Rolling bearing remaining service life prediction method oriented to multi-stage degradation under cross-working conditions

Abstract

The invention relates to the technical field of bearing residual service life prediction, in particular to a rolling bearing residual service life prediction method oriented to multi-stage degradation under cross-working conditions, which comprises the following steps: S1, recognizing a bearing degradation stage; S2, carrying out data preprocessing and feature extraction: S1.1, establishing a multivariate time sequence autoregression model, and S1.2, establishing a residual error control chart; the step S2 further comprises the steps of S2.1, data preprocessing, S2.2, multi-scale convolutional neural network module design and S2.3, a correlation alignment-model migration domain self-adaption method. According to the method, different degradation stages of the bearing can be identified on line, depth domain self-adaption between a source domain and a target domain is achieved, and the method has the advantages of being high in accuracy, high in accuracy and high in robustness. The problems that the prediction precision of the remaining service life of the rolling bearing is reduced due to the difference of different degradation stages and the generalization performance of the model is poor due to the fact that the distribution difference of degradation data under different working conditions is large are well solved, and a multi-stage remaining service life prediction result with high precision can be obtained under the cross-working-condition condition.

Description

Technical Field

[0001] The present invention relates to the technical field of bearing remaining service life prediction, and in particular to a method for predicting the remaining service life of a rolling bearing subject to multi-stage degradation under cross-operating conditions. Background Art

[0002] Remaining Useful Life (RUL) mainly refers to the remaining service life of a system after it has been running for a period of time. If the RUL of a system can be accurately predicted, the losses caused by system downtime can be greatly reduced, and the operational reliability of the system can be effectively improved.

[0003] RUL prediction methods based on a single deep learning model usually assume that the data in the source domain and the target domain have the same distribution, which is usually difficult to achieve in practical applications. In the field of RUL prediction of rolling bearings, how to migrate the prediction model trained based on the bearing data under one working condition to the RUL prediction of bearings under different working conditions is a difficult problem in the field of bearing RUL prediction. Existing domain adaptation methods usually only consider the distribution differences of bearing data over the entire life cycle when applied to bearing RUL prediction. However, the degradation process of bearings includes multiple different degradation stages. If the full life cycle data without stage division is directly used for domain adaptation, and the impact of different degradation stages is ignored, negative migration may occur, which will seriously affect the accuracy of the model and lead to large deviations in the prediction results. Summary of the invention

[0004] The object of the present invention is to provide a method for predicting the remaining service life of a rolling bearing with multi-stage degradation under cross-working conditions based on a multi-scale convolutional neural network (MSWCNN) and CoMo-TL.

[0005] Based on the above purpose, the present invention adopts the following technical solution:

[0006] A method for predicting the remaining useful life (RUL) of a rolling bearing with multi-stage degradation under cross-operating conditions comprises the following steps:

[0007] S1. Identify the degradation stage of the bearing: Establish a degradation stage division method based on the multivariate time series autoregression model (MSAR model) and the residual control chart, collect the degradation data of the equipment and establish the autoregression model, and calculate the residual of the overall data; identify the degradation mutation point based on the residual, identify the online observation data based on the obtained mutation point, realize the division of the degradation process stage, and obtain the multi-stage degradation data of the test bearing;

[0008] S2. Data preprocessing and feature extraction: Preprocess the multi-stage degradation data obtained in step S1, and then use the MSWCNN feature extractor to extract features from the preprocessed data, and input the data at different degradation stages into the MSWCNN at different stages respectively; input the extracted feature data into the regression prediction module to obtain the RUL prediction result; then minimize the covariance distance between the source domain and target domain features through CoMo-TL, align the feature distributions of the two domains, and realize accurate prediction of RUL in the target domain by the model under cross-working conditions.

[0009] Preferably, identifying the bearing degradation stage in step S1 comprises the following steps:

[0010] S1.1. Establishing MSAR model: Collect historical multivariate monitoring data of rolling bearings, and select relatively stable data to establish autoregressive model;

[0011] S1.2. Residual control chart: Use the new observation data to establish the MSAR model to obtain the residuals, draw the residual control chart, and dynamically monitor the residual sequence through the predefined control line, identify the mutation points and obtain multiple degradation stages divided by the residual control chart.

