The purpose of the current investigations is to solve the nonlinear dynamics based on the nervous stomach model (NSM) using the supervised neural networks (SNNs) along with the novel features of Levenberg-Marquardt backpropagation technique (LMBT), i.e., SNNs-LMBT. The SNNs-LMBT is implemented with three different types of sample data, authentication, testing and training. The ratios for these statistics to solve three different variants of the nonlinear dynamics of the NSM are designated 75% for training, 15% for validation and 10% for testing, respectively. For the numerical measures of the nonlinear dynamics of the NSM, the Runge-Kutta scheme is implemented to form the reference dataset. The attained numerical form of the nonlinear dynamics of the NSM through the SNNs-LMBT is implemented in the reduction of the mean square error (MSE). For the exactness, competence, reliability and efficiency of the proposed SNNs-LMBT, the numerical actions are capable using the proportional arrangements through the features of the MSE results, error histograms (EHs), regression and correlation.

The nonlinear dynamics of the nervous stomach model (NSM) have three segments, Tension (T), Food (F) and Medicine (M), i.e., TFM model. The generic form of the TFM system

The detail of each segment of the nonlinear TFM system is described together with the description of each parameter shown in the above

Parameters | Description |
---|---|

sleep factor | |

Quality of food | |

Recovered rate | |

Death rate | |

Tension rate | |

ICs |

The stomach, like other organs in the human's body has great significance. In human body every organ is linked to other organs, e.g., eye, nose and ear. If one organ doesn't work properly, the other organs also feel its effects. Likewise, the stomach is linked to many other organs that play a vital role in maintaining the human health. The researchers studied the stomach for many years and the earliest Greeks professed the natural bitter gastric materials. In the beginning of the 16th century, the idea that stomach comprises acid in the digestion procedures presented by Sánchez et al. [

The nonlinear dynamics of the NSM based TFM system is solved using the supervised neural networks (SNNs) along with the novel features of Levenberg-Marquardt backpropagation technique (LMBT), i.e., SNNs-LMBT. The obtained results have been compared with the designed database results based on the Runge-Kutta method. The data percentages to solve three different cases of the nonlinear dynamics of the NSM based TFM are designated 75% for training, 15% for validation and 10% for testing, respectively. The stochastic methods have been applied in diverse recent applications [

A computational intelligent novel SNNs-LMBT is implemented to solve the nonlinear dynamics of the NSM based TFM system.

The exact matching of the numerical outcomes with good measures based on the absolute error (AE) enhances the value of the proposed novel SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM system.

The presentations using the related soundings on MSE, regression metrics, correlation measures and error histograms (EHs) validate the performances of the proposed novel SNNs-LMBT.

The remaining paper parts are organized as: The numerical performances of the proposed novel SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM systems are described in Section 2. The numerical results through SNNs-LMBT are presented in Section 3. The concluding outcomes with latent related soundings together with the future research reports are labeled in the Section 4.

The proposed SNNs-LMBT is provided in two phases to solve the nonlinear dynamics of the NSM based TFM system.

Essential explanations of the proposed SNNs-LMBT are provided.

Implementation measures support the proposed SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM system.

The appropriate optimization based on the proposed SNNs-LMBT are provided in

This section indicates the numerical performances of the obtained results of three different cases based on the nonlinear dynamics of the NSM using the proposed SNNs-LMBT. The mathematical form of each case is given as:

Case-1: Consider the nonlinear dynamics of the NSM based TFM system with

Case-2: Consider the nonlinear dynamics of the NSM based TFM system with

Case-3: Consider the nonlinear dynamics of the NSM based TFM system with

The numerical performances are obtained using the SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM system with inputs (0, 1) and 0.01 step size. The ‘nftool’ command in Matlab is usedto solve the nonlinear dynamics of the NSM based TFM system using 15 numbers of neurons using the data as designated 75% for training, 15% for validation and 10% for testing, respectively. The achieved performance of the results using the SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM system is drawn in

The graphs of the designed SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM system are provided in ^{−10}, 1.3029 × 10^{−11} and 2.3984 × 10^{−14}, respectively. ^{−08}, 9.9615 × 10^{−08} and 9.8429 × 10^{−08}. These graphical plots indicate the precision, accuracy and convergence of the designed SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM system. The fitting curve plots for each case of the nonlinear dynamics of the NSM based TFM system are drawn in

Case | MSE | Performance | Gradient | Mu | Epoch | Time | ||
---|---|---|---|---|---|---|---|---|

[Training] | [Validation] | [Testing] | ||||||

I | 1.07 × 10^{−10} |
1.76 × 10^{−10} |
1.60 × 10^{−10} |
1.07 × 10^{−10} |
9.96 × 10^{−08} |
1.00 × 10^{−09} |
125 | 6 s |

II | 6.30 × 10^{−12} |
1.30 × 10^{−11} |
3.12 × 10^{−11} |
6.30 × 10^{−12} |
9.68 × 10^{−08} |
1.00 × 10^{−11} |
71 | 4 s |

III | 2.87 × 10^{−14} |
2.31 × 10^{−14} |
8.39 × 10^{−15} |
2.87 × 10^{−14} |
9.84 × 10^{−08} |
1.00 × 10^{−13} |
74 | 4 s |

The result comparisons are plotted in ^{−05} to 10^{−06} for case I and II, while the AE for case III lie around 10^{−06} to 10^{−08}. ^{−04} to 10^{−06} for case I and II, while the AE for case III lie around 10^{−06} to 10^{−10}. Similarly, the AE for M(y) is noticed in ^{−05} to 10^{−06} for case I and II, while the AE for case III lie around 10^{−07} to 10^{−08}. These closely matched values of AE indicate the correctness of the proposed SNNs-LMBT to solve the nonlinear dynamics of the NSM based TFM system.

The current investigations are related to solve the nonlinear dynamics of the nervous stomach model based on the three factors; Tension, Food and Medicine are using the proposed supervised neural networks along with the Levenberg-Marquardt backpropagation technique, i.e., SNNs-LMBT. The stomach is linked to many other organs that play a vital role in maintaining the human health. The SNNs-LMBT is applied to the sample data testing, training, and authentication. The percentages used for these statistics for solving three different variants of the nervous stomach model are designated 75% for training, 15% for validation and 10% for testing, respectively. To check the brilliance, excellence, exactness and precision of the SNNs-LMBT, the matching of the outcomes is obtained to solve the nonlinear dynamics of the nervous stomach model. The presentations based on the MSE convergence are applied to the testing, best curve, training and authentication for each factor in the nonlinear dynamics of the nervous stomach model. The correlation performances are proficient to authenticate the regression procedures. The gradient values using the step size are attained for each factor of the nonlinear dynamics of the nervous stomach model. Moreover, the exactness, precision, correctness is observed using the graph as well as the numerical conformations through the EHs, the MSE catalogues, regression dynamics, and convergence plots, respectively.

In future, the proposed SNNs-LMBT can be explored to solve the fractional models, lonngren-wave models, fluid systems and higher order singular models [

The authors would like to thanks the editors of CMC and anonymous reviewers for their time and reviewing this manuscript.