# Modelling of Behavior for Inhibition Corrosion of Bronze Using Artificial Neural Network (ANN)

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}SO

_{4}, using the experimental data of Electrochemical Impedance Spectroscopy (EIS). The database was divided into training, validation, and test sets randomly. The parameters process used as the inputs of the ANN models were frequency, temperature, and inhibitor concentration. The outputs for each ANN model and the components in the EIS spectrum (Z

_{re}, Z

_{im}, and Z

_{mod}) were predicted. The transfer functions used for the learning process were the hyperbolic tangent sigmoid in the hidden layer and linear in the output layer, while the Levenberg–Marquardt algorithm was applied to determine the optimum values of the weights and biases. The statistical analysis of the results revealed that ANN models for Z

_{re}, Z

_{im}, and Z

_{mod}can successfully predict the inhibition corrosion behavior of bronze in different conditions, where what was considered included variability in temperature, frequency, and inhibitor concentration. In addition, these three input parameters were keys to describe the behavior according to a sensitivity analysis.

## 1. Introduction

_{2}SO

_{4}) in the presence and absence of ketoconazole as a corrosion inhibitor at 25, 40, and 60 °C. The electrochemical evidence exhibited that ketoconazole inhibits the corrosion of the bronze, forming a protective layer with its conjugated bonds and nitrogen atoms that decrease both the charge transfer and the diffusion of aggressive species towards the metal surface. Hence, the ketoconazole acts as an adequate mixed type corrosion inhibitor [11].

## 2. Experimental

_{2}SO

_{4}) at 25, 40, and 60 °C with inhibitor concentrations of 0, 5, 10, 25, 50, and 100 ppm. All electrochemical measurements were performed in a typical three-compartment glass cell using a calomel electrode and graphite as a reference and counter electrode, respectively [11].

_{re}(Ω·cm

^{2}) and Z

_{im}(Ω·cm

^{2}); the second Z

_{mod}respect to frequency (Hz).

## 3. Artificial Neural Network Methodology

#### 3.1. Database Preparation

_{re}(Ω·cm

^{2}), Z

_{im}(Ω·cm

^{2}) and Z

_{mod}(Ω·cm

^{2}). Table 1 shows the interval work for each input and output for the ANN model.

#### 3.2. Normalization Input Data

#### 3.3. Development of ANN Models

^{®}software (R2015b, Mathworks

^{®}, Natick, MA, USA) was used for the development of the three models, evaluating different combinations of activation functions and the number of neurons was increased until the best correlation between input and output variables was achieved. The training process was purposed to minimize the prediction error of the ANN through the different connections between weights and biases; it was possible using the hyperbolic tangent sigmoid transfer function in the hidden layer and linear transfer function in the output layer.

^{2}). The database was randomly divided into training (60%), test subsets (20%), and validation (20%). Remarking that the last percentage corresponds to new data meaning than the validation values were not used during training. In order to obtain a good performance model and the optimum architecture, it was necessary to decrease differences between experimental and simulated values, increasing the number of neurons in the hidden layer gradually and determining MSE and R

^{2}at the same time to find the minimum value for MSE and maximum for R

^{2}, (Figure 1); when the MSE increased, the training was stopped because at this moment its generate overfitting in ANN and the performance associated to R

^{2}value could not improve, such as in Figure 2, where the plot represents the R

^{2}and MSE function of the number of neurons in the hidden layer for each ANN model.

#### 3.4. Statistical Analysis of Experimental and Predicted Data

^{2}presents the strength of the linear proportion of variability in a dataset, and is the most often seen number between 0 and 1, and R

^{2}near to 1 indicates that a regression line fits that data well [32]. Furthermore, the intercept-slope test (slope = 1 and intercept = 0) was achieved to validate the linearity and exactitude model [34].

#### 3.5. Sensitivity Analysis

_{j}is the relative importance of the frequency, temperature and concentration on the Z

_{re}, Z

_{im}and Z

_{mod}, N

_{i}and N

_{h}are the quantity of input and hidden neurons, respectively; W are connection weights, the superscripts ”I”, “h” and “o” refer to input, hidden and output layers, respectively; and subscripts “k”, “m” and “n” refer to input, hidden and output neurons, respectively [35].

## 4. Results and Discussion

#### 4.1. ANN Model

_{2}SO

_{4}solution with the EIS database at 24 h of exposure to electrolyte; finding that the best architectures were Z

_{re}(3:8:1), Z

_{im}(3:16:1) and Z

_{mod}(3:16:1) (see Figure 2) given that when the number of neurons is major to the values mentioned for each model, the coefficient R

^{2}decreases and the MSE is major then the performance model was lower. All ANN models developed are described by the following equation:

_{b}= Z

_{re}, Z

_{im}, and Z

_{mod}, S is the number of neurons in the hidden layer (S = 8, 16, 16), k is the number of neurons in the input layer (K = 3), W are weights and b the biases. The Table 2, Table 3 and Table 4 list the obtained parameters (W

_{i}, W

_{o}, b

_{1}, and b

_{2}) used for each ANN model; where W

_{i}represent weights in the hidden layer, W

_{0}weights of the output layer; while b

_{1}and b

_{2}correspond to biases values in the hidden and output layer in the same order.

