Recurrent neural network

Recurrent neural network (RNN) was first studied in 1980 as a network for processing sequential data. And then with the deepening and progress of theory and computing power, it became a kind of deep learning algorithm in the early 2000s and was continuously used in the field of natural language processing (NLP), such as text generation, machine translation, and semantic recognition. The core of RNN is the existence of a memory function, which can remember the previous sequence of the current sequence and thus obtain the relationship between data at different moments. The health status of equipment components is constantly degraded over time, so the data monitored by the sensors are the time-series data that can be processed by RNN. Guo et al. (Reference Guo, Li, Jia, Lei and Lin2017) used RNN to predict the RUL of bearings, and a better prediction performance was obtained. However, RNN suffers from severe gradient disappearance problems and cannot capture long-time dependence during gradient descent backpropagation. Therefore, the literature on the prediction of component RUL using RNN alone has been almost unretrieved in recent years.

Long short-term memory (LSTM) network is a special kind of RNN, mainly to solve the gradient disappearance problem during the training of long sequences; the structure is shown in Figure 2.

Figure 2. Single-layer structure of long short-term memory.

As can be seen from Figure 2, the LSTM structure has three controlled gates, which are called the forget gate, the input gate, and the output gate. The forget gate processes the information of $ {h}_{t-1} $ and $ {x}_t $ through the sigmoid cell to determine how much information is retained by the cell state $ {C}_{t-1} $ , as shown in Eq. (2):

(2) $$ {f}_t=\sigma \left({W}_f\cdot \left[{h}_{t-1},{x}_t\right]+{b}_f\right). $$

The input gate processes the information of $ {h}_{t-1} $ and $ {x}_t $ through the sigmoid unit to determine how much information to update. The updated information is obtained by processing the information of $ {h}_{t-1} $ and $ {x}_t $ by the tanh unit, as shown in Eqs. (3) and (4):

(3) $$ {i}_t=\sigma \left({W}_i\cdot \left[{h}_{t-1},{x}_t\right]+{b}_i\right), $$

(4) $$ {\overset{\sim }{C}}_t=\tanh \left({W}_C\cdot \left[{h}_{t-1},{x}_t\right]+{b}_C\right). $$

Then the cell state $ {C}_{t-1} $ is updated to $ {C}_t $ according to the input gate and forget gate, as shown in Eq. (5):

(5) $$ {C}_t={f}_t\ast {C}_{t-1}+{i}_t\ast {\overset{\sim }{C}}_t. $$

Finally, the $ {h}_t $ is determined according to $ {C}_t $ , the output gate $ {h}_{t-1} $ , and $ {x}_t $ , as shown in Eqs. (6) and (7):

(6) $$ {o}_t=\sigma \left({W}_o\cdot \left[{h}_{t-1},{x}_t\right]+{b}_o\right), $$

(7), $$ {h}_t={o}_t\ast \tanh \left({C}_t\right). $$

where * represents the Hadamard product, $ {h}_t $ is passed to the next layer, and the above operation is continued. Compared with RNN, the cell state $ {C}_t $ is added to make the memory time series data longer. By stacking LSTM units, the multilayer LSTM networks are constructed to mine the time-series information inside the data and predict the RUL. Gao (Reference Gao2020) proposed a deep learning model based on the LSTM algorithm and feedforward neural network to achieve high prediction accuracy on the C-MAPSS dataset; Sayah et al. (Reference Sayah, Guebli, Al Masry and Zerhouni2021) developed a framework for testing the robustness of deep LSTM networks to ensure the confidence and quality of the training model, and the effectiveness of the framework is validated on the C-MAPSS dataset.

Compared with RNNs, LSTMs have made great progress in prediction performance and are widely used in the research literature. To further improve the ability of LSTM to predict RUL, variants such as bi-directional LSTM (Bi-LSTM), gated recurrent unit (GRU), and bi-directional GRU (Bi-GRU) have also been increasingly applied to the component’s RUL prediction. Yu et al. (Reference Yu, Hu, Si, Zheng and Zhang2020) proposed a Bi-LSTM-based network model averaging method to predict the RUL of graphite/LiCoO₂ batteries and got improved prediction accuracy. Chen et al. (Reference Chen, Jing, Chang and Liu2019) first performed feature selection by the kernel principle component analysis method to address the problem of excessive parameters due to redundancy of sensor data features, and then fed the features into the GRU network to predict the RUL of the turbofan engine. Al-Dulaimi et al. (Reference Al-Dulaimi, Asif and Mohammadi2020) applied Bi-GRU to the C-MAPSS dataset as part of the hybrid framework and obtained a higher accuracy for the turbofan engine’s RUL.

