Quantum Gene Chain Coding Bidirectional Neural Network for Residual Useful Life Prediction of Rotating Machinery

 Abstract: In classical recurrent neural networks, the pre-and post-relationships of time series tend to be neglected so that long-term overall memory is generally inaccessible; meanwhile, the weights are transferred and updated mainly by the gradient descent method, which leads to their low prediction accuracy and high computation cost in the application of residual useful life (RUL) prediction of rotating machinery (RM). In view of this, a quantum gene chain coding bidirectional neural network (QGCCBNN) is proposed to predict RUL of RM in this paper. In our proposed QGCCBNN, the quantum bidirectional transmission mechanism is designed to establish the pre-and post-relationships of time series for readjusting the weight parameters according to the feedback from the output layer, so that higher consistency between the input information and the overall memory of the network can be realized, thus endowing QGCCBNN with better nonlinear approximation ability. Moreover, in order to improve the global optimization ability and convergence speed, the quantum gene chain coding instead of gradient descent method is constructed to transmit and update data, in which the qubit probability amplitude real number coding is adopted and the cosine and sinusoidal qubit probability amplitudes corresponding to the minimum loss function are compared with those of the current time by the phase selection matrix for the directional parallel updating of the weight parameters. On this basis, a new RUL prediction method for RM is proposed, and higher prediction accuracy as well as desirable efficiency can be obtained due to the advantages of QGCCBNN in nonlinear approximation ability and convergence speed. The experimental example for RUL prediction of a double-row roller bearing demonstrates the effectiveness of our proposed method.


