In this study, the design of a computational heuristic based on the nonlinear Liénard model is presented using the efficiency of artificial neural networks (ANNs) along with the hybridization procedures of global and local search approaches. The global search genetic algorithm (GA) and local search sequential quadratic programming scheme (SQPS) are implemented to solve the nonlinear Liénard model. An objective function using the differential model and boundary conditions is designed and optimized by the hybrid computing strength of the GA-SQPS. The motivation of the ANN procedures along with GA-SQPS comes to present reliable, feasible and precise frameworks to tackle stiff and highly nonlinear differential models. The designed procedures of ANNs along with GA-SQPS are applied for three highly nonlinear differential models. The achieved numerical outcomes on multiple trials using the designed procedures are compared to authenticate the correctness, viability and efficacy. Moreover, statistical performances based on different measures are also provided to check the reliability of the ANN along with GA-SQPS.

In scientific fields, most of the real-world problems are observed in plasma physics, solid state physics, mathematical biology, fluid dynamics, and chemical kinetics, which have been stated in the form of differential systems [

The present study is related to solving a nonlinear Liénard model (NLM) using the efficiency of artificial neural networks (ANNs) along with hybridization procedures of global search genetic algorithm (GA) and local search sequential quadratic programming scheme (SQPS). All the above-cited schemes have their own benefits and drawbacks, however, the stochastic computing efficiency based on ANN along with GA-SQPS is neither been tested nor implemented to solve the stiff-natured NLM. The NLM is mathematically given as [

The above form of the NLM is used to generalize the damped springmass system as well as these models have been applied in the physically substantial areas by taking different values of the

Due to the huge importance of the NLM, the focus of the research community was to present the significant applications of this model [

Few novel influences based on the stochastic scheme are shortened as:

Design of ANN along with GA-SQPS is presented successfully for solving the NLM.

The matching of achieved and exact results for three different problems based on the NLM establish convergence, reliability and precision.

The values of the absolute error (AE) are obtained in good measures for solving each example based on the NLM.

The consistent, accurate performance is observed through statistical observations on multiple trials of ANN along with GA-SQPS in terms of semi-interquartile range (S.I.R), mean absolute deviation (MAD), Nash Sutcliffe Efficiency (NSE) and Theil’s Inequality Coefficient (TIC) metrics.

Beside sensibly precise continuous input training interval, ease in the concept, robustness, smooth implementable training, extendibility and stability are other commendable proclamations for the designed ANN along with GA-SQPS.

The rest of the paper is organized as;

This section presents the design of the ANN along with GA-SQPS measures to solve the nonlinear Liénard differential model. The detail of fitness function is accessible using the mean square error along with the learning practice of the hybrid computing GA-SQPS.

The ANNs have been exploited to solve various applications arising in numerous fields [^{th} ^{th} component form of

An error based fitness function (

Here

The optimization processes to solve the nonlinear Liénard differential model using the designed ANN along with GA-SQPS measures are provided in this section.

GA is an optimization based global search technique work to present the solutions of the linear and nonlinear networks. It is normally implemented to regulate the precise population results for numerous steep/complex systems based on the optimal training performance. GA is applied recently as automatic clustering [

The performance based on the GA optimization converges rapidly by the hybridization with the local search scheme by taking its best values as initial inputs. Therefore, an efficient SQPS, i.e., a local search technique is implemented in the process of hybridization. SQPS is used recently in unified full-chip CMP model aware dummy fill insertion framework [

The statistical operators performances of MAD, TIC, S.I.R and ENSE are provided together with the global representations to check the reliability of the designed ANN along with GA-SQPS. The detailed pseudocode based structure of the designed ANN along with GA-SQPS for solving the NLM is presented in

_{GA}_{−Best} _{GA}_{−Best}_{Fit}^{−19}], [^{−20}],^{−21}], [StallLimit = 135],_{Fit}_{GA}_{−Best}_{GA}_{−Best}_{Fit}_{GA}_{−Best}_{GA}_{−SQPS}_{GA−Best}, Iterations & Projects._{Fit}^{−19}],^{−19}]._{Fit} _{GA}_{−SQPS} _{Fit}_{GA}_{−SQPS}_{Fit} |

The current section present the result discussions for three different problems based NLM by applying the ANN along with GA-SQPS. The graphical and numerical measures have been provided to assess the convergence and accuracy.