[0012] Preferably, step S2 of data preprocessing and feature extraction comprises the following steps:

[0013] S2.1, data preprocessing: According to the mutation points obtained in step S1, the vibration data of the source domain and the target domain are divided into multi-stage vibration data, and the vibration data of each stage of different channels are normalized, and the minimum-maximum normalization is used to scale them to [0,1] to eliminate the dimensional differences between different channels; after completing the data normalization, the RUL label is calculated;

[0014] S2.2, MSWCNN module design: The MSWCNN module is constructed based on the multi-scale convolution path, the pooling weighting mechanism and the feature fusion, and the fusion of the output features of different convolution layers is realized through the concatenation method; the present invention adopts a multi-layer MSWCNN for feature extraction. The network consists of multiple weighted attention convolution blocks of different scales, which can effectively extract deep features such as time domain and frequency domain from the vibration signal to improve the accuracy of multi-stage RUL prediction;

[0015] S2.3, CoMo-TL domain adaptation method: Combining correlation alignment (Coral) and model transfer, a deep domain adaptation framework based on CoMo-TL (Coral-model transfer) is established to achieve accurate prediction of RUL.

[0016] Preferably, the process of establishing the MSAR model in step S1.1 includes:

[0017] The historical multivariate monitoring data of rolling bearings collected: X = x 1 ,x 2 ,...,x M , where M represents the number of degraded data items, and relatively stable data are selected and expressed as Where m represents the number of sample points;

[0018] definition and is the mutation point between the three degradation stages, and the data sequences of the three degradation stages are represented as and According to the data series The first-stage MSAR model established is as follows:

[0019]

[0020] In the formula, is the mean of the data in the stationary part, p 1 represents the order of the model, represents the autocorrelation coefficient, is subject to normal distribution N 1 (0,C 1 ) error term;

[0021] All parameters in the model are unknown and parameter estimation is required; select the data sequence Formula (1) can be rewritten as in,

[0022]

[0023]

[0024]

[0025]

[0026] In the formula, M represents the number of degradation data items, p represents the order of the model, and x a Indicates the first mutation point, x c Represents the upper limit of the selected data sequence, where c <a;

[0027] According to the least squares method, we can get A 1 and C 1 Estimated value of and Defined by:

[0028]

[0029] The order of the model is p 1 Estimated value of Defined by:

[0030]

[0031] In the formula, represents the number of feasible data points;

[0032] According to the above steps, the parameters of the MSAR model in the first stage can be estimated Repeat the above steps to estimate the model parameters of the second and third stages. and

[0033] Substituting the model parameters of the three stages into formula (1) in turn, the predicted values ​​of the data in each stage can be expressed as follows:

[0034]

[0035] Preferably, the process of the residual control chart in step S1.2 includes:

[0036] The residual can be expressed as:

[0037]

[0038] In the formula, represents the residual value of the nth degradation data in the i-th stage, represents the true value of the nth degradation data in the i-th stage, represents the predicted value of the nth degradation data in the i-th stage;

[0039] Since the present invention divides the degradation process into three different stages, a three-stage residual control chart is proposed to monitor the degradation process; the residual obtained by establishing the MSAR model using new observation data, the residual control chart is drawn and the residual sequence is dynamically monitored according to the predefined control line; the control lines of the three stages are defined as:

[0040]

[0041] Where a2 and b2 are distributions between two adjacent distributions. Generally covers 99% of the distribution range, and in the residual control chart, L = 3; and in,

[0042] The intervals of the three stages are defined as (CL1, CL2), (CL2, CL3) and (CL3, CL4) respectively; the three degradation stages of the new observation data can be segmented online by using the established residual control chart;

[0043] The process of online segmentation of the three degradation stages includes: using formula (7) to monitor and update the residual of the degradation data; if the residual is within the range of [CL1, CL2), the system is considered to be in the first degradation stage; after a period of operation, the residual will be at the mutation point x a The degradation from the new system to this critical point is regarded as the first degradation stage. As the system ages, when the residual falls into the range of [CL2, CL3), until the mutation point x b Jump to [CL3, CL4), this process system is considered to be in the second degenerate stage; when the residual is at the mutation point x b If it jumps to [CL3,CL4), the system is considered to be in the third degradation stage;

[0044] Therefore, three degradation stages separated by the residual control chart can be obtained:

[0045]

[0046] Preferably, the data preprocessing process in step S2.1 includes:

[0047] The process of dividing the vibration data into multiple stages, normalizing the vibration data, and scaling it to [0, 1] to eliminate the dimension difference is defined by equation (9):

[0048]

[0049] In the formula, represents the normalized data of the zth channel in the i-th row, Represents the original data of the zth channel in the i-th row;

[0050] After normalizing the data, the RUL label is calculated; the ratio of the current RUL to the entire life cycle is used as the target label for RUL fitting. Assuming that the i-th sampling point is the i-th sample, there are a total of n sampling points, and each sampling point represents an operating cycle, that is, the bearing fails at the n-th sampling point, then the RUL label y of the i-th sample is i It can be defined by formula (10):