Convolutional neural network

Convolutional neural network (CNN) was developed based on the concept of perceptual wilderness proposed in the 1960s and formally introduced by Yann LeCun of New York University in 1988. Compared with other network structures, the most important feature of CNN is the addition of convolutional and pooling layers. The purpose of convolution is to extract the deeper features of the input, and the pooling layer after the convolution layer is used to compress the amount of data and parameters. The mathematical expression of convolution once and the simple structure diagram are shown in Eq. (8):

(8) $$ {y}_c={f}_c\left(x\cdotp {k}_c+b\right), $$

where $ {f}_c $ represents the activation function, $ x $ represents the input, $ \cdotp $ represents the convolution operation, $ {k}_c $ represents the convolution kernel, and $ b $ represents the bias. The most important feature of CNN is the use of local connectivity and weight sharing, which on the one hand reduces the number of weights making the network easy to train and optimize, and on the other hand, reduces the complexity of the network model and reduces the risk of model overfitting to some extent.

In the field of RUL prediction for equipment components, CNNs have been used in many studies for their powerful feature extraction capability. A single deep CNN (DCNN) structure or the CNN as a feature extractor for other methods is often chosen in the literature to improve the prediction accuracy of RUL. Li et al. (Reference Li, Ding and Sun2018) used DCNN to mine the high-dimensional depth features of the C-MAPSS dataset to obtain more accurate RUL prediction values. Ma and Mao (Reference Ma and Mao2020) added the idea of convolution operation in CNN to the LSTMs, and proposed a new convolutional LSTM (CLSTM) network, which can capture both temporal frequency information and learn temporal information, and achieved higher prediction accuracy compared with DCNN. Liang et al. (Reference Liang, Yuan and Gaon.d.) used the CNN as a feature extractor for lithium battery degradation data, and the features associated with lithium battery degradation were extracted, and then used the Bi-LSTM to integrate the temporal features and predict the RUL.

One-dimensional CNN (1D-CNN) is a network dedicated to time series. Distinct from CNNs applied to image processing, the convolution kernel of 1D-CNN is one-dimensional. Compared with models such as RNN and LSTM, the training convergence of 1D-CNN is faster. In recent years, it is promising to obtain RUL predicting results in the field of equipment components that are not inferior to those of LSTM. Man et al. (Reference Man, Zheng, Liu, Shen and Li2022) first used a generative adversarial network for data augmentation, and then the augmented data were trained adversarially using 1D-CNN, achieving smaller mean absolute deviations and mean square root errors for prediction results. Zheng et al. (Reference Zheng, Bao, Zhao, Chen, Liu, Sun and Wang2022) used 1D-CNN as part of the fused deep learning model to mine data for temporal features to aid RUL prediction. Wang et al. (Reference Wang, Zhang and Wang2022b) used the 1D-CNN to mine the high-dimensional time-series features within the lithium battery data and obtained more accurate battery health state assessment values. Hong et al. (Reference Hong, Ko and Hur2021) combined the concepts of Dilated CNN and EfficientNet, and added the dilated kernels in all hidden layers to cover a larger range of the convolutional kernels’ perceptual fields, which better captured the deeper features of the signal.

Deep belief network

Deep belief network (DBN) first appeared (Hinton and Salakhutdinov, Reference Hinton and Salakhutdinov2006) published in Science by Geoffrey Hinton, a deep learning guru, for image data compression. In the literature (Hinton and Salakhutdinov, Reference Hinton and Salakhutdinov2006), it is pointed out that DBN consists of a multilayer stacked restricted Boltzmann machine (RBM), which is essentially a constructed special neural network. RBM contains two layers of neurons: the visible layer (visible variables) and the hidden layer (hidden variables). The visible layer is used for data input and is also called the input layer; the hidden layer is used for mining the implicit features of the input data, so it is also called the feature extractor. RBM is a probabilistic undirected graph model, that is, there are only two states, representing the activated and inactive states. Liao et al. (Reference Liao, Jin and Pavel2016) used the RBM incorporating the regularization item to deeply extract the monitoring information related to failure degradation trends, and then predicted the RUL of bearings based on the similarity method; RBM is often used in the field of components’ RUL prediction to extract features and then combines other methods for RUL prediction. In recent years, RBM has appeared less in the papers about RUL, and most of the literature uses stacked RBM (i.e., DBN), and the specific structure of DBN is shown in Figure 3. By stacking the RBMs, the hidden layer of the first RBM is used as the hidden layer of the second RBM, which makes the implicit features extracted by the hidden layer deeper and more representative.