^{2}value is reasonably high, which indicates the predictive power of the models (see Figure 3) for Z

_{re}0.9875, 0.9944 correspond to Z

_{im}, and finally 0.9876 for Z

_{mod}(Table 5). In order to validate the ANN models, the intercept-slope test with 99% confidence was applied to demonstrate the linearity model, as mentioned before. The results are shown in Table 5, which indicates that the model is adequate to describe the behavior for inhibition corrosion of bronze considering that the slope = 1 and intercept = 0.

#### 4.2. Sensitive Analysis of Input Variables

## 5. Conclusions

_{2}SO

_{4}indicating coefficients of determination equivalent to R

^{2}= 0.9875, 0.9944, and 0.9876, for Z

_{re}, Z

_{im}, and Z

_{mod}respectively. Additionally, the models achieved the intercept-slope test requirements.

_{re}model was obtained with (3:8:1) neurons, whereas for Z

_{im}and Z

_{mod}(3:16:1) neurons were used in the (input: hidden: output) layer respectively.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Numerical method for the artificial neural network (ANN) learning process used to predict Z

_{re}, Z

_{im}and Z

_{mod}.

**Figure 2.**Mean Square Error (MSE) and Coefficient of determination (R

^{2}) in function of neurons number for each ANN model (

**a**) Z

_{re}; (

**b**) Z

_{im}and (

**c**) Z

_{mod}.

**Figure 3.**Regression performance of experimental and predicted values of electrochemical impedance spectroscopy for (

**a**) Z

_{re}, (

**b**) Z

_{im}and (

**c**) Z

_{mod}.

**Figure 4.**Comparison between experimental and simulated behavior of Electrochemical Impedance Spectroscopy (EIS) composed by Nyquist (

**a**,

**c**,

**e**) and Bode spectrum (

**b**,

**d**,

**f**) for 25, 40 and 60 °C respectively.

**Figure 5.**Relative importance (%) for input variables on the reponse to (

**a**) Z

_{re}; (

**b**) Z

_{im}and (

**c**) Z

_{mod}.

**Table 1.**Experimental intervals used to acquire the electrochemical impedance spectroscopy (EIS) values Input.

Variable | Interval | Units | |
---|---|---|---|

Input | Frequency (f) | 0.01–100,000 | Hz |

Temperature (T) | 25, 40 and 60 | °C | |

Concentration (X) | 0, 5, 10, 25, 50 and 100 | ppm | |

Output | Z_{re} | 2.43–30,730.92 | Ω·cm^{2} |

Z_{im} | 0.06–14,635.77 | Ω·cm^{2} | |

Z_{mod} | 2.54–34,028.29 | Ω·cm^{2} |

Number of Neurons (S) | Weigths | Bias | ||||
---|---|---|---|---|---|---|

Hidden Layer (S = 8, K = 3), W_{i} = (S, K) | Output Layer (l = 1) | b_{1} (S) | b_{2} (l = 1) | |||

Temperature (K = 2) | Concentration (K = 3) | Frequency (K = 1) | W_{o} (S) | |||

1 | −33.646 | −20.438 | 0.787 | −2747.15 | 38.4 | 3852.324 |

2 | −0.083 | −0.218 | −14,401.331 | 10,146.758 | 1440 | - |

3 | −35.456 | 20.288 | −0.783 | 2747.545 | 9.26 | - |

4 | 29.495 | 212.895 | 237.096 | 3870.101 | −43.1 | - |

5 | −0.084 | −0.222 | −14,575.934 | −3403.786 | 1460 | - |

6 | −3.519 | 20.429 | −0.783 | −2747.269 | −4.93 | - |

7 | 0.285 | −0.118 | −184.512 | 978.928 | 14.4 | - |

8 | −55.035 | −20.405 | 0.783 | −2747.447 | 10.8 | - |

Number of Neurons (S) | Weigths | Bias | ||||
---|---|---|---|---|---|---|

Hidden Layer (S = 16, K = 3), W_{i} = (S, K) | Output Layer (l = 1) | b_{1} (S) | b_{2} (l = 1) | |||

Temperature (K = 2) | Concentration (K = 3) | Frequency (K = 1) | W_{o} (S) | |||