[0051]

[0052] Preferably, the multi-scale convolution path in step S2.2 includes:

[0053] This path applies convolution kernels of different sizes to different convolution branches to extract spatial features of different scales from multi-channel signals. For different branches, the multi-scale convolution path introduces a pooling weighting mechanism to select important spatial features on the channel according to their relevance to the prediction task, and finally performs special fusion. In the multi-scale convolution path, for the i-th scale, the size of its receptive field is calculated by The width of the convolution kernel used in The output of can be defined by formula (11):

[0054]

[0055] In the formula, represents the convolution calculation of the i-th scale in the first convolutional layer, ReLU represents the rectified linear unit activation function, and F BN represents the batch normalization operation, X*W1 represents the convolution operation, W1 represents the weight of the first convolutional layer, and b i represents the bias, represents multi-stage data, b, c, l represent the batch size, the number of channels and the length of each channel respectively; the output feature S of the i-th scale i It can be defined by formula (12):

[0056]

[0057] Preferably, the pooling weighting mechanism in step S2.2 includes:

[0058] The pooling weighting mechanism is used to assign weights to multi-scale features in order to better capture the spatial and temporal characteristics of the input signal. The parameters of each convolution kernel channel in this mechanism represent the importance of the feature, and these parameters are adaptively updated according to the relevance of the data of different input channels to the prediction task, that is, features with strong relevance will have larger weights, and features with weak relevance will have smaller weights;

[0059] First, average pooling and maximum pooling operations are performed on the input features to generate two different spatial descriptors W avg and W max , respectively represent the average pooling feature and the maximum pooling feature, and the weight obtained by the pooling weighting mechanism for:

[0060]

[0061] In the formula, F FC represents the fully connected layer function, Softmax(·) represents the activation function to ensure that the total weight is 1, Indicates a connection operation; and They represent the average pooling feature and the maximum pooling feature of the i-th scale respectively, which can be defined by equations (14) and (15):

[0062]

[0063] The feature weights calculated by formula (13) are used to calculate the output features S of the multi-scale convolution path. i The weighting process can be defined by formula (16):

[0064]

[0065] In the formula, Represents the weighted output features.

[0066] Preferably, the feature fusion process in step S2.2 includes:

[0067] The output features of each scale after pooling weights are fused with the output features of different convolutional layers through the concatenate method. This process can be defined by formula (17):

[0068]

[0069] In the formula, S + represents the features fused by the concatenate method, F concat Represents the calculation function of the concatenate method;

[0070] The features fused by the concatenate method are further extracted by the MSWCNN module and the final feature X′ is output:

[0071] X′=Conv 2 (S + ), (18);

[0072] In the formula, Conv 2 (·) represents the convolution calculation of the second convolutional layer, which can be defined by formula (19):

[0073] Conv 2 (S + )=ReLU(F BN (S + *W2+b i )), (19);

[0074] Where W2 represents the weight of the second convolutional layer;

[0075] Based on the above feature extraction steps, the feature extraction process of the MSWCNN module can be summarized as follows:

[0076]

[0077] Preferably, the process of the CoMo-TL domain adaptation method in step S2.3 includes:

[0078] Correlation Alignment: Correlation Alignment (Coral) is a method applied to cross-modal information retrieval. Its main idea is to learn the conversion relationship between two different modalities by introducing an adaptive layer, thereby achieving cross-modal information retrieval. Coral extracts features from input data through two branches, and then maps image features and text features into a common subspace. In the common subspace, Coral uses an adaptive layer to learn the conversion relationship between two different modalities.

[0079] Define the source domain training samples as Label L s =y i ,i∈{1,2,...,L}; the unlabeled target domain data is Where d represents the number of output neurons in the network; the number of samples in the source domain and the target domain are n s and n t ; and represents the j-th dimension feature of the i-th sample in the source domain and the target domain respectively, C s and C t Represent the feature covariance matrices of the source domain and the target domain respectively; the Coral loss is defined as the covariance distance between the source domain and the target domain features, expressed by formula (21):

[0080]

[0081] in Represents the F2 norm. According to the covariance distance, the covariance calculation formula between the source domain and the target domain can be obtained:

[0082]

[0083] After obtaining the covariance between the source and target domains, the chain rule is used to calculate the gradient relative to the input features:

[0084]

[0085] The final Coral loss function minimizes the covariance matrix between the target domain and the source domain, thereby reducing the data distribution difference between the two domains. The Coral loss function can be defined by formula (24):