Introduction
Rotating machinery (RM) is widely used in automobile transmission systems, aero engines, wind turbines and other machines [1]. Because of its complex structure and various working conditions, different types of faults may affect its normal operation and ultimately give rise to the arrival of the service life threshold [2]. The damage degree of key components of RM, such as bearings, gears, rotors and impellers, determines whether the machine can operate safely and reliably for a long time [3,4]. Therefore, it is very important to monitor the operating condition of RM and predict its residual useful life (RUL) [5,6].
Recently, data-driven RUL prediction methods based on artificial intelligence algorithms have attracted much attention because of their fast and accurate characteristics [7]. These methods start from industrial scene and construct a non-linear model by establishing the relationships among the feature vector, the fault mode and the RUL prediction [8]. Furthermore, in order to accurately judge the fault state of RM and predict its RUL, this kind of methods pays much attention to the hidden information of data. Overall, data-driven RUL prediction methods can provide enough decision-making information for the maintenance of corresponding components, thus avoiding major catastrophic accidents of machines. Consequently, they possess great engineering application values, and have become a research hotspot and the development trend in the field of RUL prediction of machines [9]. For example, Ren et al. [10] used a deep autoencoder and deep neural network (DADNN) to estimate RUL of RM; Banu et al. [11] applied a fully complex-valued radial basis function neural network (FCRBFNN) to forecast the load trend of power systems; Hinchi et al. [12] adopted a convolutional long-short-term memory neural network (CLSTMNN) to predict the degradation trend of the rolling bearing; Xiao et al. [13] predicted the degradation trend and RUL of rolling Quantum Gene Chain Coding Bidirectional Neural Network for Residual Useful Life Prediction of Rotating Machinery ·3· bearings based on the least squares support vector machine (LS-SVM). However, all the above state-of-the-art methods of data-driven RUL prediction have some defects. For DADNN, its structure of deep neural network means that the data propagates hierarchically in the multi-layer network, which can result in gradient disappearance or gradient explosion. In terms of FCRBFNN, it is difficult to properly tune its massive parameters, such as the number of radial basis functions, the values of center vectors, and so on; once these parameters are selected inappropriately, the convergence error can be increased and the network may be over-fitting. As for CLSTMNN, its complex structure requires four linear layers per cell, which may lead to a slow training process. In the case of LS-SVM, it is difficult to set kernel functions and parameters accurately, so the prediction results are often uncertain, resulting in the degradation of its generalization performance.
RUL prediction is essentially an input series trend prediction problem with threshold determination. As a typical time series prediction theory, the recurrent neural network (e.g., the long short-term memory neural network (LSTMNN) [14], the gated recurrent unit neural network (GRUNN) [15], etc.), has showed a certain application prospect in RUL applications [16][17][18] because of its cumulative effect of time series. However, it has an obvious defect of lacking reverse feedback from output to input [19]. Fortunately, the bidirectional transmission mechanism can make up for the above defect [20]. Moreover, the quantum computing can contribute to improving the global optimization ability and computational performance of the recurrent neural network [21][22][23] because of its high speed and parallelism [24,25]. Based on the complementary advantages of bidirectional transmission mechanism and quantum computing, a novel recurrent neural network called quantum gene chain coding bidirectional neural network (QGCCBNN) is proposed in this paper. In QGCCBNN, the quantum bidirectional transmission mechanism is designed to establish the preand post-relationships of time series for readjusting the weight parameters according to the feedback from the output layer, so that higher consistency between the input information and the overall memory of the network can be realized, thus endowing QGCCBNN with a better nonlinear approximation ability. Meanwhile, in order to avoid the gradient disappearance, gradient explosion, slow convergence speed and large time cost caused by the gradient descent method in classical recurrent neural networks, the quantum gene chain coding is constructed to transmit and update data, in which the qubit probabilistic amplitude real number coding is adopted and the cosine and sinusoidal qubit probability amplitudes corresponding to the minimum loss function are compared with those of the current time by the phase selection matrix for the directional parallel updating of the weight parameters. Consequently, QGCCBNN has better global optimization ability and faster convergence speed.
Based on the advantages of QGCCBNN in nonlinear approximate ability, global optimization performance and convergence speed, we propose a new method for RUL prediction of RM. In the proposed method, a trend index (TI) is firstly extracted from the vibration acceleration data of RM as the performance degradation feature, then the TI is put into QGCCBNN for predicting the degradation trend of RM, and finally the failure probability model is established to predict RUL of RM based on the degradation trend curve. Theoretical analysis of QGCCBNN and experimental results of RUL prediction of a double-row roller bearing show that our proposed method can achieve higher prediction accuracy and shorter computational time.
The main motivation of this paper is to solve the problem of low accuracy and high computation cost of RUL prediction of RM caused by poor nonlinear approximation ability and slow convergence speed of classical recurrent neural networks. On account of this, we propose a novel neural network termed as quantum gene chain coding bidirectional neural network (QGCCBNN) to predict RUL of RM with higher accuracy and desirable efficiency.
The rest of this paper is organized as follows. In Section 2, the background knowledge about QGCCBNN is briefly introduced, and the QGCCBNN algorithm is proposed and derived in detail. Then, the RUL prediction method for RM based on QGCCBNN is described in Section 3. Section 4 presents the experimental results along with corresponding comparative analysis to demonstrate the effectiveness of our RUL prediction method. Finally, some conclusions are drawn in Section 5.
where ' φ is the updated state by the quantum phase-shift Obviously, the function of quantum phase-shift gate is to realize phase rotation.
On the other hand, the controlled non-gate, whose operation objects are control qubits and target qubits, is also an important part of quantum gate group. Assume a control qubit as where  is an exclusive OR operation and 12 a a a n is the output of the multi-bit controlled non-gate () n C U . In short, the function of controlled non-gate is to change the probability amplitude by the control qubits and target qubits.

