38E−02  0.38E−02  0.31E−02  6.10E−06  3.72E−11  1.47E−05  
0.9  0.10E−01  0.10E−01  0.10E−01  0.10E−01  5.88E−06  3.45E−11  1.04E−04  
PS  0.1  5.35E−06  1.16E−03  2.62E−04  1.94E−04  2.40E−04  5.75E−08  6.86E−08 
0.3  6.54E−06  2.15E−03  3.46E−04  2.88E−04  3.18E−04  1.28E−07  1.20E−07  
0.5  8.28E−06  1.83E−03  8.89E−04  9.11E−04  4.08E−04  1.66E−07  7.91E−07  
0.7  7.19E−06  4.63E−03  3.67E−03  3.73E−03  5.62E−04  3.16E−07  1.34E−05  
0.9  7.58E−06  1.09E−02  9.99E−03  1.01E−02  0.11E−02  1.16E−06  9.99E−05  
PSSQP  0.1  1.66E−08  2.00E−04  1.73E−05  7.01E−06  2.65E−05  7.01E−10  2.98E−10 
0.3  8.63E−05  0.27E−02  0.14E−03  0.13E−03  2.64E−05  6.99E−10  2.09E−08  
0.5  0.74E−03  0.11E−02  0.10E−02  0.10E−02  3.65E−05  1.33E−09  1.06E−06  
0.7  0.37E−02  0.39E−02  0.38E−02  0.38E−02  3.38E−05  1.14E−09  1.48E−05  
0.9  0.10E−01  0.10E−01  0.10E−01  0.10E−02  3.05E−05  9.29E−10  0.10E−03  
GA  0.1  2.04E−07  9.47E−01  0.18E−01  7.54E−05  1.18E−02  0.14E−01  0.34E−03 
0.3  0.71E−05  9.17E−02  0.18E−01  0.22E−03  1.16E−02  0.13E−01  0.33E−03  
0.5  0.36E−03  8.75E−02  0.18E−01  0.10E−02  1.13E−02  0.11E−01  0.34E−03  
0.7  0.14E−02  8.22E−02  0.20E−01  0.39E−02  1.07E−02  0.11E−01  0.40E−03  
0.9  0.5068E−02  7.57E−02  0.24E−01  0.10E−01  1.00E−02  0.10E−01  0.61E−03  
GASQP  0.1  4.04E−08  4.27E−05  6.15E−06  4.62E−06  6.24E−06  3.89E−11  3.72E−11 
0.3  0.12E−02  0.18E−03  0.14E−03  0.14E−03  1.01E−05  1.02E−10  2.00E−08  
0.5  0.10E−02  0.10E−02  0.10E−02  0.10E−02  8.79E−06  7.72E−11  1.05E−06  
0.7  0.38E−02  0.38E−02  0.38E−02  0.38E−02  8.27E−06  6.84E−11  1.48E−05  
0.9  0.10E−01  0.10E−01  0.10E−01  0.10E−01  7.59E−06  5.76E−11  0.10E−03 
Problem III: (Case III: n = 5)
Comparative studies of the results of proposed methodologies for Problem III
X  Exact  Reported solution  Proposed approximate solution \(\hat{y}({\text{x}})\)  

y(x)  Analytical results  ChNN (Mall and Chakraverty 2014)  SQP  PS  GA  PSSQP  GASQP  
0  1.000  1.000  1.000  1.000  1.0013  1.0003  1.000  1.000 
0.1  0.998  0.998  0.998  0.99835  1.001  0.9987  0.9983  0.99834 
0.2  0.993  0.993  0.994  0.99342  0.9964  0.994  0.9935  0.99341 
0.3  0.985  0.985  0.990  0.98535  0.9878  0.9861  0.9854  0.98534 
0.4  0.973  0.974  0.971  0.97437  0.977  0.9751  0.9744  0.97436 
0.5  0.958  0.961  0.968  0.96078  0.9635  0.9615  0.9608  0.96077 
0.6  0.940  0.945  0.941  0.94492  0.9471  0.9455  0.9449  0.94492 
0.7  0.918  0.927  0.930  0.92716  0.9289  0.9276  0.9271  0.92715 
0.8  0.893  0.908  0.908  0.90785  0.9097  0.9082  0.9078  0.90785 
0.9  0.865  0.887  0.883  0.88737  0.8888  0.8876  0.8873  0.88736 
1  0.833  0.866  0.865  0.86603  0.8673  0.8662  0.866  0.86603 
Comparative studies based on values of absolute errors (AE) for Problem III
X  Proposed methods (AE)  Reported (AE)  

