# Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions

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## Abstract

**:**

## 1. Introduction

- A pioneering framework using the integrated computational ANN-GA-SQPM is provided to solve the singular model involving the Neumann–Robin, Dirichlet, and Neumann boundary conditions.
- The performance of the computational ANN-GA-SQPM is observed using a small and large number of neurons.
- The matching of the results that were obtained by the proposed computational ANN-GA-SQPM with the exact solutions authenticate the value in terms of convergence and precision.
- The absolute error (AE) is found in good measure for each problem of the singular model.
- The verification of the ANN-GA-SQPM is authorized from the statistical exploration on multiple executions for 10 neurons based on the performance of Variance Account For (VAF), Nash Sutcliffe Efficiency (NSE), and Theil’s Inequality Coefficient (TIC).
- Besides the equitable precise solutions of the system, the easy understanding, smooth operations, robustness, and comprehensive stability are other valued merits.

## 2. Methodology: ANN-GA-SQPM

#### 2.1. ANNs Modeling

#### 2.2. Optimization Process: GA-SQPM

## 3. Results and Discussion

**Problem 1:**Consider the following singular Lane–Emden nonlinear model along with Dirichlet boundary conditions, which are written as:

**Problem 2:**Consider the following singular Lane–Emden nonlinear model along with the Dirichlet boundary conditions, which are written as:

**Problem 3:**Consider the following singular Lane–Emden nonlinear model along with Neumann boundary conditions is written as:

**Problem 4:**Consider the following singular Lane–Emden nonlinear model along with Neumann–Robin boundary conditions used in the modelling of isothermal gas spheres is given as:

## 4. Investigation through Multiple Executions of ANN-GA-SQPM

^{−08}to 10

^{−09}, 10

^{−11}to 10

^{−12}, 10

^{−05}to 10

^{−06}, and 10

^{−10}to 10

^{−11}. The mean values for problem 1 are found around 10

^{−06}to 10

^{−07}, 10

^{−08}to 10

^{−09}, 10

^{−04}to 10

^{−05}, and 10

^{−07}to 10

^{−08}. The worst values even lie around 10

^{−04}to 10

^{−05}, 10

^{−06}to 10

^{−07}, 10

^{−03}to 10

^{−04}, and 10

^{−06}to 10

^{−07}. For problem 2, the FIT, ENSE, RMSE, and EVAF best values lie around 10

^{−09}to 10

^{−10}, 10

^{−12}to 10

^{−14}, 10

^{−05}to 10

^{−07}, and 10

^{−11}to 10

^{−13}. The mean values for problem 2 are found around 10

^{−05}to 10

^{−07}, 10

^{−09}to 10

^{−10}, 10

^{−04}to 10

^{−05}, and 10

^{−06}to 10

^{−08}. The worst values even lie around 10

^{−04}to 10

^{−05}, 10

^{−07}to 10

^{−08}, 10

^{−03}to 10

^{−05}, and 10

^{−06}to 10

^{−07}. For problem 3, the FIT, ENSE, RMSE, and EVAF best values lie around 10

^{−10}to 10

^{−11}, 10

^{−12}to 10

^{−13}, 10

^{−06}to 10

^{−08}, and 10

^{−12}to 10

^{−14}. The mean values for problem 3 are found around 10

^{−05}to 10

^{−06}, 10

^{−08}to 10

^{−10}, 10

^{−04}to 10

^{−05}, and 10

^{−06}to 10

^{−07}. The worst values even lie around 10

^{−04}to 10

^{−05}, 10

^{−06}to 10

^{−08}, 10

^{−02}to 10

^{−03}, and 10

^{−06}to 10

^{−08}. For problem 4, the FIT, ENSE, RMSE, and EVAF best values lie around 10

^{−10}to 10

^{−11}, 10

^{−13}to 10

^{−14}, 10

^{−06}to 10

^{−07}, and 10

^{−13}to 10

^{−14}. The mean values for problem 4 are found around 10

^{−05}to 10

^{−07}, 10

^{−08}to 10

^{−09}, 10

^{−06}to 10

^{−07}, and 10

^{−07}to 10

^{−08}. The worst values even lie around 10

^{−04}to 10

^{−05}, 10

^{−06}to 10

^{−08}, 10

^{−04}to 10

^{−05}, and 10

^{−05}to 10

^{−07}. These optimal close values for each operator enhance the worth of the propose ANN-GA-SQPM for solving the nonlinear singular Lane–Emden system.