Quantum gene chain coding algorithm
Assume that the number of chromosomes is n and that the number of genes, which are arranged on the chromosomes, is m . Then, the genome can be described as where t is the iteration number of the genome. According to the thought of gene pairing, a gene can be divided into two parts (i.e., t i  and t i  ). In other words, it can also be represented by quantum bits. Thus, a quantum chromosome consisting of two quantum gene chains can be represented as follows: where the number of genes is m (i.e., the length of chromosomes.) and =1,2, , jn is the number of the j -th chromosome in the genome () t Q . Since a probability amplitude can be transformed into a trigonometric function, Eq. (4) can be further converted as follows: 12 12 cos( ) cos( ) cos( ) sin( ) sin( ). sin( ) = cos arccos ,cos arccos , ,cos arccos where cos(arccos( ))    of the global optimal solution s can be updated synchronously in each iteration, and thus the optimization process of the global optimal solution s can be accelerated. Therefore, quantum gene chain coding has the ability of expanding search range and parallel computing. Fig. 2 shows the model structure of QGCCBNN. In order to realize higher consistency between the input information and the overall memory of the network and obtain better nonlinear approximation ability, the quantum bidirectional transmission mechanism is designed and introduced to QGCCBNN for establishing the pre-and post-relationships of time series and readjusting the weight parameters according to the feedback from the output layer. First, the quantized time series t x is transmitted from the input layer to hidden layer by quantum phase-shift gate and controlled non-gate. Then, in a similar way, the hidden

Theory model of QGCCBNN
, whose quantum state is Thus, the transfer relationship between the input and the output in each layer of QGCCBNN can be deduced as follows.
First of all, the normalized input series can be written as

Output layer
Hidden layer

Input layer
Phase of quantum phase-shift gate (weight parameter) are respectively the lower and upper bounds of the normalized input series. Then, according to the model structure of QGCCBNN shown in Fig. 2 and Eqs. (1-3) (i.e., the definition of quantum phase-shift gate and controlled non-gate), the hidden layer state at time t can be computed as . (17) Now, our newly-designed quantum bidirectional transmission mechanism has been realized by Eqs. (13)(14)(15)(16)(17), which can readjust the weight parameters (i.e., the phases of quantum phase-shift gates) according to the feedback from the output layer, so QGCCBNN can realize higher consistency between the input information and the overall memory of the network.
Finally, the output layer state at time t can be computed as Therefore, the probability amplitude of the state 1 is selected as the output of each layer, and then the outputs of the hidden layer and the output layer can be respectively expressed as

Training algorithm for QGCCBNN
In our proposed QGCCBNN, the weight parameters (i.e., , the phases of quantum phase-shift gates) that need to be updated are represented by the quantum phase matrices as follows: a. The quantum phase matrix which transfers data from the input layer to the hidden layer: b. The quantum phase matrix which transfers data from the hidden layer of the previous time to the current hidden layer: 11 12 1 c. The quantum phase matrix which transfers feedback data from the output layer to the hidden layer: In order to update the matrices t θ , t δ , t γ , and t φ , all their elements are firstly extracted into a parameter vector as Then t p can be updated by the quantum gene chain coding mentioned in subsection 2.1.2. The process of updating t p with quantum gene chain coding is described in detail as follows.
For convenience, the parameter vector can be as written as: Moreover, according to Eqs. (5-7), a quantum chromosome, for calculating t p , can be described as follows: where , cos( ) cos( ) cos( ) cos( ) cos( )     and t φ can be updated in parallel. Thus, based on the above analysis, the quantum phase matrices t θ , t δ , t γ , and t φ can be directionally updated in parallel by the phase selection matrix M .

cos( )
To sum up, in the proposed QGCCBNN, the quantum bidirectional transmission mechanism (see Eqs. (13)(14)(15)(16)(17)) is designed to establish the pre-and post-relationships of time series for readjusting the weight parameters according to the feedback from the output layer, so QGCCBNN can realize higher consistency between the input information and the overall memory of the network, and show better nonlinear approximation ability. Meanwhile, the quantum gene chain coding, in which the qubit probability amplitude real number coding (see Eqs. (21)(22)(23)(24)) is adopted and the cosine and sinusoidal qubit probability amplitudes corresponding to the minimum loss function are compared with those of the current time by the phase selection matrix for the directional parallel updating the quantum phase matrices (see Eq. (26)), is constructed to transmit and update weight parameters for improving the global optimization ability and convergence speed of QGCCBNN. It is believed that the above advantages of QGCCBNN in nonlinear approximation ability and convergence speed can offer higher prediction accuracy and desirable computational efficiency to the proposed RUL prediction method based on QGCCBNN.