The exact form of the solution is

The exact solution of

The exact solution of

The results based on NLM for problems 1 to 3 have been performed through the numerical performance of global and local search GA-SQPS. These optimization procedures of GA-SQPS is provided in

The graphical measures are provided in ^{−07} to 10^{−10}, 10^{−05} to 10^{−07} and 10^{−06} to 10^{−08}, respectively. The mean AE values were found around 10^{−02}–10^{−04}, 10^{−03}–10^{−04} and 10^{−02}–10^{−03} for problems 1, 2 and 3. The performance procedures for the best, mean and worst results for the NLM is provided in ^{−07} to 10^{−10}, 10^{−10} to 10^{−11} and 10^{−13}–10^{−15}, the values based on the mean operators of MAD, TIC and ENSE lie as 10^{−03}–10^{−05}, 10^{−06}–10^{−08} and 10^{−03}–10^{−04}, while the worst MAD, TIC and ENSE operators found as 10^{−01}–10^{−02}, 10^{−04}–10^{−05} and 10^{−02}–10^{−03}. For 2^{nd} problem, the best MAD, TIC and ENSE operators are calculated as 10^{−05}–10^{−06}, 10^{−08}–10^{−09} and 10^{−04}–10^{−05}, the MAD, TIC and ENSE operator mean values are 10^{−03}–10^{−04}, 10^{−07}–10^{−08} and 10^{−02}–10^{−04}, while the MAD, TIC and ENSE worst operator measures found as 10^{−02}–10^{−03}, 10^{−05}–10^{−06} and 10^{−03}–10^{−04}. For 3^{rd} problem, the best MAD, TIC and ENSE operators found as 10^{−05}–10^{−06}, 10^{−09}–10^{−10} and 10^{−10}–10^{−11}, the MAD, TIC and ENSE mean operators are 10^{−03}–10^{−04}, 10^{−06}–10^{−08} and 10^{−02}–10^{−04}, while the MAD, TIC and ENSE operator worst performances found as 10^{−01}–10^{−02}, 10^{−03}–10^{−05} and 10^{−02}–10^{−03}. These best presentations through the comparison of the results in the form of AE along with the statistical operators indicate the correctness of the proposed scheme for the NLM.

The statistical representations for Fitness (FIT), TIC, MAD and ENSE together with the histograms are provided in ^{−07}–10^{−09}, 10^{−08}–10^{−11} and 10^{−04}–10^{−08} for 1–3 problems. ^{−08} to 10^{−11} for problem 1 to 3. ^{−04} to 10^{−08} for 1 to 3 problem. ^{−09}–10^{−14} for 1^{st} problem, 10^{−07}–10^{−08} for 2^{nd} problem and 10^{−04} to 10^{−07} for 3^{rd} problem. In another sense, one can observe based on these performances that almost 85% of executions achieved a very reasonable and precise accuracy level of the statistical measures. The optimization tool built-in command MATLAB has been used in this study for the optimization procedure as well as other simulation studies.

In order to observe the precision as well as accuracy level of the proposed ANN along with GA-SQPS, the statistical performances based on minimum (MIN), Maximum (MAX), median (MED), Mean, S.I.R and Standard deviation (STD) are considered for 50 independent executions in

Problem 1 based on the NLM | ||||||
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MIN | MAX | MED | MEAN | S.I.R | STD | |