Figure 3. Network structure of deep belief network.

DBN opens the door to deep learning model extensions, which can be used not only for classification problems but also for regression problems and is widely used in various fields such as image processing, NLP, remote sensing classification, and RUL prediction. Hu et al. (Reference Hu, Pei, Si, Du, Pang and Wang2019) toke the DBN as part of a bearing diagnostic model, primarily for mining the deep hidden features behind the monitoring data. Peng et al. (Reference Peng, Jiao, Dong and Pi2019) proposed an unsupervised health index (HI) construction method, extracted the hidden features corresponding to the system fault states based on DBNs, and constructed HI according to the distance between the degradation and fault states. Che et al. (Reference Che, Wang, Fu and Ni2019) developed a PHM model based on multiple deep learning algorithms and embedded the DBNs into the model, obtaining low error rates and biases on the turbofan engine dataset. Li et al. (Reference Li, Chen and Xiang2020b) used the DBN to extract the engine timing information processed by LSTM, built the corresponding degradation HI, and then predicted the RUL based on the degradation curve and threshold. Zhang et al. (Reference Zhang, Tang and Zhang2022) proposed a joint training model to visualize the degraded features extracted by DBN, and the joint training guarantees the computational efficiency of the prediction process, which shows great performance on the engine dataset.

Autoencoder

The concept of autoencoder (AE) was first introduced in 1986 by Rumelhart in the literature (Rumelhart et al., Reference Rumelhart, Hinton and Williams1986). It is a typical unsupervised learning algorithm.AE consists of an encoder and a decoder. In the encoding stage, the encoder maps the input data into a low-dimensional space by encoding and compressing them, whereas in the decoding stage, the decoder achieves unsupervised feature learning by reducing the original input data. Similar to RBM, stacked AE (SAE) can be achieved by stacking multiple layers of AE, and its structure is shown in Figure 4.

Figure 4. Structure of stacked autoencoder.

Similar to DBN, SAE can mine the deep features of data and has been applied to many practical aspects of life. These applications make AE a broader development prospect. Al-Khazraji et al. (Reference Al-Khazraji, Nasser, Hasan, Al Mhdawi, Al-Raweshidy and Humaidi2022) made full use of the unsupervised feature extraction capability of AE and used the extracted features from AE as the input of DBN, achieving a lower root mean square error (RMSE) for the RUL prediction of the engine dataset. Wu et al. (Reference Wu, Liu, Zuo, Cheng, Yang, Ma and Kong2021b) proposed a prediction model for different degradation stages, and embedded AE into the model, obtaining higher RUL prediction accuracy on Hilbert transformed bearing signals. Nguyen et al. (Reference Nguyen, Ahmad and Kim2022) used a pre-training and fine-tuning SAE approach to train the network parameters of the DNN, built degenerate HI from the original data, and then predicted the RUL by computing the Kullback–Leibler divergence with the labeled samples. Chen et al. (Reference Chen, Qin, Wang and Zhou2021) used a deep convolutional AE based on quadratic functions to solve the problem that the prior knowledge is required to construct HI. By directly extracting relevant information from the original data to construct HI, they obtained great bearings’ RUL prediction. Yu-hui et al. (Reference Yu-hui, Jun-qing, Ya, Yu and Tang-binn.d.) incorporated time scales into the process of extracting HI, and used the AE to extract nonlinear features of the engine, showing good RUL prediction on the C-MAPSS dataset. Lu et al. (Reference Lu, Liu, Wei, Chen, Zhang and Li2021) proposed a generative adversarial learning framework, and used LSTM as a generator and AE as a discriminator to overcome the problem of superposition of prediction errors in the bearings’ RUL prediction process.