Process of RUL prediction method
In the following, QGCCBNN is applied to the RUL prediction of RM. The process of our RUL prediction method is shown in Fig. 4, which is described in detail as follows: Step 1: The original vibration acceleration data of RM running are collected.
Step 2: The power spectral entropy (PSE) [26,27] is constructed based on the original vibration data.
Step 3: The trend index (TI) is constructed by PSE as the performance degradation feature.
Step 4: TI is input into QGCCBNN for training QGCCBNN according to the training process in subsection 3. 3.
Step 5: The future performance degradation trend of RM is predicted by the well-trained QGCCBNN according to the prediction process in subsection 3.4.
Step 6: According to the predicted TI curve, the failure probability model is established to predict the failure time point (i.e., failure probability threshold point) and the RUL of RM.

Description of Trend Index (TI)
TI is constructed by PSE of a time series, and its construction process can be described as follows.
Firstly, the original time series   Since TI can quantitatively measure the variation tendency of time series, it can be used to characterize the performance degradation feature of RM.

Training process of QGCCBNN
The training process of QGCCBNN mentioned in Step 4 in subsection 3.1 is now elaborated in this subsection.

Suppose the input series composed of TI is
x .
x can be decomposed into the training sample matrix X and the target output vector  y , which are respectively written as where each row of X represents a training sample, and the dimension of a training sample is n ; each element in  y represents the corresponding training target of each training sample. Obviously, the total number of training samples is --+1 b a n . Thus, we can summarize the training process of QGCCBNN as follows: Step 1: All qubit phases of the quantum gene chains are initialized, namely the values in the interval of [0,2 ] π are randomly assigned to each qubit phase.
Step 2: The training sample matrix X is input into QGCCBNN to compute the actual output vector y .
Step 3: All quantum phase matrix parameters quantum phase matrices are updated by the training algorithm in subsection 2.3, the updated values are taken as the initial values in the next iteration.
Step 4: Steps 2 and 3 are repeated to update the actual output vector y until the mean square error is less than the set threshold or the training steps reach the maximum value.

Performance degradation trend prediction by the well-trained QGCCBNN
Now, the well-trained QGCCBNN can be applied to predict the performance degradation trend. Here, the multi-step-ahead prediction method is adopted to conduct the prediction process.
First, the training sample ( ) Second, the newest testing sample ( )

RUL prediction by the failure probability model
In this subsection, according to TIs predicted by QGCCBNN, the failure probability model is established for RUL prediction of RM.
Firstly, the continuous performance degradation process of RM can be expressed as In order to obtain λ and 2 σ , the maximum likelihood estimation method [28,29] is introduced, and the likelihood function ( , ) ψ λ σ can be represented as  ( ) Then, according to the maximum likelihood estimation method, the derivative of ( , ) l λ σ can be expressed as (38) Finally, according to Eq. (38), λ and 2 σ can be computed as follows: (39) After λ and 2 σ are obtained, the next step is to compute the failure probability ( ) Ft at time t , which can be expressed as where w is the threshold of TI. Now, the failure probability model has been constructed. Assume that the starting prediction time is start t , the NT  is the interval between two adjacent time points in power spectral entropy curve.