0 | 7.1801E-10 | 6.3433E-02 | 1.1808E-06 | 9.1625E-04 | 3.1352E-06 | 7.5805E-03 |

0.05 | 3.4027E-10 | 6.8192E-02 | 1.0219E-06 | 9.8453E-04 | 3.0456E-06 | 8.1493E-03 |

0.1 | 2.4784E-08 | 7.1993E-02 | 1.1815E-06 | 1.0393E-03 | 2.9925E-06 | 8.6036E-03 |

0.15 | 3.5463E-08 | 7.5060E-02 | 1.1622E-06 | 1.0835E-03 | 2.7579E-06 | 8.9702E-03 |

0.2 | 5.7298E-09 | 7.7566E-02 | 1.6914E-06 | 1.1196E-03 | 3.1028E-06 | 9.2696E-03 |

0.25 | 6.4170E-10 | 7.9581E-02 | 2.2558E-06 | 1.1486E-03 | 2.5876E-06 | 9.5105E-03 |

0.3 | 1.8354E-08 | 8.1071E-02 | 2.2730E-06 | 1.1702E-03 | 2.2627E-06 | 9.6885E-03 |

0.35 | 1.3211E-08 | 8.1953E-02 | 2.5263E-06 | 1.1829E-03 | 2.6208E-06 | 9.7939E-03 |

0.4 | 2.2913E-09 | 8.2214E-02 | 2.4241E-06 | 1.1864E-03 | 3.1175E-06 | 9.8251E-03 |

0.45 | 4.4739E-09 | 8.1966E-02 | 2.2484E-06 | 1.1825E-03 | 3.1785E-06 | 9.7956E-03 |

0.5 | 1.3619E-08 | 8.1340E-02 | 1.9611E-06 | 1.1730E-03 | 2.8202E-06 | 9.7207E-03 |

0.55 | 3.3247E-08 | 8.0349E-02 | 1.7123E-06 | 1.1581E-03 | 2.9199E-06 | 9.6023E-03 |

0.6 | 2.2385E-08 | 7.8935E-02 | 1.6185E-06 | 1.1371E-03 | 2.8555E-06 | 9.4334E-03 |

0.65 | 8.3347E-08 | 7.7118E-02 | 1.4602E-06 | 1.1105E-03 | 2.4927E-06 | 9.2163E-03 |

0.7 | 7.1063E-08 | 7.5022E-02 | 1.4809E-06 | 1.0804E-03 | 2.4540E-06 | 8.9658E-03 |

0.75 | 1.0672E-07 | 7.2750E-02 | 1.7591E-06 | 1.0480E-03 | 2.8649E-06 | 8.6943E-03 |

0.8 | 5.0016E-08 | 7.0292E-02 | 2.3777E-06 | 1.0131E-03 | 2.8329E-06 | 8.4005E-03 |

0.85 | 6.5628E-08 | 6.7574E-02 | 2.7571E-06 | 9.7450E-04 | 2.8030E-06 | 8.0756E-03 |

0.9 | 8.7921E-08 | 6.4589E-02 | 2.7358E-06 | 9.3220E-04 | 3.6340E-06 | 7.7188E-03 |

0.95 | 6.5819E-08 | 6.1465E-02 | 2.4349E-06 | 8.8786E-04 | 4.1125E-06 | 7.3454E-03 |

1 | 1.0204E-08 | 5.8337E-02 | 2.0280E-06 | 8.4327E-04 | 4.5129E-06 | 6.9715E-03 |

Problem 2 based on the NLM | ||||||
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MIN | MAX | MED | MEAN | S.I.R | STD | |

