In modern railway train systems, the safety of fault diagnosis technology is crucial. Currently, data-driven mainstream algorithms have achieved good results in fault diagnosis. However, current research typically focuses on the equipment data set itself, with training and test sets often being in the same distribution. Due to the diversity of equipment and operating conditions, applying these methods to industrial practice can be challenging. The purpose of this study is to address the challenges in applying data-driven fault diagnosis methods to industrial practice, given the diversity of railway train equipment and operating conditions.
To address this issue, the authors propose an unsupervised domain adaptation method for fault diagnosis in subway train transmission systems, which combines convolutional neural networks (CNN) and a novel domain adaptation algorithm, coral-maximum mean discrepancy (Cor-MMD).
First, feature extraction is performed on data sets of subway train transmission systems under various operating conditions, and the data sets are divided into training, validation and test sets. Next, the CNN is trained using the training set to obtain model weight files. Then, based on the weight files along with the labeled training set and unlabeled test set, a deep domain adaptation model is trained and the model is optimized by minimizing the Cor-MMD loss.
Finally, the accuracy of the model was tested on the test set, and the effectiveness of our method was verified through comparative experiments.
1. Introduction
With the development of technology, subways have become a major mode of transportation. Therefore, the reliability and safety of subway trains are critical. The subway train’s transmission system, including traction motors, drive gearboxes and axle boxes, is an important component of the train’s power supply, capable of transmitting kinetic energy from the motor to the wheelset. Its health status will directly impact the operational safety and reliability of the train. Any failure in any part of the train’s transmission system could lead to accidents, causing economic losses and casualties (Ding et al., 2024). Thus, studying fault diagnosis of the train’s transmission system is crucial for ensuring the safe and reliable operation of the train.
Over the past few decades, research on rail train fault diagnosis has been ongoing and has made significant progress. In data-driven rail train fault diagnosis, Cheng et al. (2020) proposed using Deep Slow Feature Analysis and Belief Rule Base for multidimensional data fault detection of wheels; Wang et al. (2020) proposed a fault diagnosis method for the magnetic levitation train suspension system that combines model-based fault diagnosis with data-based implementation methods; Ma et al. (2023) proposed a data-driven model space method based on back propagation neural networks for fault diagnosis of high-speed train air brake pipes. In deep learning-based rail train fault diagnosis, Yin and Zhao (2016) proposed constructing a deep belief network model using multiple stacked Restricted Boltzmann Machines for fault diagnosis of on-board equipment (VOBE) of high-speed trains; Du et al. (2024) proposed a running track grounding fault diagnosis method based on MS-1DCNN; Kou et al. (2020) proposed a multidimensional end-to-end CNN model for fault diagnosis of rotating equipment in high-speed train bogies.
These works typically divide data sets into training and testing sets, extract features from different samples, input the training set into neural networks for multiple rounds of training, retain the best-performing network weights and after completing the training, use the test set to evaluate the performance of the used methods. However, these studies often focus on algorithm research under a same-distribution data set, although their algorithmic results are excellent, they are often difficult to apply to industrial practice due to the diversity of equipment and working conditions. To address this issue, many scholars have conducted in-depth research on domain adaptation. Ding et al. (2023) proposed a new deep imbalanced domain adaptation through cost-sensitive learning and category alignment to overcome class imbalance label shift, improving the accuracy of bearing fault diagnosis; Lu et al. (2020) proposed a deep learning architecture based on unsupervised domain adaptation, which uses adaptive weighting strategies and maximum mean discrepancy (MMD) to adapt to marginal and conditional distribution differences, enhancing the performance of intelligent fault diagnosis; Tong et al. (2018) proposed a domain adaptation transferable features method for bearing fault diagnosis under different working conditions. These studies provide valuable insights and techniques in the field of fault diagnosis, but when facing diverse and complex changing working conditions, the robustness and generalization ability of models still need to be improved.
To solve the aforementioned problems, we propose a deep domain adaptation network (CoMMDA) based on Coral and Multi-kernel Maximum Mean Discrepancy (MK-MMD) for fault diagnosis. CoMMDA extracts features through convolutional neural networks (CNNs) and uses Coral to align source and target domain features, making the features more regular. It also uses MK-MMD to minimize the gap between the source and target domains, mapping them to the same feature space to enhance the generalization of the model. The main contributions of this paper are as follows:
A fault diagnosis framework for the subway train transmission system based on CNNs and domain adaptation methods is proposed. Specifically, we first performed time-frequency domain feature extraction on vibration and current signals from the data set, then fed the features into a CNN network and combined domain adaptation strategies for unsupervised training on the basis of the CNN network, achieving better results on out-of-distribution test sets.