[0086]

[0087] Where t represents the number of Coral loss layers in the deep network, and λ represents the weight for weighing the adaptation and classification accuracy in the source;

[0088] Deep domain adaptation framework based on CoMo-TL: The present invention combines Coral and model migration and proposes a CoMo-TL (Coral-model migration) method to solve the problem of accurately predicting the RUL of rolling bearings when the data distribution is inconsistent with the target domain data and the target domain is not labeled; CoMo-TL consists of Coral-based deep domain adaptation and model-based migration; in the first part, the pre-processed state monitoring data in the source domain and the target domain are simultaneously input into the feature extraction module, and Coral domain adaptation is performed after a multi-scale weighted convolution layer to obtain the Coral loss. By minimizing the Coral loss of the source domain and target domain features, the feature distributions of the two domains are automatically aligned, thereby achieving domain adaptation; in the second part, the network parameters of the feature extraction module of the model trained in the source domain data are migrated to the target domain data to achieve accurate prediction of the RUL.

[0089] In the feature extraction module, the input comes from the data Then the input features are mapped to the subspace Where Z s =φ1(D s ); at the same time, the target domain data Input feature extraction module and map to the same subspace Where Z t =φ1(D t ); After feature extraction, the source domain labeled data Z s Input the regression prediction module, map it to the subspace where the RUL result is located, and output the prediction result in By minimizing Z s and Z t Coral loss and the average loss error between the source domain predicted RUL and the true RUL, update the overall trainable parameters of the source domain model;

[0090] Model optimization goal: The loss function of the model in the present invention is composed of the loss function L of the multi-stage RUL prediction module pred And the Coral loss function L between the source domain and the target domain Coral Composition, among which L Coral It has been defined by formula (24), and the loss function L of the multi-stage RUL prediction module can be calculated by the mean absolute error (MAE): pred :

[0091]

[0092] In the formula, n represents the number of samples, y i represents the true RUL of the i-th sample, represents the predicted RUL of the i-th sample;

[0093] Therefore, the final loss function L of the model proposed in this invention is final It can be defined by formula (26):

[0094] L final =L pred +L coral , (26).

[0095] The beneficial effects of the present invention include:

[0096] The present invention proposes a degradation process stage division method for an autoregressive model, which can identify different degradation stages of bearings online; the method can identify degradation mutation points through residuals, and can identify mutation points in three degradation stages of the target bearing by importing online observation data, and then identify different degradation stages of the bearing, thereby realizing the division of bearing degradation stages; by identifying different degradation stages of the bearing, it is possible to more accurately predict the RUL of the bearing at different degradation stages, effectively improving the RUL prediction accuracy.

[0097] The present invention adopts MSWCNN to perform deep feature extraction on multi-stage vibration signals in source domain and target domain, minimizes the covariance distance between source domain and target domain through CoMo-TL method, aligns feature distribution, and thus realizes deep domain adaptation. It effectively solves the problems of decreased RUL prediction accuracy due to differences in different degradation stages of rolling bearings and poor model generalization performance due to large differences in degradation data distribution under different working conditions, and can obtain multi-stage RUL prediction results with higher accuracy under cross-working conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0098] Figure 1 This is a flow chart of Embodiment 1 of the present invention;

[0099] Figure 2 Schematic diagram of residual control diagram of embodiment 1 of the present invention;

[0100] Figure 3 This is a flowchart of degradation stage division based on residual control diagram according to Embodiment 1 of the present invention;

[0101] Figure 4 This is a structural diagram of the MSWCNN of Embodiment 1 of the present invention;

[0102] Figure 5 This is a multi-scale convolution path structure diagram of Example 1 of the present invention;

[0103] Figure 6 This is a structural diagram of a pooling weighting mechanism according to Embodiment 1 of the present invention;

[0104] Figure 7 The CoMo-TL-based deep domain adaptation framework of Embodiment 1 of the present invention;

[0105] Figure 8 This is the experimental device of Example 2 of the present invention;

[0106] Fig. 9 is the residual control diagram of bearing 1_4 of embodiment 2 of the present invention;

[0107] Fig.10 is the residual control diagram of bearing 1_5 of embodiment 2 of the present invention;

[0108] Fig.11 is the residual control diagram of bearing 2_4 of embodiment 2 of the present invention;

[0109] Fig.12 is the residual control diagram of bearing 2_5 of embodiment 2 of the present invention;

[0110] Fig.13 is the residual control diagram of bearing 3_3 in embodiment 2 of the present invention;