Attention mechanism

The concept of attention mechanism (AM) was first proposed in the field of computer vision, and in 2014 Google Mind added AM to RNN models for image classification in the literature (Mnih et al., Reference Mnih, Heess and Graves2014), making AM widely popular in the academic community. Since then, AM has been widely used in computer vision, NLP, machine translation, and other fields. In the last 2 years, some literature has also used AM for RUL prediction. AM mimics the attention allocation effect of human observation, and its core idea is to shift attention from global to focused regions. In deep neural networks with a large number of model parameters, AM cannot only save computational resources and allocate them to more important features but also focus on the key information of the input and reduce the attention to other redundant information. In summary, AM is usually embedded as a module in the deep neural network model, as shown in Figure 5. In addition to the conventional AM, some variants have also emerged, which are the soft/hard AM, key–value pair AM, and multi-head self-AM (i.e., Transformer) (Vaswani, Reference Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, Kaiser and Polosukhin2017).

Figure 5. Diagram of the attention module.

(Yuan et al., Reference Yuan, Huang and Cheng2022) used the AM for feature extraction, which weighted multiple features of the original data, helping the RUL prediction model focus on important features and improving prediction performance. Remadna et al. (Reference Remadna, Terrissa, Al Masry and Zerhouni2022) added the AM to the deep convolutional variational AE to make full use of the feature processing capability of AM. It assigned higher weights to important information in the encoding stage while providing a low-dimensional representation of the features, which increases the explainability of the model. Fan et al. (Reference Fan, Chai and Chen2022) proposed a new AM framework that can save computational overhead while selecting information relevant to the task of predicting RUL. The information extracted by AM is then fed to the fully convolutional network to predict the engines’ RUL. Zhao et al. (Reference Zhao, Zhong, Fu, Tang and Pecht2020) implement soft thresholding of signal features with the help of AM. They constructed a network to learn the optimal selection of thresholding and then combined it with a deep residual network to predict the bearings’ RUL. Chadha et al. (Reference Chadha, Shah, Schwung and Ding2022) added AM as a layer to the Transformer framework to identify feature patterns that degrade with RUL by segmenting the time domain features of individual sensor data and then fused these patterns to predict the engines’ RUL. Wang et al. (Reference Wang, Amogne, Chou and Tseng2022a) placed the AM layer behind the Bi-LSTM network structure for capturing the feature information most relevant to the RUL degradation, and gave greater weights to the captured information sequences to assist the Lithium batteries’ RUL online prediction. Jiangyan et al. (Reference Jiangyan, Ma and Wu2024) combined the local modeling ability of CNN and the global modeling ability of Transformer to improve the overall architecture of the RUL prediction model.

Transfer learning

The concept of TL first appeared in the pre-conference workshop of Neural Information Processing Systems (NIPS-95) in 1995. In 2009, Sinno Jialin Pan and Qiang Yang classified TL into three major categories based on the data labeling of the source and target domains, namely inductive TL, transductive TL, and unsupervised TL, as shown in Figure 6.

Figure 6. Transfer learning classification.

For the equipment components’ RUL prediction problem, most of the literature uses the transductive TL based on domain adaptive. Define the source domain $ {D}_s $ and the source task $ {f}_s $ , and the target domain $ {D}_T $ and the target task $ {f}_T $ . Usually, we have a sufficient amount of labeled samples in the source domain, that is, $ {D}_s={\left\{{X}_s,{y}_s\right\}}^{N_s} $ , where $ {X}_s=\left\{{x}_1,{x}_2,{x}_3\dots {x}_n\right\} $ denotes the feature space of the source domain samples, $ {X}_T\sim P\left({X}_T\right) $ represents the data distribution of source domain, $ {y}_s $ denotes the corresponding RUL labels, and $ {N}_s $ denotes the number of samples in the source domain. For the target domain $ {D}_T={\left\{{X}_T\right\}}^{N_T} $ , that is, the dataset to predict the RUL, $ {X}_T=\left\{{x}_1,{x}_2,{x}_3\dots {x}_n\right\} $ denotes the feature space of the target domain samples, $ {X}_T\sim P\left({X}_T\right) $ denotes the data distribution of the target domain, $ {N}_T $ denotes the number of samples in the target domain, and $ {D}_T $ contains only a small number of labeled samples, most of them or all of them have no RUL labels and need to rely on TL to obtain $ {y}_T $ . The goal of TL is to use the knowledge in the source domain $ {D}_s $ and the source task $ {f}_s $ to serve and help the learning of the target prediction function $ f $ in $ {D}_T $ , that is, $ {y}_T\approx f\left({X}_T\right) $ , in case $ {D}_s\ne {D}_T $ or $ {f}_s\ne {f}_T $ .