1 | 6.614 | −23.974 | 0 | −764.894 | −3.78 | 650.218 |

2 | −0.036 | −0.011 | 9166.232 | −1805.728 | −915 | - |

3 | −0.191 | 0.322 | 21.357 | −102.331 | 0.665 | - |

4 | −332.336 | −192.892 | 0.786 | −363.086 | 329 | - |

5 | −135.744 | −142.424 | 0.003 | 650.751 | 142 | - |

6 | 1302.857 | 1486.882 | 361.305 | 368.812 | −1950 | - |

7 | −1.379 | −4.616 | −188.514 | 247.849 | 16.9 | - |

8 | 236.332 | −946.128 | 93.239 | 97.318 | 173 | - |

9 | 11.089 | −9.867 | −0.003 | −585.071 | 0.621 | - |

10 | 6.134 | −3.169 | −0.371 | −21.576 | 0.216 | - |

11 | −14.635 | −17.402 | −0.018 | −291.533 | 10.9 | - |

12 | −0.316 | 0.135 | −262.205 | 187.906 | 23.3 | - |

13 | −0.042 | −0.012 | 9380.565 | 629.753 | −937 | - |

14 | −4.066 | 2.754 | 0.02 | −396.08 | −0.54 | - |

15 | 5.255 | −17.41 | −0.015 | 287.948 | 2.15 | - |

16 | 1.68 | 3.64 | 169.097 | 510.83 | −14.6 | - |

Number of Neurons (S) | Weigths | Bias | ||||
---|---|---|---|---|---|---|

Hidden Layer (S = 16, K = 3), W_{i} = (S, K) | Output Layer (l = 1) | b_{1} (S) | b_{2} (l = 1) | |||

Temperature (K = 2) | Concentration (K = 3) | Frequency (K = 1) | W_{o} (S) | |||

1 | −33.176 | 17.960 | 0.012 | −547.000 | −17.189 | 552.201 |

2 | 3.805 | −69.199 | 1271.784 | 527.000 | −61.965 | - |

3 | 0.083 | 1.466 | 251.299 | −107.000 | −22.409 | - |

4 | −0.018 | −0.092 | −6863.647 | −1800.000 | 685.730 | - |

5 | −0.944 | 25.642 | −0.678 | −235.000 | −7.706 | - |

6 | −22.081 | −11.516 | 0.534 | −49.100 | 23.759 | - |

7 | −0.017 | −0.100 | −7271.011 | 654.000 | 726.734 | - |

8 | −33.898 | −11.118 | 0.529 | 49.200 | 34.268 | - |

9 | −0.940 | 21.966 | −0.679 | 236.000 | −6.602 | - |

10 | −0.601 | −0.365 | 37.056 | −168.000 | 0.104 | - |

11 | 4.029 | 145.354 | 114.095 | 547.000 | −22.550 | - |

12 | −17.393 | −10.582 | 15.370 | 0.084 | 8.632 | - |

13 | 20.647 | −2.031 | 1124.730 | −117.000 | −110.286 | - |

14 | −272.457 | −176.891 | 278.738 | −0.078 | 123.102 | - |

15 | −0.020 | −0.083 | −6356.422 | 2930.000 | 634.291 | - |

16 | −1982.569 | 417.592 | −660.569 | 0.112 | 735.831 | - |

Output Variable | Architecture | R^{2} | MSE | Intercept Slope Test | |||
---|---|---|---|---|---|---|---|

a_{min} | a_{max} | b_{max} | b_{min} | ||||

Z_{re} | 3:8:1 | 0.9875 | 0.00659 | 0.0251 | −0.0079 | 1.0049 | 0.9862 |

Z_{im} | 3:16:1 | 0.9944 | 0.00475 | 0.0186 | −0.0013 | 1.0002 | 0.9878 |

Z_{mod} | 3:16:1 | 0.9876 | 0.00686 | 0.0198 | −0.0059 | 1.0009 | 0.9873 |

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**MDPI and ACS Style**

Millán-Ocampo, D.E.; Parrales-Bahena, A.; González-Rodríguez, J.G.; Silva-Martínez, S.; Porcayo-Calderón, J.; Hernández-Pérez, J.A.
Modelling of Behavior for Inhibition Corrosion of Bronze Using Artificial Neural Network (ANN). *Entropy* **2018**, *20*, 409.
https://doi.org/10.3390/e20060409

**AMA Style**

Millán-Ocampo DE, Parrales-Bahena A, González-Rodríguez JG, Silva-Martínez S, Porcayo-Calderón J, Hernández-Pérez JA.
Modelling of Behavior for Inhibition Corrosion of Bronze Using Artificial Neural Network (ANN). *Entropy*. 2018; 20(6):409.
https://doi.org/10.3390/e20060409

**Chicago/Turabian Style**

Millán-Ocampo, D. Elusaí, Arianna Parrales-Bahena, J. Gonzalo González-Rodríguez, Susana Silva-Martínez, Jesús Porcayo-Calderón, and J. Alfredo Hernández-Pérez.
2018. "Modelling of Behavior for Inhibition Corrosion of Bronze Using Artificial Neural Network (ANN)" *Entropy* 20, no. 6: 409.
https://doi.org/10.3390/e20060409