# A Novel Deep Learning Model for Mechanical Rotating Parts Fault Diagnosis Based on Optimal Transport and Generative Adversarial Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Proposed Fault Diagnosis Method

#### 2.1. OT-Caps Model Architecture

- (1)
- The network architecture has been improved to enable the capsule network to directly process the one-dimensional original vibration signal, from where the complexity of data processing is reduced;
- (2)
- The Generative Adversarial Networks (GAN) and optimal transport theory replace the original routing iteration algorithm to calculate the characteristic distribution error. Compared with the traditional model, the complexity and the calculation time are reduced, and the real-time performance of fault diagnosis is improved.

#### 2.1.1. OT-Caps Network Objective Loss Function

_{k}can only be 0 or 1. m

^{+}is the upper bound, which punishes the false positives, that is, false samples are mistaken as true; m

^{−}is the lower bound, which punishes the false negatives, that is, positive samples are mistaken as true. λ is the proportional coefficient, which is used to adjust the proportion of the two bounds. The total loss is the sum of the losses of all samples. Here m

^{+}= 0.9, m

^{−}= 0.1, λ = 0.5. In addition, if k exists, $\Vert {v}_{k}\Vert $ will not be less than 0.9. If k does not exist, $\Vert {v}_{k}\Vert $ will not be greater than 0.1. The importance of penalizing false positives is twice of penalizing false negatives.

#### 2.1.2. OT-Caps Network Training Optimization Algorithm

- (1)
- Input learning rate: the parameters include the attenuation coefficient for moment estimation ${\rho}_{1}$, ${\rho}_{2}$, the constant term $\sigma $, the initialize neural network coefficients $\delta $, the initialize first and second moment variables $s=0$, $r=0$, and the number of iterations $t=1$;
- (2)
- Use the training set data to train the model and output the loss value e;
- (3)
- Calculate the gradient and update the number of iterations:$$g\leftarrow {\nabla}_{\theta}L\left({f}_{\theta}(x),y\right)t\leftarrow t+1;$$
- (4)
- Update the first moment variable:$$s\leftarrow {\rho}_{1}s+\left(1-{\rho}_{1}\right)g;$$
- (5)
- Update the second moment variable:$$r\leftarrow {\rho}_{2}r+\left(1-{\rho}_{2}\right)g\odot g;$$
- (6)
- Correct the deviation of the first and second moments:$$\widehat{s}\leftarrow \frac{s}{1-{\rho}_{1}^{t}},\widehat{r}\leftarrow \frac{r}{1-{\rho}_{2}^{t}};$$
- (7)
- Calculation update factor value:$$\mathsf{\Delta}\theta =-LR\frac{\widehat{s}}{\sqrt{\widehat{r}}+\delta};$$
- (8)
- Update factor: $\theta \leftarrow \theta +\mathsf{\Delta}\theta $, repeat steps 2 to 8.