Domain adaptive methods in TL can be divided into three types: reconstruction methods (Ghifary et al., Reference Ghifary, Kleijn, Zhang, Balduzzi and Li2016), discrepancy-based methods (Du et al., Reference Du, Li, Su, Zhu and Lu2021), and adversarial methods (Ganin et al., Reference Ganin, Ustinova and Ajakan2016). The reconstruction method tries to reconstruct data with more differentiated and common features (Bousmalis et al., Reference Bousmalis, Trigeorgis, Silberman, Krishnan and Erhan2016); the discrepancy-based method measures the difference between the source and target domains by statistical metrics, and the commonly used numerical scales are the maximum mean discrepancy and the maximum density divergence (MDD) (Li et al., Reference Li, Chen, Ding, Zhu, Lu and Shen2020a); the adversarial method trains discriminators and feature extractors in an adversarial manner to reduce the gap between the source and target domain (Li et al., Reference Li, Chen, Xiang and Cai2021).

Generally, TL is used as an embedded technique, on the basis of the deep learning method mentioned in the “Deep learning-based RUL prediction methods” section, to address mainly the problems of few labeled samples and cross-condition problems in the field of RUL prediction.

Semi-supervised learning

Equipment RUL prediction belongs to regression problems, and conventional methods generally require sufficient labeled RUL information in the training set, which is known as supervised learning, such as RNNs and CNNs, which can mine the feature information affecting labeled RUL and predict unknown RUL; on the other hand, unsupervised deep learning introduces an initial pre-training phase to extract features related to the degradation from the original unlabeled training data, such as RBM and AE. They can automatically extract deep high-dimensional information by coding and decoding from the original unlabeled training data. Therefore, even in the case of a few labeled training data, the combination of unsupervised and supervised learning, that is, semi-supervised learning can also provide accurate RUL prediction results.

In actual industrial production, to ensure the proper operation of equipment, equipment parts are usually replaced before their failure, so the sensors collect a large amount of pseudo-failed hover data. And semi-supervised learning can make full use of these pseudo-failed hovering data to extract information related to degradation from them. He et al. (Reference He, Tian and Zuo2022a) considered not only the failed labeled data but also the hovering unlabeled data in the process of actual industrial production when extracting fault information, which improves the robustness and generalization capability of the RUL prediction model. Krokotsch et al. (Reference Krokotsch, Knaak and Ghümann2022) proposed an improved semi-supervised learning framework for the lack of available data for the whole life cycle of equipment components. They used a self-supervised learning approach to extract valid knowledge from the failure-labeled data and pseudo-failed unlabeled data, where pseudo-labels are generated for model training. Ellefsen et al. (Reference Ellefsen, Bjørlykhaug, Æsøy, Ushakov and Zhang2019) used semi-supervised learning to pre-train the framework and used the genetic algorithm to obtain the optimal hyperparameters. Their results showed that the prediction results of semi-supervised learning outperform those of supervised learning. Nieves Avendano et al. (Reference Nieves Avendano, Vandermoortele, Soete, Moens, Ompusunggu, Deschrijver and Van Hoecke2022) leveraged structure learning and pseudo-label (Lee, Reference Lee2013). They added the pseudo-labels to a large amount of unlabeled data by classifier iterations and at the same time built the monotonic HIs. Then these HIs were used in the enhanced RUL prediction model.

In summary, pseudo-hovering unlabeled data from the real manufacturing process can be fully utilized by semi-supervised learning. By pre-training or using self-supervised techniques to add pseudo-labels to the hovering data, the relevant feature space and training set can be increased. Also, if the hovering unlabeled data include a variety of operating or failure conditions, TL techniques can also be used to solve the cross-domain issue.

Bayesian neural networks

The performance degradation of equipment components under different operating environments and scenarios will exhibit a certain degree of uncertainty, including aleatoric (data) uncertainty and epistemic (model) uncertainty, as shown in Figure 7. Aleatoric uncertainty refers to systematic error, chance error, coarse error, and unknown error in the data due to the limitations of observation means and processing algorithms. Epistemic uncertainty refers to the error of prediction methods or models unable to fully simulate the dynamics of real-world systems. Meanwhile, the accumulative error in the process of building deep learning networks for equipment component degradation also leads to the uncertainty of RUL prediction results.

Figure 7. Aleatoric (data) uncertainty and epistemic (model) uncertainty.