## 5. Performance Operators

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Comparison of results along with the AE through ANN-GA-SQPM to solve the singular model of Lane–Emden type. (

**a**–

**d**) for the solution dynamics, while (

**e**–

**h**) for AE.

Start of GA |

Inputs: |

The chromosome with the same entries of the system are signified as: |

$\mathit{W}=[\mathit{r},\mathit{w},\mathit{s}]$ |

Population: The chromosomes set is designated as: |

$\mathit{r}=[{r}_{1},{r}_{2},{r}_{3},\dots {r}_{k}],\hspace{0.17em}\hspace{0.17em}\mathit{w}=[{w}_{1},{w}_{2},{w}_{3},\dots ,{w}_{k}]$ $\mathrm{and}$ $\mathit{s}=[{s}_{1},{s}_{2},{s}_{3},\dots ,{s}_{k}]$ |

$\mathit{P}={[{\mathit{W}}_{1},\hspace{0.17em}\hspace{0.17em}{\mathit{W}}_{2},,\dots ,{\mathit{W}}_{k}]}^{t}$ |

Output: The best weights of GA are W_{Best-GA} |

Initialization |

Create W that is a W_{Best-GA} of real numbers to signify a chromosome. Initialize the W with real entries. Adjust the ‘Generation’ & ‘declarations’ values of ‘gaoptimset’ & GA routines |

Fitness formulation |

Accomplish the ${\xi}_{Fit}$ in P to show all W for Equations (5)–(8) |

Termination |

Stop the process to accomplish |

•
${\xi}_{Fit}$= 10 ^{−18}, TolFun = 10^{−21}, Generations = 100,
,• TolCon = 10 ^{−22}, Population Size = 270, StallGenLimit = 120 |

Go to [storage], when stopping standards obtains. |

Ranking |

Rank W of P for brilliance of ${\xi}_{Fit}$ |

Storage |

Save W_{Best-GA}, iterations, ${\xi}_{Fit}$ and time for the current trials of GAs |

End of GA |

GA-SQPM Start |

Inputs |

W_{Best-GA} is the start point |

Output |

W_{GA-SQPM} represents the best values |

Initialize |

Adjust W_{GA-SQPM} represents an initial input |

Termination |

Stop the procedure, when ${\xi}_{Fit}$= 10 ^{−}^{18′}, generations = 1000, TolFun = 10^{−21}, |

TolX=10^{−}^{19′}, TolCon =10^{−18′}, MaxEvalsFun= 229,000 |

While [Terminate] |

Fitness Calculations |

Calculate ${\xi}_{Fit}$ of the present W using Equations (5–8). |

Amendments |

Invoke ‘fmincon’ for the SQPM. Adjust W for each generation of SQPM. Calculate |

Calculate ${\xi}_{Fit}$ of updated W using Equations (5–8) |

Accumulate |

Store W_{GA-SQPM}, time, ${\xi}_{Fit}$ and number of generations for the current trials of SQPM. |

End of GA-SQPM Procedure |

**Table 2.**Results comparison of Problem 1 based on ANN-GA-SQPM for three, 10, and 15 neurons, or nine, 30, and 45 variables with reference solutions.