#### 2.2. OT-Caps Network Feature Distribution Error Acquisition Algorithm

#### 2.2.1. GAN Network Computing Feature Distribution

#### 2.2.2. Measurement of Data Distribution Error

## 3. Experiment Method

#### 3.1. OT-Caps Fault Diagnosis Algorithm Real-Time Comparison Verification

#### 3.1.1. OT-Caps Network Training Optimization Algorithm

#### 3.1.2. Comparison of Fault Recognition Accuracy

#### 3.2. OT-Caps Transfer Capability Comparison Verification

#### 3.2.1. Data Preprocessing

#### 3.2.2. Comparison of Experimental Results

#### 3.3. Gearbox Failure Test

#### 3.3.1. Test Equipment

#### 3.3.2. Test Result

#### 3.4. Bench Test Verification of Integrated Transmission System

#### 3.4.1. Test Equipment

#### 3.4.2. Test Result

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Liu, X.; Song, P.; Yang, C.; Hao, C.; Peng, W. Prognostics and Health Management of Bearings Based on Logarithmic Linear Recursive Least-Squares and Recursive Maximum Likelihood Estimation. IEEE Trans. Ind. Electron.
**2018**, 65, 1549–1558. [Google Scholar] [CrossRef] - Lei, Y.; Li, N.; Gontarz, S.; Lin, J.; Radkowski, S.; Dybala, J. A Model-Based Method for Remaining Useful Life Prediction of Machinery. IEEE Trans. Reliab.
**2016**, 65, 1314–1326. [Google Scholar] [CrossRef] - Miao, X.; Li, S.; Zhu, Y.; An, Z. A Novel Real-Time Fault Diagnosis Method for Planetary Gearbox Using Transferable Hidden Layer. IEEE Sens. J.
**2020**, 20, 8403–8412. [Google Scholar] [CrossRef] - Yan, R.; Shen, F.; Sun, C.; Chen, X. Knowledge Transfer for Rotary Machine Fault Diagnosis. IEEE Trans. Reliab.
**2020**, 20, 8374–8393. [Google Scholar] [CrossRef] - Li, X.; Zhang, X.; Li, C.; Zhang, L. Rolling element bearing fault detection using support vector machine with improved ant colony optimization. Measurement
**2013**, 46, 2726–2734. [Google Scholar] [CrossRef] - Li, J.; Huang, R.; He, G.; Wang, S.; Li, G.; Li, W. A Deep Adversarial Transfer Learning Network for Machinery Emerging Fault Detection. IEEE Sens. J.
**2020**, 20, 8413–8422. [Google Scholar] [CrossRef] - Chen, L.; Wang, Z.; Qin, W.; Ma, J. Fault diagnosis of rotary machinery components using a stacked denoising autoencoder-based health state identification. Signal Process.
**2017**, 30, 377–388. [Google Scholar] - Qian, G.; Lu, S.; Pan, D.; Tang, H.; Liu, Y.; Wang, Q. Edge Computing: A Promising Framework for Real-Time Fault Diagnosis and Dynamic Control of Rotating Machines Using Multi-Sensor Data. IEEE Sens. J.
**2019**, 19, 4211–4220. [Google Scholar] [CrossRef] - Soualhi, A.; Medjaher, K.; Zerhouni, N. Bearing Health Monitoring Based on Hilbert–Huang Transform, Support Vector Machine, and Regression. IEEE Trans. Instrum. Meas.
**2015**, 64, 52–62. [Google Scholar] [CrossRef] [Green Version] - Li, N.; Lei, Y.; Lin, J.; Ding, S. An Improved Exponential Model for Predicting Remaining Useful Life of Rolling Element Bearings. IEEE Trans. Ind. Electron.
**2015**, 62, 7762–7773. [Google Scholar] [CrossRef] - Shalalfeh, L.; AlShalalfeh, A.A. Early Warning Signals for Bearing Failure Using Detrended Fluctuation Analysis. Appl. Sci.
**2020**, 10, 8489. [Google Scholar] [CrossRef] - Feng, J.; Lei, Y.; Jing, L.; Xin, Z.; Na, L. Deep neural networks: A promising tool for fault characteristic mining and intelligent diagnosis of rotating machinery with massive data. Mech. Syst. Signal Process.
**2016**, 72, 303–315. [Google Scholar] - Jia, F.; Lei, Y.; Guo, L.; Lin, J.; Xing, S. A neural network constructed by deep learning technique and its application to intelligent fault diagnosis of machines. Neurocomputing
**2018**, 272, 619–628. [Google Scholar] [CrossRef] - Ince, T.; Kiranyaz, S.; Eren, L.; Askar, M.; Gabbouj, M. Real-Time Motor Fault Detection by 1-D Convolutional Neural Networks. IEEE Trans. Ind. Electron.
**2016**, 63, 7067–7075. [Google Scholar] [CrossRef] - He, M.; He, D. Deep Learning Based Approach for Bearing Fault Diagnosis. IEEE Trans. Ind. Appl.
**2017**, 53, 3057–3065. [Google Scholar] [CrossRef] - Chen, Y.; Peng, G.; Xie, C.; Zhang, W.; Li, C.; Liu, S. ACDIN: Bridging the gap between artificial and real bearing damages for bearing fault diagnosis. Neurocomputing
**2018**, 294, 61–71. [Google Scholar] [CrossRef] - Wei, Z.; Li, C.; Peng, G.; Chen, Y.; Zhang, Z. A deep convolutional neural network with new training methods for bearing fault diagnosis under noisy environment and different working load. Mech. Syst. Signal Process.