Application of the RUL prediction method based on QGCCBNN
In this section, the full-life vibration acceleration data of double-row roller bearings collected by the University of Cincinnati [30] are used to verify the effectiveness of our QGCCBNN-based RUL prediction method. Fig. 5 illustrates the bearing test rig and sensor placement that are used to collect the vibration acceleration data of bearings. Four ZA-2115 double-row roller bearings made by Rexnord are installed on the rotating shaft of the bearing test bench. A motor is used to drive the rotating shaft through a friction belt with a constant speed of 2000 r/min. Spring mechanisms are used to apply 6000 pounds of radial load onto the rotating shaft and bearings. High-sensitivity ICP accelerometers are installed on the bearings to collect the vibration acceleration data every 10 min. The sampling frequency is 20 kHz, and the sampling length is 20480 points.

RUL prediction of the second bearing on test bed
In this experiment, the outer race failure occurs in the second bearing after continuous running for 9840 min (i.e., about seven days). Here, PSE is constructed based on the vibration acceleration data (i.e., 984 samples). As shown in Fig. 6, PSE remains stable from the starting sample point to the 805th point, which means that the bearing is in the normal operation period. After that, PSE decreases slowly from the 806th to the 950th point, indicating that the bearing is in the initial degradation stage (i.e., early fault stage); from the 951st to the last point, the PSE curve becomes more and more irregular, which reveals that the outer race defect of the bearing is expanding rapidly, namely, serious failure occurs.
In order to quantitatively measure the variation tendency of PSE, the next step is to construct the trend index (TI) as the performance degradation feature. As shown in Fig. 7, a total of 729 TIs corresponding to 984 samples are obtained. From the starting point to the 550 th point, TI changes slowly, indicating that the bearing is in the normal operation stage. From the 551 st to the 695 th point, TI starts to ascend, indicating that the bearing is in the stage of initial degradation. After that, the value of TI rises sharply, which means that the bearing is in the stage of accelerated degradation, namely, the outer race failure occurs.

Prediction results
Because the initial degradation stage and its subsequent degradation stage are the most important time intervals in performance degradation trend observation and RUL prediction, the TI points from the 551th to the 650th are taken as training samples, and the 651st TI point and its subsequent data (i.e., the remaining 79 TI points) are the testing samples. Here, the parameters of the proposed RUL prediction method based on QGCCBNN are set as follows: in the theory model of QGCCBNN, the input-layer dimension is

·12·
The next step is to predict RUL of the second bearing. As shown in Fig. 9 is the interval between two adjacent time points shown in Fig. 10.