A new domain adaptation method Coral-Maximum Mean Discrepancy (Cor-MMD), integrating Coral and MK-MMD, is proposed. For the bottleneck layer output of the CNN, we used Coral to align source and target domain features, making the features more regular. We also used MK-MMD to minimize the difference between the source and target domains, mapping them to the same feature space to enhance the generalization of the model.
To verify the effectiveness of CoMMDA, we conducted fair comparative experiments on the subway train transmission system data set. Experimental results show that our method improves the accuracy by about 1.13% compared to the best-performing method, validating the effectiveness of our approach.
2. Related work
2.1 Maximum mean discrepancy
In the domain adaptation problem of transfer learning, there are at least two domains, namely the transferred domain and the domain to be learned. The former is called the source domain and the latter is called the target domain. In this work, we mainly discuss the unsupervised domain adaptation problem of the subway train transmission system, that is, for a given labeled source domain data and target domain data , the source domain and the target domain have different joint probability distributions Ps (x, y) ≠ Pt(x, y), that is, using the source domain data to learn a prediction function f:xt → yt on the target domain, so that there is a minimum prediction error on the target domain.
For MMD, we regard the source domain and the target domain as two probability distributions p and q. We use ℋk to represent the reproducing kernel Hilbert space (RKHS) defined by the significant kernel k, where the average embedding of the probability distribution p is expressed as μk (p), and μk (p) is the only element in the space ℋk, such that for any function f ∈ ℋk in the space ℋk, there is:
MK-MMD is defined as the squared distance between the kernel average embeddings of distributed sums in RKHS space:
where ϕ(⋅) defines a mapping from the original data to RKHS. At the same time, if and only if Mk (p, q) = 0, p = q. For two sets of data samples s and t from distributions p and q, the empirical estimate of MK-MMD is:
There, the multi-core MMD defines kernel k(x, x′) = 〈ϕ(x), ϕ(x′)〉 as a combination of m kernels {ku}, and βu is the weight of each kernel:
2.2 Deep coral
The Covariance Alignment (CORAL) (Sun et al., 2016) is a method that reduces the discrepancy between the source and target domains by aligning the features. CORAL minimizes domain shift by comparing the second-order statistics of the source and target distributions. Assuming Cs and Ct are the covariance matrices of the source and target domains, respectively, the covariance alignment method learns a second-order feature transformation A that minimizes the feature distance between the source and target domains:
Then, the features of the source domain and target domain can be transformed respectively as:
where Es and Et are identity matrices of the same size corresponding to the source domain and target domain, respectively. Deep CORAL (Sun and Saenko, 2016) applies this method to deep neural networks and has achieved promising results.
3. Method
The proposed model consists of a CNN along with Coral alignment and MK-MMD. The overall framework is illustrated in Figure 1. First, the CNN is trained using the training set data to obtain the model weight file. Then, the weight file is loaded into a new CNN. Subsequently, the output layer is removed from this CNN, and a bottleneck layer and a new output layer are added, incorporating Coral loss and MK-MMD loss, to form the deep domain adaptation network. By minimizing the Coral loss, MK-MMD loss and classification loss, the entire network structure is continuously optimized to ensure that the model performance reaches its optimal state.
3.1 Convolutional neural networks
Transfer learning typically leverages pre-trained weights from CNNs, such as VGGNet and ResNet which accelerates the convergence and performance improvement of model domain adaptation training. To make it more applicable to fault diagnosis, we obtain the model weight file by training the CNN with the training set. Subsequently, we use these weights in conjunction with the target domain test set for domain adaptation transfer.
We designed the CNN as shown in Figure 1, with specific parameters listed in Table 1. Here, the padding for the convolutional layers is set to 2 because the sample matrices are relatively small; increasing the padding allows for capturing more key features. During the domain adaptation training process, we first import the model weight file into the CNN. Then, we remove the output layer of this CNN and add a bottleneck layer and a new output layer.
3.2 Coral-maximum mean discrepancy
During the domain adaptation training process, we use Cor-MMD to compute the domain adaptation loss, thereby optimizing the deep domain adaptation network. Specifically, the domain adaptation loss consists of the Coral alignment loss and the minimized maximum mean discrepancy (MMD) loss from MK-MMD, as shown in equation (7):
For Coral, we used the loss computation method of DeepCoral, as shown in equation (8). For MK-MMD, the loss calculation is given by equations (3) and (9):
Then, combining the classification loss, the total loss function of CoMMDA is:
where ℓ is the multi-class cross-entropy loss, and α, β are the weights corresponding to the transfer losses. By minimizing ℒ, the model will continuously optimize and achieve the best performance.