In the actual industrial production process, different systems with different working conditions do not have the same requirements for the risk level of equipment components. Therefore, in the RUL prediction of components, compared with the single-value RUL prediction, uncertainty analysis, and modeling can be performed to obtain the probability distribution or confidence interval of RUL. It cannot only avoid the error generated by single-value prediction to a certain extent but also provide users with more reference value for suggestions and decisions, which is often more practical in practical applications.

Since the standard deep neural network structure produces a deterministic single-value prediction result for a given input and does not have the ability to express uncertainty, Buntine and Hutter (Reference Buntine and Hutter2010) added the Bayesian method to the neural network modeling process and eventually developed the Bayesian neural network method. It makes deep learning feasible for modeling uncertainty and allows neural networks to describe uncertainty factors while having powerful representation capabilities. According to the Bayesian method, the parameters such as weights and biases in the neural network structure can be considered as random variables. The deep learning model that sets the network parameters as random variables is called the deep Bayesian model (deep Gaussian process). The comparison of the structures of conventional artificial neural networks and Bayesian artificial neural network models (deep Gaussian process) is shown in Figure 8.

Figure 8. The top represents conventional artificial neural networks, while the bottom represents Bayesian artificial neural networks (deep Gaussian process).

In the Bayesian neural network structure, the objective function for model training is the likelihood function determined from the sample data, and the posterior probability distribution of the weights and bias parameters are adjusted during the training process. Due to the complexity and difficulty of training deep Bayesian neural network models and the difficulty of converging to the optimal solution, many improved Bayesian methods have been applied in deep neural network research. In previous studies, variational inference (Blei et al., Reference Blei, Kucukelbir and McAuliffe2017) has been applied to many Bayesian neural network methods for approximation. Gal and Ghahramani (Reference Gal and Ghahramani2016) proposed the theory that the conventional neural network structure by adding dropout to each layer and multiple Monte Carlo dropout inference is mathematically equivalent to a deep Gaussian process model (i.e., Bayesian neural network). Therefore, most of the literature approximates the deep Bayesian model by multiple Monte Carlo inferences to achieve uncertainty in RUL prediction. Zhu et al. (Reference Zhu, Chen, Peng and Ye2022) set the dropout in the convolutional layer of CNN from the perspective of active learning and obtained the bearings’ RUL prediction uncertainty by multiple Monte Carlo dropout inferences. Then they selected labeled samples from the alternative sample pool according to the uncertainty, which reduced the number of training samples under the condition of the same prediction accuracy. Biggio et al. (Reference Biggio, Wieland, Chao, Kastanis and Fink2021) evaluated a variety of deep Gaussian process scaling and solved the problem that deep Gaussian processes could not scale when there was a very large number of data in the real industries. Meanwhile, their method could provide both accurate RULs and uncertainty estimation. Lin and Li (Reference Lin and Li2022) proposed a new iterative calibration method to address the problem that Bayesian uncertainty prediction methods cannot accurately capture model bias and noise bias. By combining the isotonic regression and standard deviation scaling for metric quantification of uncertainty, they integrated them into the Bayesian deep learning framework. Chen et al. (Reference Chen, Shi, Lu, Zhu and Jiang2022) proposed a framework for using RUL prediction uncertainty to implement component maintenance decisions, which addressed the current problem of separating the study of RUL prediction and maintenance decisions in PHM technology and also was in line with actual industrial processes. Xu et al. (Reference Xu, Guo and Saleh2021c) combined the deep learning method and non-smooth Gaussian process to fully capture the temporal dependence of the raw data, then they predicted the RUL and quantified their uncertainty. By validating the C-MAPSS dataset, they got low prediction errors, as well as valid uncertainty intervals.

Digital twin-driven

The concept of DT was first introduced based on the National Aeronautics and Space Administration (NASA) Apollo program in 2003 by Dr. Michael of the University of Michigan. Its name has changed from “mirror space” to “information mirror,” and “DT” was first proposed and quoted in 2011. After several years of application and precipitation, DT was named one of the 10 key technologies in 2019. From different perspectives, there are different understandings of DT. Among them, five commonly used interpretations of DT as shown in Table 2.