Discussion
In this subsection, the advantages of the RUL prediction method based on QGCCBNN are verified by comparative analysis.
First, the nonlinear approximation capability of QGCCBNN is compared with that of deep autoencoder and deep neural network (DADNN), gated recurrent unit neural network (GRUNN), or fully complex-valued radial basis function neural network (FCRBFNN). In order to compare the four prediction methods under the same conditions, the performance degradation feature in the latter three prediction methods is also represented by TI, which is the same as that of our proposed method based on QGCCBNN. At the same time, the maximum training step size, the number of neurons in each layer, and the mean square error threshold of the latter three methods are set as the same values as those of QGCCBNN respectively. The learning curves of these methods are shown in Fig. 11. As can be seen, at the training step th 20 Ns = and its subsequent training steps, the mean square error of QGCCBNN is much less than that of DADNN, GRUNN or FCRBFNN, namely, the nonlinear approximation capability of QGCCBNN is much better than that of the latter three methods. The underlying reason is that the quantum bidirectional transmission mechanism is designed in QGCCBNN to establish the pre-and post-relationships of time series for avoiding the lack of reverse feedback from the output layer to the input layer in the latter three methods. This means that QGCCBNN can remember the overall regulation of time series, thus achieving better nonlinear approximation capability.
Second, the prediction accuracy of the RUL prediction method based on QGCCBNN is compared with that of the prediction methods based on DADNN, GRUNN, FCRBFNN or LS-SVM. The performance degradation feature of the latter four methods is also expressed by TI. The maximum training step size, the number of neurons in each layer, and the mean square error threshold of DADNN, GRUNN and FCRBFNN are set as the same as those of QGCCBNN, respectively. In LS-SVM, the Gaussian kernel is set as its kernel function, and its regularization parameter is optimized by s-folder cross validation algorithm. Moreover, the failure probability models adopted by the latter four methods are identical to that of QGCCBNN. Figs. 12-15 show the predicted TI,   .83 h, respectively. In other words, our method is the most accurate and reliable. This can be attributed to the better nonlinear approximation capability of QGCCBNN brought by the quantum bidirectional transmission mechanism and its superior global optimization ability given by quantum gene chain coding. Therefore, it is expected that the above advantages of QGCCBNN can improve the prediction accuracy of our RUL prediction method.
Then, in order to quantitatively evaluate the prediction accuracy and stability of the five RUL prediction methods, the mean squared error (MSE), mean absolute error (MAE), root mean squared error (RMSE), and coefficient of determination ( 2 R ) are taken as evaluation indicators to statistically measure the levels of prediction accuracy and stability of the five methods, which are respectively defined as where n is the number of predicted sample points, i y is the predicted TI value of the i -th sample point, ; the closer the value of 2 R to 1, the higher the prediction accuracy of the method. Statistical analyses on the MSE , MAE , RMSE , and 2 R values of the five prediction methods are respectively conducted for 50 times, and the average values of MSE , MAE , RMSE , and 2 R of these methods in the 50 repeated predictions, i.e., MSE , MAE , RMSE and 2 R , are respectively calculated, as shown in Table 1. are the closest to 1, which indicates that QGCCBNN has higher prediction accuracy and more reliable stability. Finally, the computational time (i.e., the sum of training time, TI prediction time and RUL prediction time) of our RUL prediction method based on QGCCBNN is compared with that of the prediction method using DADNN, GRUNN, FCRBFNN or LS-SVM. The computational time is measured by MATLAB 2016 in the following hardware configuration environment: 8G RAM and 3.2 GHz Intel CPU. The average computational time of the 50-fold cross-validation executions for each prediction method in the RUL prediction experiment is recorded on the same full-time data, as plotted in Fig. 16. The computational time of QGCCBNN is only 14.85 s, and those of DADNN, GRUNN, FCRBFNN, and LS-SVM are respectively 32.68 s, 29.59 s, 23.74 s, and 19.93s. Obviously, the computational time of QGCCBNN is much shorter than that of DADNN, GRUNN, FCRBFNN or LS-SVM, which can be attributed to the superiority of parallel computing brought by quantum gene chain coding.

·15·
In conclusion, we propose a novel recurrent neural network named quantum gene chain coding bidirectional neural network (QGCCBNN) to predict RUL of RM. The main contributions of this paper are as follows: (1) In our QGCCBNN, a quantum bidirectional transmission mechanism is designed to establish the preand post-relationships of time series for readjusting the weight parameters according to the feedback from the output layer, so QGCCBNN can realize higher consistency between the input information and the overall memory of the network, and then has better nonlinear approximation ability. To the best of our knowledge, the quantum bidirectional transmission mechanism is the first attempt for neural network data transmission and the experiment results confirm its effectiveness.
(2) In the process of training QGCCBNN, quantum gene chain coding is innovatively constructed to transmit and update data to avoid the gradient disappearance, gradient explosion, slow convergence speed and large time cost caused by traditional gradient descent method. As a consequence, QGCCBNN has better global optimization ability and faster convergence speed.
(3) Due to the advantages of QGCCBNN in nonlinear approximation ability, global optimization ability and convergence speed, the RUL prediction method based on QGCCBNN can realize higher prediction accuracy and lower computation cost.
(4) Comparative analysis on the RUL prediction results of a double-row roller bearing by different methods demonstrates that our proposed method is effective in RUL prediction of RM and superior to the other methods.