0 | 1.7302E-07 | 3.6327E-03 | 7.0372E-06 | 1.4608E-04 | 5.6257E-06 | 5.4135E-04 |

0.05 | 1.7891E-06 | 3.7776E-03 | 4.7283E-06 | 1.5545E-04 | 6.2104E-06 | 5.7515E-04 |

0.1 | 5.8262E-07 | 3.9212E-03 | 2.6224E-06 | 1.6482E-04 | 8.3853E-06 | 6.0992E-04 |

0.15 | 5.4155E-08 | 4.0506E-03 | 1.1021E-06 | 1.7447E-04 | 1.0946E-05 | 6.4506E-04 |

0.2 | 5.1690E-07 | 4.1594E-03 | 3.5857E-06 | 1.8567E-04 | 1.1396E-05 | 6.7920E-04 |

0.25 | 2.3866E-06 | 4.2467E-03 | 5.8671E-06 | 1.9663E-04 | 1.1626E-05 | 7.1201E-04 |

0.3 | 8.2746E-07 | 4.3152E-03 | 8.1877E-06 | 2.0717E-04 | 1.1995E-05 | 7.4315E-04 |

0.35 | 8.9591E-07 | 4.3700E-03 | 1.0517E-05 | 2.1755E-04 | 1.2307E-05 | 7.7263E-04 |

0.4 | 7.0989E-07 | 4.5908E-03 | 1.2835E-05 | 2.2785E-04 | 1.2025E-05 | 8.0077E-04 |

0.45 | 2.5323E-07 | 4.7998E-03 | 1.4898E-05 | 2.3811E-04 | 1.1499E-05 | 8.2817E-04 |

0.5 | 9.1944E-07 | 5.0028E-03 | 1.7423E-05 | 2.4906E-04 | 1.1291E-05 | 8.5539E-04 |

0.55 | 2.3375E-07 | 5.2088E-03 | 1.9802E-05 | 2.6026E-04 | 1.5849E-05 | 8.8350E-04 |

0.6 | 4.4174E-07 | 5.4271E-03 | 2.2324E-05 | 2.7182E-04 | 1.7238E-05 | 9.1359E-04 |

0.65 | 1.2717E-06 | 5.6665E-03 | 2.4897E-05 | 2.8396E-04 | 1.8671E-05 | 9.4668E-04 |

0.7 | 2.3874E-06 | 5.9348E-03 | 2.7536E-05 | 2.9689E-04 | 2.0019E-05 | 9.8368E-04 |

0.75 | 3.8665E-06 | 6.2383E-03 | 3.0249E-05 | 3.1081E-04 | 2.1175E-05 | 1.0254E-03 |

0.8 | 5.7209E-06 | 6.5814E-03 | 3.3116E-05 | 3.2591E-04 | 2.1760E-05 | 1.0724E-03 |

0.85 | 7.8909E-06 | 6.9673E-03 | 3.5987E-05 | 3.4239E-04 | 2.1480E-05 | 1.1252E-03 |

0.9 | 7.2039E-06 | 7.3972E-03 | 3.8913E-05 | 3.6048E-04 | 2.1159E-05 | 1.1842E-03 |

0.95 | 7.6656E-06 | 7.8712E-03 | 4.1952E-05 | 3.8048E-04 | 2.0935E-05 | 1.2495E-03 |

1 | 9.6419E-06 | 8.3877E-03 | 4.5191E-05 | 4.0261E-04 | 2.1076E-05 | 1.3212E-03 |

Problem 3 based on the NLM | ||||||
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MIN | MAX | MED | MEAN | S.I.R | STD | |