Table 2. Five-dimensional interpretations of DT

However, according to Table 2 regardless of the interpretation and definition, they all share a common feature: it connects the physical and virtual worlds and serves the physical world according to demand. The DT relies on knowledge mechanism, simulation interaction, big data, and other technologies to establish digital models, and uses technologies such as the IoT to map real-time data from the physical space to the DT space. On the above basis, combining artificial intelligence (AI), data mining, edge computing, and other technologies, common applications such as the description of the DT, diagnosis and prediction, intelligent decision making, and maintenance are empowered to various vertical industries such as aerospace satellite, aerospace, oil, and gas industry, environmental protection, architectural design, military science, medical and pharmaceutical, smart city (He et al., Reference He, Ai, Wang, Tao, Pan, Qiu and Yang2022b), smart factory (Tao et al., Reference Tao, Zhang, Cheng, Ren, Wang, Qi and Li2020), national grid, equipment security, and other various fields.

Mechanical equipment maintenance from the initial “maintenance after failure” to “preventive regular maintenance,” and now “PdM,” DT will promote the maintenance of equipment components upgrade, to achieve “accurate condition-based maintenance” (Tao et al., Reference Tao, Zhang, Liu and Nee2018b). The aim is to achieve the goal of collaborative optimization at all levels, ensuring the safe operation of machinery and equipment and minimizing maintenance costs. DT has to combine with other advanced technologies (such as deep learning and complex high-frequency networks) to ensure real-time, high-speed, and low latency, completely break the information silo, and play its great advantage (Tao et al., Reference Tao, Zhang, Liu and Nee2018a). Lv and Xie (Reference Lv and Xie2021) proposed that when using DT technology to assist physical entity models to collaborate, precise maintenance and guarantee for faulty physical entity equipment or components are also necessary. The flow chart of DT-driven mechanical equipment maintenance is shown in Figure 9. Among them, the use of DT combined with AI prediction to improve fault diagnosis and prediction accuracy is a major direction (Li et al., Reference Li, Lei and Mao2022a).

Figure 9. Flow of digital twin-driven mechanical equipment assurance.

Xiong et al. (Reference Xiong, Wang, Fu and Xu2021) proposed that traditional model-based or data-based PdM methods are often reactive and heuristic and cannot diagnose faults and assess potential risks promptly, much less provide safe emergency strategies and guarantees. To this end, they combined DT techniques with deep LSTM neural networks, achieved RUL accurate prediction, and provided complete health management solutions. Wu and Li (Reference Wu and Li2021) built an aero-engine DT model based on IoT and sensor data, where they used LSTM deep networks to dynamically update the DT model to assist the RUL prediction of physical entity models. And then they extended the DT model to remote operation and health maintenance management of equipment components. Anis et al. (Reference Anis, Taghipour and Lee2020) a proposed DT framework-based LSTM-ED network for the need for real-time data updates and online learning. And they used the LSTM-ED network to reconstruct the time series of sensor data for real-time RUL prediction and update. Guo et al. (Reference Guo, Yang, Chen, Luo and Hu2021) established DT models based on real-time sensor data for real-time RUL prediction, providing digital solutions for accurate maintenance of machinery and equipment, and guiding spare parts inventory optimization.

In summary, DT technology plays a major role in real-time prediction and accurate maintenance of equipment components: from the digital model establishment to physical real-world connection, to real-time model update, real-time fault diagnosis and prediction, real-time inventory optimization, and real-time health monitoring (Hollenbeck and Chen, Reference Hollenbeck and Chen2022). In the future, it is necessary to explore effective deep-learning methods for fault diagnosis and RUL prediction. Deep learning methods that can solve the challenges of multiple cross-working conditions, few samples, online learning, and uncertainty are to be developed to embed into the DT model to provide accurate, fast, and real-time maintenance for the operational safety of machinery and equipment.

C-MAPSS dataset

This part uses the NASA C-MAPSS dataset to compare and summarize the effectiveness of various deep learning methods in predicting RUL. The C-MAPSS dataset is widely used for predicting the RUL of turbofan engines, which simulates the actual degradation process of turbofan engines over flight cycles, from a healthy state to run to failure. The dataset is divided into four different subsets, as shown in Table 3. There are two types of engine failure modes, among which FD001 and FD002 only contain HPC fault mode, while FD003 and FD004 contain HPC and Fan fault mode; The operating conditions are also divided into two types, with FD001 and FD002 only containing a single operating condition, and FD003 and FD004 containing multiple operating conditions. In each subset, the training set, the testing set, and the real RUL labels are provided. Each training and testing set contains 21-dimensional sensor data for monitoring engine operation, as well as 3-dimensional operation setting data. For a more detailed introduction to this dataset, please refer to Saxena and Goebel (Reference Saxena and Goebel2008).

