# Predictability of Different Machine Learning Approaches on the Fatigue Life of Additive-Manufactured Porous Titanium Structure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Data

#### 2.2. Machine Learning Models

#### 2.2.1. Multiple Linear Regression (MLR)

_{0}is called the intercept, and w

_{1}to w

_{n}are called the regression coefficients.

#### 2.2.2. Artificial Neural Networks (ANN)

_{i}represents the output, x

_{j}is the input, w

_{ij}is the weight, and f

_{i}is the activation function. When the sum $\sum {w}_{ij}{x}_{j}$ is larger than a limit, the output is activated by the activation function. Furthermore, the values of weight and limit could be revised for the sake of minimizing the output errors.

#### 2.2.3. Support Vector Regression (SVR)

_{p}is a penalty parameter, ${\xi}_{i}$ and ${\xi}_{i}^{*}$ are relaxation factors, and $\epsilon $ is the error tolerance.

_{p}is the kernel parameter.

#### 2.2.4. Random Forests (RFs)

#### 2.3. Model Evaluation

^{2}) and Mean-Squared Error (MSE), and their mathematical expressions are provided below:

#### 2.4. Overall Strategy

^{2}and MSE as evaluation metrics. Based on this analysis, suitable hyperparameter values were recommended for each machine learning model.

## 3. Results

#### 3.1. Data Analysis

#### 3.2. Fatigue-Life Prediction

#### 3.3. Performance of the Models

## 4. Discussion

#### 4.1. Effects of MLR Parameters on Predicted Results and Prediction Accuracy

#### 4.2. Effects of ANN Parameters on Predicted Results and Prediction Accuracy

#### 4.3. Effects of SVR Parameters on Predicted Results and Prediction Accuracy

#### 4.4. Effects of RF Parameters on Predicted Results and Prediction Accuracy

^{2}and MSE) of the RF model on the training set, test set, and overall data. As illustrated in Figure 20a,b, when n_estimators is set to three and max_depth is gradually increased, the R-squared of the RF model gradually increases on both the training set and the test set. Additionally, Figure 20a,b indicates that when max_depth is set to 5 and n_estimators is gradually increased, the R-squared on the training set gradually increases while the R-squared on the test set gradually decreases. Moreover, Figure 20c shows that as both max_depth and n_estimators increase, the R-squared of the RF model on the overall data exhibits a gradual increase, while the MSE decreases.

#### 4.5. Remarks

## 5. Conclusions

- The MLR model’s predictions of fatigue life for AM titanium porous components are not significantly affected by variations in the training sets used.
- To achieve accurate predictions of fatigue life for AM titanium porous components using the ANN model, it is recommended to create the first hidden layer with three or four neurons.
- For the SVR model, gamma equal to 0.0001 and C equal to 30 are recommended for the fatigue-life prediction of AM titanium porous components.
- For accurate predictions of fatigue life in AM titanium porous components using the RF model, it is suggested to set the n_estimators equal to three and the max_depth equal to seven.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Machine learning process flowchart of predicting fatigue life with the synergic effect of yield stress and fatigue stress.