SQP  PS  GA  PSSQP  GASQP  ChNN (Mall and Chakraverty 2014)  
0  4.13E−09  3.361E−06  6.481E−07  6.901E−06  6.096E−08  0.00E+00 
0.1  1.331E−05  5.361E−05  3.561E−05  6.251E−05  9.283E−06  2.001E−04 
0.2  8.653E−05  0.171E−03  0.160E−03  0.106E−03  7.541E−05  0.123E−03 
0.3  0.352E−03  0.437E−03  0.459E−03  0.113E−03  0.338E−03  0.461E−03 
0.4  0.104E−02  0.114E−02  0.115E−02  0.825E−03  0.102E−02  0.324E−02 
0.5  0.245E−02  0.248E−02  0.254E−02  0.228E−02  0.241E−02  0.761E−02 
0.6  0.492E−02  0.497E−02  0.499E−02  0.477E−02  0.497E−02  0.384E−02 
0.7  0.882E−02  0.888E−02  0.888E−02  0.867E−02  0.888E−02  0.321E−02 
0.8  0.145E−01  0.145E−01  0.145E−01  0.143E−01  0.145E−01  0.234E−03 
0.9  0.228E−01  0.224E−01  0.224E−01  0.222E−01  0.223E−01  0.443E−02 
1  3.270E−02  3.274E−02  3.274E−02  3.259E−02  3.269E−02  9.000E−04 
Results of statistics based on 100 independent runs of each algorithm for Problem III
Methods  x  Min  Max  Mean  STD  Variance  MSE 

SQP  0.1  8.691E−08  1.020E−03  2.640E−05  1.030E−04  1.070E−08  2.640E−05 
0.3  0.132E−03  0.723E−03  0.333E−03  5.730E−05  3.290E−09  3.330E−04  
0.5  0.134E−02  0.259E−02  0.242E−02  1.150E−04  1.320E−08  2.420E−03  
0.7  0.783E−02  0.893E−02  0.880E−02  1.030E−04  1.050E−08  8.800E−03  
0.9  0.218E−01  0.224E−01  0.223E−01  5.940E−05  3.530E−09  2.240E−02  
PS  0.1  3.812E−06  5.934E−01  0.937E−02  0.661E−01  0.440E−02  0.937E−02 
0.3  5.215E−06  4.442E−01  0.763E−02  0.511E−01  0.260E−02  0.763E−02  
0.5  2.342E−05  3.891E−01  0.828E−02  0.436E−01  0.220E−02  0.828E−02  
0.7  0.406E−03  3.695E−01  0.137E−01  0.391E−01  0.150E−02  0.137E−01  
0.9  0.158E−01  3.648E−01  0.266E−01  0.355E−01  0.130E−02  0.266E−01  
PSSQP  0.1  3.824E−06  5.933E−01  0.937E−02  0.600E−01  0.440E−02  0.937E−02 
0.3  5.232E−06  4.442E−01  0.763E−01  0.657E−01  0.260E−02  0.763E−02  
0.5  2.341E−05  3.891E−01  0.828E−02  0.699E−01  0.190E−02  0.828E−02  
0.7  0.406E−03  3.695E−01  0.137E−01  0.722E−01  0.150E−02  0.137E−01  
0.9  0.158E−01  3.648E−01  0.266E−01  0.722E−01  0.130E−02  0.266E−01  
GA  0.1  1.224E−06  4.256E−01  0.946E−02  0.600E−01  0.360E−02  0.946E−02 
0.3  5.632E−05  4.670E−01  0.103E−01  0.657E−01  0.430E−02  0.103E−01  
0.5  4.856E−05  4.981E−01  0.132E−01  0.699E−01  0.490E−02  0.132E−01  
0.7  0.634E−01  5.172E−01  0.201E−01  0.722E−01  0.520E−02  0.201E−01  
0.9  0.201E−02  5.224E−01  0.338E−01  0.722E−01  0.520E−02  0.338E−01  
GASQP  0.1  6.86E−09  2.680E−05  7.920E−06  5.42E−06  2.940E−11  7.920E−06 
0.3  0.293E−03  0.387E−03  0.338E−03  1.45E−05  2.110E−10  3.380E−04  
0.5  0.241E−02  0.246E−02  0.244E−03  8.49E−06  7.200E−11  2.440E−03  
0.7  0.879E−02  0.883E−02  0.881E−02  7.79E−06  6.060E−11  8.820E−03  
0.9  0.223E−01  0.223E−01  0.223E−01  7.23E−06  5.220E−11  2.240E−02 
Comparison through statistics