Table 3. C-MAPSS dataset description

The training data samples and the corresponding RUL label can be used for training the deep learning model. After the model training is completed, the RUL of each turbofan engine can be predicted in the test set, and the predicted results are compared with the actual RUL values provided in the dataset to verify the effectiveness of the prediction model.

Evaluation index

To evaluate the effectiveness of the model, RMSE and score function (S-score) are usually used for evaluation.

RMSE

RUL prediction is a typical regression problem, and RMSE is one of the most commonly used evaluation indexes in regression problems (Chen et al., Reference Chen, Jing, Chang and Liu2019; Gao, Reference Gao2020; Sayah et al., Reference Sayah, Guebli, Al Masry and Zerhouni2021), which is used to reflect the deviation between the actual value and the predicted value, as shown in Eq. (8):

(8) $$ RMSE=\sqrt{\frac{1}{N}\sum \limits_{i=1}^N{\left({RUL}_{pre}-{RUL}_{actual}\right)}^2}, $$

where $ N $ represents the total number of samples, $ {RUL}_{pre} $ represents the predicted RUL, and $ {RUL}_{actual} $ represents the actual RUL.

S-score

In the field of RUL prediction, many research papers use the score function as the evaluation index, and this evaluation indicator was also adopted by the 2008 International PHM Data Challenge (Li et al., Reference Li, Ding and Sun2018; Al-Dulaimi et al., Reference Al-Dulaimi, Asif and Mohammadi2020; Zheng et al., Reference Zheng, Bao, Zhao, Chen, Liu, Sun and Wang2022). Compared with RMSE, the biggest feature of the score function is that it imposes more severe penalties for predicting values that are too large. This meets the requirements of practical engineering, as in important fields such as aerospace, predicting failure time later than the actual failure time may lead to incorrect maintenance decisions, resulting in very serious losses (Che et al., Reference Che, Wang, Fu and Ni2019; Li et al., Reference Li, Chen and Xiang2020b). The mathematical expression (Eqs. (9) and (10)] of the score function is as follows:

(9) $$ S- score=\sum \limits_{i=1}^N{s}_i, $$

(10) $$ {s}_i=\left\{\begin{array}{c}{e}^{-\frac{RUL_{pre}-{RUL}_{actual}}{13}}-1,\hskip0.72em {RUL}_{pre}-{RUL}_{actual}<0,\hskip0.6em \\ {}{e}^{\frac{RUL_{pre}-{RUL}_{actual}}{10}}-1,\hskip0.6em {RUL}_{pre}-{RUL}_{actual}\ge 0,\end{array}\right. $$

where $ e $ represents the natural logarithm.

Method analysis

To further compare the application of deep learning methods in RUL prediction, Table 4 summarizes some deep learning methods validated on the C-MAPSS dataset in recent years and presents the experimental results of various methods on the first subset (FD001). Table 4 also provides a brief introduction to the used methods. Although various methods may not be consistent in data processing, the C-MAPSS dataset specifies the training and testing sets, so RUL prediction methods based on different deep learning could be compared on this publicly available dataset.

Table 4. RUL prediction method based on deep learning and its performance on C-MAPSS (FD001)

In Table 4, there are 11 deep learning methods applied to the C-MAPSS dataset in the past 3 years. It can be seen that all the deep learning methods mentioned in the “Deep learning-based RUL prediction methods” section have been applied to this dataset. Among them, CNN is mainly used for extracting spatial features, RNN is mainly used for extracting temporal features, and DBN and AE are used for unsupervised feature dimensionality reduction. Also, it could be noted that adding AM to the deep learning model decreases the RMSE compared with those not added, because AM could help the model pay more attention to the important features which related to the RUL. Also, by combining the advantages of different methods, more in-depth feature expressions related to equipment degradation can be learned, thus improving the prediction accuracy, such as the hybrid single model, and the combination of AM and deep learning models. It is worth noting that the improvement of RUL prediction accuracy by various models in Table 4 is limited. We believe that a simple combination of multiple methods is not the key to improving prediction performance. How to flexibly utilize different methods for complementary advantages and thereby improve model prediction accuracy is a problem worth studying. In addition, more data generated in actual industrial production will also promote model validation and facilitate AI-driven production, where there will be more paper really concerning the challenging problems in practical scenarios.