$\mathsf{\Omega}$ | Exact | Approximate Results $\widehat{\mathit{z}}(\mathsf{\Omega})$ | ||
---|---|---|---|---|

$\widehat{\mathit{z}}(\mathsf{\Omega})$ | 9 Variables | 30 Variables | 45 Variables | |

0 | 0.69314718056 | 0.68224453346 | 0.69315090195 | 0.69314891025 |

0.05 | 0.69065030036 | 0.67524281241 | 0.69064571170 | 0.69064852047 |

0.1 | 0.68319684971 | 0.66423337183 | 0.68320463422 | 0.68319977498 |

0.15 | 0.67089657163 | 0.64935335193 | 0.67091717339 | 0.67090442363 |

0.2 | 0.65392646741 | 0.63075628397 | 0.65395396612 | 0.65393705056 |

0.25 | 0.63252255874 | 0.60861130029 | 0.63255161225 | 0.63253391938 |

0.3 | 0.60696948432 | 0.58310217609 | 0.60699764007 | 0.60698071676 |

0.35 | 0.57758883993 | 0.55442621042 | 0.57761623704 | 0.57759992784 |

0.4 | 0.54472717544 | 0.52279295679 | 0.54475520147 | 0.54473850462 |

0.45 | 0.50874445756 | 0.48842281690 | 0.50877439795 | 0.50875637605 |

0.5 | 0.47000362925 | 0.45154551423 | 0.47003584924 | 0.47001621283 |

0.55 | 0.42886168591 | 0.41239846634 | 0.42889548155 | 0.42887471647 |

0.6 | 0.38566248081 | 0.37122507743 | 0.38569646043 | 0.38567556481 |

0.65 | 0.34073129354 | 0.32827297406 | 0.34076400945 | 0.34074402602 |

0.7 | 0.29437106060 | 0.28379220851 | 0.29440158495 | 0.29438315629 |

0.75 | 0.24686007793 | 0.23803345472 | 0.24688828252 | 0.24687142635 |

0.8 | 0.19845093872 | 0.19124622183 | 0.19847736564 | 0.19846157782 |

0.85 | 0.14937045665 | 0.14367710991 | 0.14939582945 | 0.14938048998 |

0.9 | 0.09982033528 | 0.09556813137 | 0.09984493662 | 0.09982983598 |

0.95 | 0.04997836981 | 0.04715511977 | 0.05000168636 | 0.04998732054 |

1 | 0 | 0.00133375423 | 0.00002119692 | 0.00000831368 |

**Table 3.**Results comparison of Problem 2 using ANN-GA-SQPM based on three, 10, and 15 neurons or nine, 30, and 45 variables neurons with the reference solutions.

$\mathsf{\Omega}$ | Exact | Approximate Results $\widehat{\mathit{z}}(\mathsf{\Omega})$ | ||
---|---|---|---|---|

$\widehat{\mathit{z}}(\mathsf{\Omega})$ | 9 Variables | 30 Variables | 45 Variables | |