4. Experiment
4.1 Introduction of experimental environment and data set
This work was completed on an RTX 2080 Ti (11GB) GPU, using an experimental environment with PyTorch 2.0.0, Python 3.8 (Ubuntu 20.04) and CUDA 11.8. The experimental data set was provided by the National Key Laboratory of Advanced Rail Transit Autonomous Operation at Beijing Jiaotong University (Ding et al., 2024), derived from a subway train transmission system fault simulation test bench. The experimental data were collected from the subway bogie fault simulation test bench shown in Figure 2, which is a scaled-down and adjusted version of a real subway bogie at a ratio of 1:2. Each transmission chain on the test bench includes a motor, a reduction gearbox and an axle box. The transmission chain is driven by a three-phase asynchronous AC motor. The load is applied through a hydraulic device. The motor bearing model is SKF6205-2RSH; the gears in the reduction gearbox are helical gears, with 16 teeth on the driving gear and 107 teeth on the driven gear; the supporting bearing model for the driving gear is Harbin Bearing Factory 32305; the axle box bearing model is Harbin Bearing Factory 352213.
Experimental platform for fault simulation of subway train transmission systems
Experimental platform for fault simulation of subway train transmission systems
We directly used the PHM2024 Challenge Preliminary Data set based on this data set, which contains nine operating conditions and is divided into Training and Test sets. In the Training set, there are three samples for each type of fault, while in the Test set, there are six samples for each type of fault, as detailed in Table 2. Different motor speeds simulate various train operating speeds, and different lateral loads simulate straight-line travel and cornering of the train. A positive lateral load indicates the loading direction is toward the motor side of the tested transmission chain, while a negative value indicates the loading direction is toward the gearbox side of the tested transmission chain.
Working conditions of the PHM2024 challenge preliminary data set
| Dataset | Working condition | |||||
|---|---|---|---|---|---|---|
| Sample 1 | Sample 2 | Sample 3 | Sample 4 | Sample 5 | Sample 6 | |
| Training | 20Hz−10kN | 40Hz−10kN | 60Hz−10kN | NULL | NULL | NULL |
| Test | 20Hz 0kN | 20Hz 10kN | 40Hz 0kN | 40Hz 10kN | 60Hz 0kN | 60Hz 10kN |
Source(s):
Authors’ own work
The experiment collected two types of signals: triaxial vibration and three-phase current, totaling 21 channels, with each channel sampled at a frequency of 64 kHz. The deployment of measurement points is shown in Figure 2. The correspondence of sensor channels is listed in Table 3.
Summary of sensor channel information
| Channel | Components | Deployment location | Signal type |
|---|---|---|---|
| CH1 | Motor | Motor (drive end) | Tri-axial acceleration |
| CH2 | |||
| CH3 | |||
| CH4 | Motor (fan end) | Tri-axial acceleration | |
| CH5 | |||
| CH6 | |||
| CH7 | Motor (cable) | Three-phase current | |
| CH8 | |||
| CH9 | |||
| CH10 | Gearbox | Gearbox (input axle) | Tri-axial acceleration |
| CH11 | |||
| CH12 | |||
| CH13 | Gearbox (output axle) | Tri-axial acceleration | |
| CH14 | |||
| CH15 | |||
| CH16 | Axle box (left) | Axle box (end cover) | Tri-axial acceleration |
| CH17 | |||
| CH18 | |||
| CH19 | Axle box (right) | Axle box (end cover) | Tri-axial acceleration |
| CH20 | |||
| CH21 |
Source(s):
Table courtesy of Ding et al. (2024)
The health conditions of the components (motor, gearbox and axle box) in the data set include normal and 16 typical single faults. Figure 3 [1] illustrates the faulty parts used in the experiments.
4.2 Feature extraction
In the training set, one sample consists of 640,000 sampling points, while in the test set, one sample consists of 64,000 sampling points. For the data samples, we split them into individual samples with a stride of 2048. The training set from the experimental data set is divided into training and validation samples at a ratio of 8:2, and all samples from the test set of the experimental data set are extracted as test samples. This process ultimately yields 12,730 training samples, 3,182 validation samples and 3,162 test samples. Feature extraction is performed on each channel of the samples to obtain corresponding feature samples (size 21 * 20), where the 20 features are as listed in Table 4.
Twenty features and their definitions
| Features name | Definition | Features name | Definition |
|---|---|---|---|
| Root mean square fea1 | Pulse factor fea11 | fea3/fea4 | |
| Peak-to-peak fea2 | max(xi) – min(xi) | Margin factor fea12 | fea3/fea7 |
| Peak fea3 | max(|xi|) | Kurtosis factor fea13 | |
| Mean amplitude fea4 | Center of gravity frequency fea14 | ||
| Variance fea5 | Mean square frequency fea15 | ||
| Standard deviation fea6 | Root mean square frequency fea16 | ||
| Root amplitude fea7 | Frequency variance fea17 | ||
| Skewness factor fea8 | Frequency standard deviation fea18 | ||
| Crest factor fea9 | fea3/fea1 | Power spectral entropy fea19 | |
| Waveform factor fea10 | fea1/fea4 | Singular spectral entropy fea20 |
Source(s):
Authors’ own work
When selecting features, we generally aim to choose only one among highly redundant features to avoid the selection of ineffective features. Pearson correlation analysis can assess the linear relationship between two variables, and through this, it can reflect the redundancy among features. We conducted a Pearson correlation analysis for the 20 features of each channel, removing those with a correlation coefficient absolute value greater than 0.9 to eliminate highly redundant features. After a comprehensive evaluation of the features from each channel, we ultimately selected features 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 18, 19 and 20, forming the final feature-extracted samples (size 21 * 14).