The comparative studies for the given five proposed artificial intelligence solvers are presented for solving LEEs in terms of fitness, mean square error (MSE) and root mean absolute error (RMAE) which is plotted in Figs. 5, 6 and 7, respectively. Furthermore, MSE results of scattered data are shown in Fig. 8.

The selection of appropriate number of neurons in the construction of ANN models has a significant role in the accuracy and complexity of the algorithms. The performance measuring indices bases on MAE, MSE, and RMAE are used to determine/evaluate the most suitable number of neurons in the proposed ANN models.

The histogram studies which show the relative frequency of obtaining performance indices values in certain range. Behavior of proposed methodologies through histogram plots are analyzed for all three problems
These confidence levels indicate that the performance of all six methods based on the fitted normal distribution and SQP showed higher accuracy than the other five in Problem I, in Problem II PS showed higher accuracy than the other five. But in the Problem III hybrid technique PSSQP showed higher accuracy than the other five solvers.
It can be easily observed from these figures that, the result obtained by technique SQP in Problem I. PS in Problem II and hybrid PSSQP in problem III is better than the results obtained by others algorithms. It is observed that for N = 30, our techniques show approximately better results from reported results and obtain the potential to minimize the errors.
Conclusions

The best advantage of the solver based on computational techniques with SQP, PS and GA algorithm to represent the approximate solution of Lane–Emden type differential equations as shown in Fig. 4.

The multiruns of each algorithm independently provide a strong evidence for the accuracy of the proposed method.

The problem is still open for future work with the combination of different activation functions like Bessel’s polynomial and BPolynomial etc.

The potential area of investigations to exploring in the existing numerical solver to deal with singularity along with strong nonlinear problems like nonlinear Lane–Emden equation based systems.

In future one may explore in Runge–Kutta numerical methods with adjusted boundary conditions as a promising future research direction in the domain of nonlinear singular systems for effective solution of Lane–Emden equation arising in astrophysics models for which relatively few solvers are available.
Declarations
Authors’ contributions
IA has designed the main idea of mathematical modeling through neural networks with application of hybrid computational methods for the solutions of nonlinear singular Lane–Emden type differential equation arising in astrophysics models, moreover IA has contributed in conclusion. MAZR carried out the setting of tables and figures and has contributed in learning methodologies. He further contributed in manuscript darting. MB has contributed in data analysis for proposed problems and its performed the statistical analysis. FA carried out the results of proposed problems and performed the solvers and participated in the results and discussion section of the study. All authors read and approved the final manuscript.
Acknowledgements
The author would like to thank SirajulIslam Ahmad for valuable comments in this research work.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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