0 | −1.38629436112 | −1.38629436112 | −1.38619926024 | −1.38630126484 |

0.05 | −1.38629592362 | −1.38629592362 | −1.38611885693 | −1.38631522992 |

0.1 | −1.38631936081 | −1.38631936081 | −1.38617352354 | −1.38633499755 |

0.15 | −1.38642091561 | −1.38642091561 | −1.38633380823 | −1.38642949431 |

0.2 | −1.38669428114 | −1.38669428114 | −1.38665342802 | −1.38669799402 |

0.25 | −1.38727044709 | −1.38727044709 | −1.38725068090 | −1.38727265740 |

0.3 | −1.38831731357 | −1.38831731357 | −1.38829571544 | −1.38832031881 |

0.35 | −1.39003890406 | −1.39003890406 | −1.39000152175 | −1.39004329283 |

0.4 | −1.39267396808 | −1.39267396808 | −1.39261717627 | −1.39267897863 |

0.45 | −1.39649373274 | −1.39649373274 | −1.39642230631 | −1.39649806188 |

0.5 | −1.40179854766 | −1.40179854766 | −1.40172202250 | −1.40180115494 |

0.55 | −1.40891317854 | −1.40891317854 | −1.40884175300 | −1.40891377590 |

0.6 | −1.41818054998 | −1.41818054998 | −1.41812154180 | −1.41817964531 |

0.65 | −1.42995382621 | −1.42995382621 | −1.42990946826 | −1.42995237211 |

0.7 | −1.44458685387 | −1.44458685387 | −1.44455392788 | −1.44458570299 |

0.75 | −1.46242316947 | −1.46242316947 | −1.46239459698 | −1.46242261605 |

0.8 | −1.48378398240 | −1.48378398240 | −1.48375199787 | −1.48378364332 |

0.85 | −1.50895575599 | −1.50895575599 | −1.50891569341 | −1.50895489962 |

0.9 | −1.53817818786 | −1.53817818786 | −1.53813127387 | −1.53817637114 |

0.95 | −1.57163349586 | −1.57163349586 | −1.57158645455 | −1.57163106694 |

1 | −1.60943791243 | −1.60943791243 | −1.60939677357 | −1.60943565431 |

**Table 4.**Results comparison of Problem 3 using ANN-GA-SQPM based on three, 10, and 15 neurons or nine, 30, and 45 variables with the reference solutions.

$\mathsf{\Omega}$ | Exact | Approximate Results $\widehat{\mathit{z}}(\mathsf{\Omega})$ | ||
---|---|---|---|---|

$\widehat{\mathit{z}}(\mathsf{\Omega})$ | 9 Variables | 30 Variables | 45 Variables | |

0 | −1.38629436112 | −1.48446054700 | −1.38630356972 | −1.38631710010 |

0.05 | −1.38691916589 | −1.48757921354 | −1.38692665406 | −1.38694119063 |

0.1 | −1.38879124132 | −1.49113183052 | −1.38879709413 | −1.38881050847 |

0.15 | −1.39190359988 | −1.49517102282 | −1.39190920225 | −1.39191976243 |

0.2 | −1.39624469197 | −1.49975342347 | −1.39625061225 | −1.39625835324 |

0.25 | −1.40179854766 | −1.50493923270 | −1.40180467700 | −1.40181070914 |

0.3 | −1.40854497005 | −1.51079149158 | −1.40855102806 | −1.40855665180 |

0.35 | −1.41645977545 | −1.51737500782 | −1.41646560212 | −1.41647178419 |

0.4 | −1.42551507427 | −1.52475487352 | −1.42552068561 | −1.42552789275 |

0.45 | −1.43567958630 | −1.53299452399 | −1.43568510504 | −1.43569335640 |

0.5 | −1.44691898294 | −1.54215330699 | −1.44692454151 | −1.44693355582 |

0.55 | −1.45919624910 | −1.55228356454 | −1.45920191864 | −1.45921127714 |

0.6 | −1.47247205736 | −1.56342727723 | −1.47247782108 | −1.47248710477 |

0.65 | −1.48670514705 | −1.57561238316 | −1.48671091461 | −1.48671979939 |

0.7 | −1.50185270175 | −1.58884895695 | −1.50185835049 | −1.50186665735 |

0.75 | −1.51787071891 | −1.60312551155 | −1.51787614407 | −1.51788384920 |

0.8 | −1.53471436624 | −1.61840575515 | −1.53471952326 | −1.53472673514 |

0.85 | −1.55233832039 | −1.63462618173 | −1.55234324496 | −1.55235015636 |

0.9 | −1.57069708412 | −1.65169487952 | −1.57070187969 | −1.57070870155 |

0.95 | −1.58974527909 | −1.66949189317 | −1.58975006558 | −1.58975694855 |

1 | −1.60943791243 | −1.68787136420 | −1.60944273339 | −1.60944968143 |

**Table 5.**Results comparison of Problem 4 using ANN-GA-SQPM based on three, 10 and 15 neurons or nine, 30, and 45 variables with the reference solutions.