**2018**, 100, 439–453. [Google Scholar] - Yang, H.; Zhao, F.; Jiang, G.; Sun, Z.; Mei, X. A Novel Deep Learning Approach for Machinery Prognostics Based on Time Windows. Appl. Sci.
**2019**, 9, 4813. [Google Scholar] [CrossRef] [Green Version] - Jiang, G.; He, H.; Yan, J.; Xie, P. Multiscale Convolutional Neural Networks for Fault Diagnosis of Wind Turbine Gearbox. IEEE Trans. Ind. Electron.
**2019**, 66, 3196–3207. [Google Scholar] [CrossRef] - Li, X.; Ding, Q.; Sun, J. Remaining useful life estimation in prognostics using deep convolution neural networks. Reliab. Eng. Syst. Saf.
**2018**, 172, 1–11. [Google Scholar] [CrossRef] [Green Version] - Kong, Z.; Cui, Y.; Xia, Z.; Lv, H. Convolution and Long Short-Term Memory Hybrid Deep Neural Networks for Remaining Useful Life Prognostics. Appl. Sci.
**2019**, 9, 4156. [Google Scholar] [CrossRef] [Green Version] - Shen, C.; Xie, J.; Wang, D.; Jiang, X.; Shi, J.; Zhu, Z. Improved hierarchical adaptive deep belief network for bearing fault diagnosis. Appl. Sci.
**2019**, 9, 3374. [Google Scholar] [CrossRef] [Green Version] - Hoang, D.; Kang, H. Rolling element bearing fault diagnosis using convolutional neural network and vibration image. Cogn. Syst. Res.
**2018**, 53, 42–50. [Google Scholar] [CrossRef] - Shao, H.; Jiang, H.; Lin, Y.; Li, X. A novel method for intelligent fault diagnosis of rolling bearings using ensemble deep auto-encoders. Mech. Syst. Signal Process.
**2018**, 102, 278–297. [Google Scholar] [CrossRef] - Sabour, S.; Frosst, N.; Hinton, G. Dynamic routing between capsules. In Proceedings of the 31st Conference on Neural Information Processing Systems, Long Beach, CA, USA, 4–9 December 2017; pp. 3859–3869. [Google Scholar]
- Wang, Z.; Zheng, L.; Du, W.; Cai, W.; Zhou, J.; Wang, J.; Han, X.; He, G. A novel method for intelligent fault diagnosis of bearing based on capsule neural network. Complexity
**2019**, 2019, 1–17. [Google Scholar] [CrossRef] [Green Version] - Zhu, Z.; Peng, G.; Chen, Y.; Gao, H. A convolutional neural network based on a capsule network with strong generalization for bearing fault diagnosis. Neurocomputing
**2019**, 323, 62–75. [Google Scholar] [CrossRef] - Wang, Y.; Ning, D.; Feng, S. A Novel Capsule Network Based on Wide Convolution and Multi-Scale Convolution for Fault Diagnosis. Appl. Sci.
**2020**, 10, 3659. [Google Scholar] [CrossRef] - Kao, I.H.; Wang, W.J.; Lai, Y.H.; Perng, J.W. Analysis of permanent magnet synchronous motor fault diagnosis based on learning. IEEE Trans. Instrum. Meas.
**2019**, 68, 310–324. [Google Scholar] [CrossRef] - Zhang, Y.; Xing, K.; Bai, R.; Sun, D.; Meng, Z. An enhanced convolutional neural network for bearing fault diagnosis based on time-frequency image. Measurement
**2020**, 157, 107667. [Google Scholar] [CrossRef] - Zhao, B.; Yuan, Q. Improved generative adversarial network for vibration-based fault diagnosis with imbalanced data. Measurement
**2021**, 169, 108522. [Google Scholar] [CrossRef] - Hinton, G.; Sabour, S.; Frosst, N. Matrix capsules with EM routing. In Proceedings of the 6th international conference on learning representations, ICLR, Vancouver, BC, Canada, 30 April–3 May 2018; pp. 1–15. [Google Scholar]
- Irpino, A.; Verde, R. Dynamic clustering of interval data using a Wasserstein-based distance. Pattern Recognit. Lett.
**2008**, 29, 1648–1658. [Google Scholar] [CrossRef] - Han, T.; Liu, C.; Yang, W.; Jiang, D. Learning transferable features in deep convolutional neural networks for diagnosing unseen machine conditions. ISA Trans.
**2019**, 93, 341–353. [Google Scholar] [CrossRef] [PubMed] - Wu, C.; Jiang, P.; Ding, C.; Feng, F.; Chen, T. Intelligent fault diagnosis of rotating machinery based on one-dimensional convolutional neural network. Comput. Ind.
**2019**, 108, 53–61. [Google Scholar] [CrossRef] - Han, T.; Liu, C.; Yang, W.; Jiang, D. A novel adversarial learning framework in deep convolutional neural network for intelligent diagnosis of mechanical faults. Knowl. Based Syst.
**2019**, 165, 474–487. [Google Scholar] [CrossRef] - Zhang, W.; Peng, G.; Li, C.; Chen, Y.; Zhang, Z. A new deep learning model for fault diagnosis with good anti-noise and domain adaptation ability on raw vibration signals. Sensors
**2017**, 17, 425. [Google Scholar] [CrossRef] [PubMed] - Li, H.; Guo, X.; Ouyang, B.D.; Wang, X. Neural Network Encapsulation. In Proceedings of the European Conference on Computer Vision (ECCV), Munich, Germany, 8–14 September 2018; pp. 252–267. [Google Scholar]