**Figure 6.**The 34 sets of experimental data with the fatigue life of the porous samples against (

**a**) the yield stress and (

**b**) the fatigue stress.

**Figure 7.**Visualization of experimental data: (

**a**–

**e**) box line plots of density, porosity, yield stress, fatigue stress, and fatigue life; (

**f**–

**i**) density plots of density, porosity, yield stress, and fatigue stress against fatigue life, respectively.

**Figure 9.**Comparison of MLR predictive fatigue life and experimental fatigue life: (

**a**) training set, (

**b**) test set, (

**c**) all data, and (

**d**) 95% confidence interval.

**Figure 10.**Comparison of ANN predictive fatigue life and experimental fatigue life: (

**a**) training set, (

**b**) test set, (

**c**) all data, and (

**d**) 95% confidence interval.

**Figure 11.**Comparison of SVR predictive fatigue life and experimental fatigue life: (

**a**) training set, (

**b**) test set, (

**c**) all data, and (

**d**) 95% confidence interval.

**Figure 12.**Comparison of RF predictive fatigue life and experimental fatigue life: (

**a**) training set, (

**b**) test set, (

**c**) all data, and (

**d**) 95% confidence interval.

**Figure 14.**Visualization of MLR prediction accuracy with different hyperparameters: (

**a**) training, set, (

**b**) test set, and (

**c**) all data.

**Figure 15.**ANN visualization with different neurons in the first hidden layer: (

**a**) 3, (

**b**) 4, (

**c**) 5, and (

**d**) 6.

**Figure 16.**Visualization of ANN prediction accuracy with different hyperparameters: (

**a**) training set, (

**b**) test set, and (

**c**) all data.

**Figure 17.**SVR visualization with different gamma and C: (

**a**) 0.001 and 10, (

**b**) 0.001 and 50, (

**c**) 0.001 and 416, (

**d**) 0.0001 and 30, (

**e**) 0.005 and 30, and (

**f**) 0.001 and 30.

**Figure 18.**Visualization of SVR prediction accuracy with different hyperparameters: (

**a**) training set, (

**b**) test set, and (

**c**) all data.

**Figure 19.**RF visualization with different n_estimators and max_depth: (

**a**) 3 and 3, (

**b**) 3 and 5, (

**c**) 3 and 7, (

**d**) 5 and 5, (

**e**) 7 and 5, and (

**f**) 9 and 5.

**Figure 20.**Visualization of RF prediction accuracy with different hyperparameters: (

**a**) training set, (

**b**) test set, and (

**c**) all data.

MLR | Hyperparameters |
---|---|

Model 1 | random_states = 39 |

Model 2 | random_states = 50 |

Model 3 | random_states = 74 |

Model 4 | random_states = 110 |

ANN | Hyperparameters |
---|---|

Model 1 | The first hidden layer has 3 neurons |

Model 2 | The first hidden layer has 4 neurons |

Model 3 | The first hidden layer has 5 neurons |

Model 4 | The first hidden layer has 6 neurons |

SVR | Hyperparameters |
---|---|

Model 1 | gamma = 0.001 and C = 10 |

Model 2 | gamma = 0.001 and C = 50 |

Model 3 | gamma = 0.001 and C = 416 |

Model 4 | gamma = 0.0001 and C = 30 |

Model 5 | gamma = 0.005 and C = 30 |

Model 6 | gamma = 0.01 and C = 30 |

RF | Hyperparameters |
---|---|

Model 1 | n_estimators = 3 and max_depth = 3 |

Model 2 | n_estimators = 3 and max_depth = 5 |

Model 3 | n_estimators = 3 and max_depth = 7 |

Model 4 | n_estimators = 5 and max_depth = 5 |

Model 5 | n_estimators = 7 and max_depth = 5 |

Model 6 | n_estimators = 9 and max_depth = 5 |

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**MDPI and ACS Style**

Gao, S.; Yue, X.; Wang, H.
Predictability of Different Machine Learning Approaches on the Fatigue Life of Additive-Manufactured Porous Titanium Structure. *Metals* **2024**, *14*, 320.
https://doi.org/10.3390/met14030320

**AMA Style**

Gao S, Yue X, Wang H.
Predictability of Different Machine Learning Approaches on the Fatigue Life of Additive-Manufactured Porous Titanium Structure. *Metals*. 2024; 14(3):320.
https://doi.org/10.3390/met14030320

**Chicago/Turabian Style**

Gao, Shuailong, Xuezheng Yue, and Hao Wang.
2024. "Predictability of Different Machine Learning Approaches on the Fatigue Life of Additive-Manufactured Porous Titanium Structure" *Metals* 14, no. 3: 320.
https://doi.org/10.3390/met14030320