$\mathsf{\Omega}$ | Exact | Approximate Results $\widehat{\mathit{z}}(\mathsf{\Omega})$ | ||
---|---|---|---|---|

$\widehat{\mathit{z}}(\mathsf{\Omega})$ | 9 Variables | 30 Variables | 45 Variables | |

0 | 1.00000000000 | 0.99488248094 | 1.00000783826 | 1.00000105336 |

0.05 | 0.99958359357 | 0.99454671429 | 0.99959083004 | 0.99958445423 |

0.1 | 0.99833748846 | 0.99326955610 | 0.99834175320 | 0.99833784278 |

0.15 | 0.99627096277 | 0.99110494864 | 0.99627165084 | 0.99627085144 |

0.2 | 0.99339926780 | 0.98810559022 | 0.99339689664 | 0.99339889181 |

0.25 | 0.98974331861 | 0.98432281012 | 0.98973898250 | 0.98974290152 |

0.3 | 0.98532927816 | 0.97980646561 | 0.98532421813 | 0.98532898386 |

0.35 | 0.98018805078 | 0.97460485937 | 0.98018334848 | 0.98018795036 |

0.4 | 0.97435470369 | 0.96876467525 | 0.97435109794 | 0.97435478017 |

0.45 | 0.96786783699 | 0.96233093074 | 0.96786565238 | 0.96786801419 |

0.5 | 0.96076892283 | 0.95534694435 | 0.96076809221 | 0.96076910439 |

0.55 | 0.95310163425 | 0.94785431626 | 0.95310179112 | 0.95310174015 |

0.6 | 0.94491118252 | 0.93989292085 | 0.94491179604 | 0.94491117389 |

0.65 | 0.93624367977 | 0.93150090948 | 0.93624420395 | 0.93624356674 |

0.7 | 0.92714554082 | 0.92271472236 | 0.92714555080 | 0.92714537307 |

0.75 | 0.91766293548 | 0.91356910830 | 0.91766222690 | 0.91766277924 |

0.8 | 0.90784129900 | 0.90409715099 | 0.90783993125 | 0.90784120821 |

0.85 | 0.89772490592 | 0.89433030113 | 0.89772317564 | 0.89772489740 |

0.9 | 0.88735650942 | 0.88429841320 | 0.88735484682 | 0.88735655314 |

0.95 | 0.87677704604 | 0.87402978614 | 0.87677583278 | 0.87677708125 |

1 | 0.86602540378 | 0.86355120724 | 0.86602471655 | 0.86602539000 |

**Table 6.**The complexity performance through ANN-GA-SQPM for each example of the singular model of Lane–Emden type.

Problem | Implementation Time | Iterations | Count of Function | |||
---|---|---|---|---|---|---|

Min | SD | Min | SD | Min | SD | |

1 | 522.78467 | 2576.40890 | 433.30000 | 110.29834 | 27589.96000 | 7079.39701 |

2 | 861.83918 | 5412.52809 | 472.52000 | 73.426410 | 29664.62000 | 4479.75041 |

3 | 881.232020 | 26.2453000 | 467.16000 | 104.07147 | 30003.56000 | 6967.51609 |

4 | 437.11324 | 2576.23782 | 338.94000 | 147.12974 | 22032.32000 | 9767.47337 |

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**MDPI and ACS Style**

Nisar, K.; Sabir, Z.; Asif Zahoor Raja, M.; Ag Ibrahim, A.A.; J. P. C. Rodrigues, J.; Refahy Mahmoud, S.; Chowdhry, B.S.; Gupta, M.
Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions. *Sensors* **2021**, *21*, 6498.
https://doi.org/10.3390/s21196498

**AMA Style**

Nisar K, Sabir Z, Asif Zahoor Raja M, Ag Ibrahim AA, J. P. C. Rodrigues J, Refahy Mahmoud S, Chowdhry BS, Gupta M.
Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions. *Sensors*. 2021; 21(19):6498.
https://doi.org/10.3390/s21196498

**Chicago/Turabian Style**

Nisar, Kashif, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Ag Asri Ag Ibrahim, Joel J. P. C. Rodrigues, Samy Refahy Mahmoud, Bhawani Shankar Chowdhry, and Manoj Gupta.
2021. "Artificial Neural Networks to Solve the Singular Model with Neumann–Robin, Dirichlet and Neumann Boundary Conditions" *Sensors* 21, no. 19: 6498.
https://doi.org/10.3390/s21196498