**Figure 4.**Installation diagram of test gearbox and sensor. (

**a**) The structure of the test gearbox; (

**b**) Input shaft acceleration sensor installation position; (

**c**) Output shaft acceleration sensor installation location.

**Figure 10.**Transmission box failure test system. (

**a**) gear transmission box; (

**b**) Internal structure of gear transmission box; (

**c**) Test equipment composition; (

**d**) Test system composition.

**Figure 11.**Collect raw vibration data. (

**a**) Gearbox bearing degradation process; (

**b**) New bearing degradation process.

Number | Layer | Convolution Kernel/Step Size/Channel | Number of Parameters | Output Dimension |
---|---|---|---|---|

1 | Conv1 | 64/8/16 | 1072 | (16,128) |

2 | Conv2 | 1/1/32 | 608 | (32,65) |

3 | Caps1 | - | 768 | (32,4,8) |

4 | Caps2 | - | 3840 | (32,8,1) |

5 | output | - | 2560 | (10,8) |

Network Type | Training Sample | Time/s | Test Samples | Time/s | Test Accuracy |
---|---|---|---|---|---|

CapsuleNet | 7500 | 331.25 | 1000 | 17 | 84.78% |

OT-Caps | 7500 | 24.43 | 1000 | 0.13 | 99.45% |

Network Type | DTS-CNN | 1-DCNN | DACNN | OT-Caps |
---|---|---|---|---|

Diagnosis accuracy | 99.37% | 99.34% | 99.3% | 99.45% |

Rotating Speed/rpm | Load/hp | Damage Size/Inch | Dataset Name |
---|---|---|---|

1772 | 1 | 7\14\21 | A |

1750 | 2 | 7\14\21 | B |

1730 | 3 | 7\14\21 | C |

Algorithm | A→B | A→C | B→A | B→C | C→A | C→B | Mean |
---|---|---|---|---|---|---|---|

SVC | 71.93 | 72.90 | 76.33 | 75.30 | 98.03 | 94.77 | 81.55 |

KNN | 83.27 | 87.33 | 78.57 | 83.17 | 97.80 | 91.97 | 87.02 |

AlexNET | 98.93 | 92.27 | 95.07 | 94.40 | 88.40 | 96.87 | 94.32 |

ResNet | 99.70 | 94.40 | 94.87 | 94.33 | 88.70 | 98.47 | 94.58 |

ICN | 98.23 | 97.17 | 99.80 | 94.71 | 94.93 | 98.10 | 97.15 |

ACDIN | 98.30 | 75.33 | 91.20 | 82.30 | 68.00 | 88.80 | 83.99 |

WDCNN | 99.50 | 86.20 | 92.40 | 89.80 | 76.03 | 83.90 | 87.97 |

OT-Caps | 100.00 | 99.89 | 99.10 | 100.0 | 95.38 | 99.63 | 99.00 |

Network Type | Training Sample | Time/s | Test Samples | Time/s | Test Accuracy |
---|---|---|---|---|---|

CapsuleNet | 3500 | 148.38 | 850 | 33.09 | 99.85% |

OT-Caps | 3500 | 11.71 | 850 | 0.17 | 100% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, X.; Liu, X.; Song, P.; Li, Y.; Qie, Y.
A Novel Deep Learning Model for Mechanical Rotating Parts Fault Diagnosis Based on Optimal Transport and Generative Adversarial Networks. *Actuators* **2021**, *10*, 146.
https://doi.org/10.3390/act10070146

**AMA Style**

Wang X, Liu X, Song P, Li Y, Qie Y.
A Novel Deep Learning Model for Mechanical Rotating Parts Fault Diagnosis Based on Optimal Transport and Generative Adversarial Networks. *Actuators*. 2021; 10(7):146.
https://doi.org/10.3390/act10070146

**Chicago/Turabian Style**

Wang, Xuanquan, Xiongjun Liu, Ping Song, Yifan Li, and Youtian Qie.
2021. "A Novel Deep Learning Model for Mechanical Rotating Parts Fault Diagnosis Based on Optimal Transport and Generative Adversarial Networks" *Actuators* 10, no. 7: 146.
https://doi.org/10.3390/act10070146