4.3 Domain adaptation training
This paper compares the proposed method with six existing methods. Details of MK-MMD (Long et al., 2015) and Deep Coral (Sun and Saenko, 2016) can be found in Section 2. The specifics of the other four methods are as follows:
LMMD (Zhu et al., 2021): This algorithm builds upon MMD by further incorporating class information from the source and target domains, optimizing domain adaptation effectiveness through minimizing the discrepancy between classes in the source and target domains.
DAAN (Yu et al., 2019): In addition to addressing marginal distribution differences, DAAN introduces the concept of conditional distribution differences and dynamically adjusts network parameters to accommodate varying domain discrepancies. Moreover, DAAN incorporates sub-domain discriminators to further enhance the model’s adaptability.
BNM (Cui et al., 2020): BNM focuses on tackling the issue of scarce labels by maximizing the batch nuclear norm to ensure predictive discriminability and diversity, thereby improving domain adaptation performance.
DANN (Ganin and Lempitsky, 2015): DANN innovatively applies the concept of GANs to the field of transfer learning, achieving good generalization on unseen target domains via adversarial training.
First, set the learning rate to 0.001, use Adam as the optimizer and cross-entropy as the loss function, with a batch size of 32. Input the feature samples of the training set into the CNN for 300 epochs of training to obtain the weight file. The model loss changes in the training process are shown in Figure 4.
Subsequently, using the previously trained CNN as the base network, load the weight file, remove the final output layer and add a bottleneck layer and a fully connected output layer to form a deep domain adaptation network. Similarly, we use Adam as the optimizer, with a batch size of 32 and a learning rate of 0.003, and use the loss function given by equation (12). We input the labeled training set samples and the unlabeled testing set samples for 300 epochs of training, calculating the testing set accuracy after each epoch. The accuracies over the 300 training epochs are as follows:
Among these, the model’s accuracy on the validation set and the test set is as follows (results rounded):
As illustrated in Figure 5, the proposed method converges at the fastest rate and achieves the best performance. The experimental results indicate that while a CNN not subjected to domain adaptation training performs well on the validation set with the same distribution, its performance drops significantly on the test set with a different distribution. In contrast, all domain adaptation algorithms exhibit improved performance on the test set compared to the CNN, underscoring the strong generalization capability of these algorithms.
Detailed results for all methods are shown in Table 5. Notably, CoMMDA outperforms all other methods, achieving an accuracy improvement of approximately 1.13% over the best-performing MK-MMD. This substantiates the effectiveness of the proposed method. Moreover, the high fault diagnosis accuracy on the test set with a different distribution highlights the superior generalization ability of this algorithm. The capacity to maintain high fault diagnosis accuracy across varying distributions can effectively enhance the potential for practical applications of the model, enabling it to better handle complex conditions while ensuring generalization and robustness.
The accuracy of different models (V is validation set and T is test set)
| Method | Accuracy | Method | Accuracy |
|---|---|---|---|
| CNN(V) | 99.96 | Deep Coral(T) (Sun and Saenko, 2016) | 80.20 |
| CNN(T) | 68.66 | BNM(T) (Cui et al., 2020) | 82.86 |
| MK-MMD(T) (Long et al., 2015) | 90.20 | DANN(T) (Ganin and Lempitsky, 2015) | 86.40 |
| LMMD(T) (Zhu et al., 2021) | 85.99 | CoMMDA(T) | 91.33 |
| DAAN(T) (Yu et al., 2019) | 75.30 |
Source(s):
Authors’ own work
The enhanced generalization and robustness provided by the proposed method are crucial for real-world applications, where data distributions can vary widely, and reliable fault diagnosis is essential.
5. Conclusion
In this work, we focus primarily on out-of-distribution test sets and introduce transfer learning into fault diagnosis. Based on a CNN and a domain adaptation strategy that combines Coral and MMD (Cor-MMD), we propose a fault diagnosis framework for subway train transmission systems. Through experimental comparisons, we have demonstrated the effectiveness and generalization capability of the proposed method, which will significantly enhance the robustness of the model when dealing with complex real-world operating conditions. In the future, we aim to further explore domain adaptation strategies and hope to achieve even better model performance.