0 | 2.1534E-07 | 1.6424E-01 | 8.8862E-06 | 8.7643E-03 | 3.1187E-05 | 3.1426E-02 |

0.05 | 4.2019E-07 | 1.6300E-01 | 8.9334E-06 | 8.6753E-03 | 3.1378E-05 | 3.1136E-02 |

0.1 | 2.4513E-07 | 1.5997E-01 | 7.9213E-06 | 8.4790E-03 | 3.2051E-05 | 3.0484E-02 |

0.15 | 2.8576E-07 | 1.5518E-01 | 7.8354E-06 | 8.1761E-03 | 3.4157E-05 | 2.9475E-02 |

0.2 | 3.6012E-07 | 1.4868E-01 | 7.6024E-06 | 7.7692E-03 | 3.7129E-05 | 2.8124E-02 |

0.25 | 2.5432E-07 | 1.4057E-01 | 7.5275E-06 | 7.2632E-03 | 3.9778E-05 | 2.6451E-02 |

0.3 | 1.4575E-07 | 1.3094E-01 | 6.4040E-06 | 6.6649E-03 | 4.1350E-05 | 2.4481E-02 |

0.35 | 7.4969E-08 | 1.1994E-01 | 7.8612E-06 | 5.9827E-03 | 4.1531E-05 | 2.2245E-02 |

0.4 | 5.7261E-07 | 1.0770E-01 | 8.1248E-06 | 5.2274E-03 | 2.8618E-05 | 1.9777E-02 |

0.45 | 2.8892E-07 | 9.4382E-02 | 7.5199E-06 | 4.4116E-03 | 2.8471E-05 | 1.7118E-02 |

0.5 | 4.2034E-08 | 8.0137E-02 | 6.2860E-06 | 3.5486E-03 | 3.2206E-05 | 1.4309E-02 |

0.55 | 7.7385E-08 | 6.5122E-02 | 7.2406E-06 | 2.6567E-03 | 2.6535E-05 | 1.1404E-02 |

0.6 | 2.1992E-08 | 4.9488E-02 | 7.7698E-06 | 1.8206E-03 | 2.0812E-05 | 8.4639E-03 |

0.65 | 1.9363E-07 | 3.3379E-02 | 6.7901E-06 | 1.2347E-03 | 1.6867E-05 | 5.6341E-03 |

0.7 | 1.0930E-07 | 1.6930E-02 | 8.3385E-06 | 1.1406E-03 | 1.9976E-05 | 3.5022E-03 |

0.75 | 7.2498E-07 | 2.4203E-02 | 1.0703E-05 | 1.1245E-03 | 2.0195E-05 | 3.9162E-03 |

0.8 | 6.9259E-07 | 3.7394E-02 | 1.1970E-05 | 2.0373E-03 | 2.4465E-05 | 6.1833E-03 |

0.85 | 1.4595E-08 | 5.0265E-02 | 1.2462E-05 | 2.9488E-03 | 2.8236E-05 | 9.0399E-03 |

0.9 | 2.1956E-07 | 6.2720E-02 | 1.2206E-05 | 3.8425E-03 | 3.1904E-05 | 1.2040E-02 |

0.95 | 1.0596E-06 | 7.4680E-02 | 1.1778E-05 | 4.7159E-03 | 3.5565E-05 | 1.5057E-02 |

1 | 9.2954E-07 | 8.6073E-02 | 1.0857E-05 | 5.5676E-03 | 3.8833E-05 | 1.8043E-02 |

The convergence measure performances are further accompanied based on the global operators of FIT, TIC, MAD and ENSE for fifty executions given in ^{−04}–10^{−06}, 10^{−03}–10^{−04}, 10^{−06}–10^{−07} and 10^{−03}–10^{−06}. The S.I.R performances of these operators found as 10^{−03}–10^{−06}, 10^{−02}–10^{−04}, 10^{−06}–10^{−07} and 10^{−03}–10^{−05}. These performances based global operators confirm the exactness of the designed scheme.

Index | Problem | G.FIT | G.TIC | G.MAD | G.ENSE | ||||
---|---|---|---|---|---|---|---|---|---|

MIN | S.I.R | MIN | S.I.R | MIN | S.I.R | MIN | S.I.R | ||

1 | 1.57E-04 | 1.31E-03 | 1.07E-03 | 8.83E-03 | 5.06E-07 | 4.19E-06 | 6.18E-04 | 5.17E-03 | |

2 | 2.41E-06 | 8.85E-06 | 2.57E-04 | 8.74E-04 | 1.24E-07 | 4.25E-07 | 6.66E-06 | 3.32E-05 | |

3 | 6.58E-04 | 2.72E-03 | 4.86E-03 | 1.72E-02 | 1.92E-06 | 6.19E-06 | 1.70E-03 | 7.76E-03 |

The computational cost of the proposed ANN along with GA-SQPS is inspected through the parameter variation of the typical time, completed cycles/iterations and executed function count. The complexity investigations of each problem of the NLM based on the numerical measures are provided in

Problem | Iterations | Implemented time | Function counts | |||
---|---|---|---|---|---|---|

MIN | STD | MIN | STD | MIN | STD | |

1 | 109.8143 | 35.0169 | 205.0000 | 8.7654 | 13396.3857 | 943.5160 |

2 | 116.5227 | 36.1397 | 204.3000 | 5.8566 | 14380.7000 | 2437.5300 |

3 | 113.4971 | 39.8903 | 205.0000 | 9.1095 | 13965.8857 | 2330.4541 |

The present work is related to solving the nonlinear Liénard model numerically through the computational intelligent ANN procedures and GA-SQPS. The nonlinear Liénard differential model is observed in the generalization of a damped spring mass and the Van der Pol system. The nonlinear Liénard equations are implemented in the modeling of fluid dynamics, vacuum tube technology/radio and oscillating circuits. The optimization of the objective function has been performed through the approximation capability of ANN along with GA-SQPS. The proposed ANN along with GA-SQPS is applied for three problems based on the NLM. The correctness of the scheme is observed by comparing the proposed results with the exact solutions. The detail of the AE, performance measures through different indices, convergent plots and weight vectors have also been provided. The accurate and specific presentations of the scheme are observed as 6 to 8 decimal places of accuracy level from the exact obtainable solutions for each example of NDM. The statistical presentations of MIN, MED, MAX, S.I.R, MEAN and STD measures certify the convergence of the designed scheme for the numerical treatment of the nonlinear Liénard differential model.

In upcoming work, the NLM can be numerically treated by using the swarming optimizations scheme based on the hidden layers of the Meyer and Morlet wavelet neural networks. Moreover, these procedures can be used to solve various nonlinear and fractional order models [