[0111] Fig.14 is the residual control diagram of bearing 3_5 of embodiment 2 of the present invention;

[0112] Fig.15 It is a multi-stage RUL prediction diagram of bearing 1_4 of embodiment 2 of the present invention;

[0113] Fig.16 It is a multi-stage RUL prediction diagram of bearing 1_5 of embodiment 2 of the present invention;

[0114] Fig.17 It is a multi-stage RUL prediction diagram of the bearing 2_4 of Example 2 of the present invention;

[0115] Fig.18 It is a multi-stage RUL prediction diagram of the bearing 2_5 of Example 2 of the present invention;

[0116] Fig.19 It is a multi-stage RUL prediction diagram of the bearing 3_3 of Example 2 of the present invention;

[0117] Fig. 20 It is a multi-stage RUL prediction diagram of the bearing 3_5 of Example 2 of the present invention;

[0118] Fig.21 This is a comparison diagram of four RUL prediction methods for bearings 1_4 according to Example 2 of the present invention;

[0119] Fig. 22This is a comparison diagram of four RUL prediction methods for bearings 1_5 according to Example 2 of the present invention;

[0120] Fig.23 4 is a comparison diagram of four RUL prediction methods for bearing 2_4 according to embodiment 2 of the present invention;

[0121] Fig.24 This is a comparison diagram of four RUL prediction methods for bearing 2_5 of embodiment 2 of the present invention;

[0122] Fig.25 This is a comparison diagram of four RUL prediction methods for the bearing 3_3 of Example 2 of the present invention;

[0123] Fig.26 This is a comparison diagram of four RUL prediction methods for bearing 3_5 in Example 2 of the present invention;

[0124] Fig. 27 is an overall performance evaluation diagram of four RUL prediction methods for bearings 1_4 according to embodiment 2 of the present invention;

[0125] Fig.28 is an overall performance evaluation diagram of four RUL prediction methods for bearings 1_5 according to embodiment 2 of the present invention;

[0126] Fig.29 is an overall performance evaluation diagram of four RUL prediction methods for bearings 2_4 according to embodiment 2 of the present invention;

[0127] Fig.30 4 is an overall performance evaluation diagram of four RUL prediction methods for bearing 2_5 of embodiment 2 of the present invention;

[0128] Fig.31 is an overall performance evaluation diagram of four RUL prediction methods for bearing 3_3 of embodiment 2 of the present invention;

[0129] Fig.32 This is an overall performance evaluation diagram of four RUL prediction methods for bearing 3_5 of embodiment 2 of the present invention. DETAILED DESCRIPTION

[0130] Example 1

[0131] The following is a further explanation of the present invention in conjunction with specific embodiments. Figure 1 As shown, the present invention is a method for predicting the RUL of a rolling bearing with multi-stage degradation under cross-operating conditions, comprising the following steps:

[0132] S1. Identify the degradation stage of the bearing: Establish a degradation stage division method based on the MSAR model and residual control chart, collect the degradation data of the equipment and establish an autoregressive model, and calculate the residual of the overall data; identify the degradation mutation point based on the residual, identify the online observation data based on the obtained mutation point, realize the division of the degradation process stage, and obtain the multi-stage degradation data of the test bearing;

[0133] Wherein, step S1 further includes the following steps:

[0134] S1.1. Establishing MSAR model: The historical multivariate monitoring data of rolling bearings collected: X = x 1 ,x 2 ,...,x M , where M represents the number of degraded data items, and relatively stable data are selected and expressed as Where m represents the number of sample points;

[0135] definition and is the mutation point between the three degradation stages, and the data sequences of the three degradation stages are represented as and According to the data series The first-stage MSAR model established is as follows:

[0136]

[0137] In the formula, is the mean of the data in the stationary part, p 1 represents the order of the model, represents the autocorrelation coefficient, is subject to normal distribution N 1 (0,C 1 ) error term;

[0138] All parameters in the model are unknown and parameter estimation is required; select the data sequence Formula (1) can be rewritten as 1 =V 1 A 1 +E 1 ;in,

[0139]

[0140]

[0141]

[0142] According to the least squares method, we can get A1 and C 1 Estimated value of and Defined by:

[0143]

[0144] The order of the model is p 1 Estimated value of Defined by:

[0145]

[0146] In the formula, represents the number of feasible data points;

[0147] According to the above steps, the parameters of the MSAR model in the first stage can be estimated Repeat the above steps to estimate the model parameters of the second and third stages. and

[0148] Substituting the model parameters of the three stages into formula (1) in turn, the predicted values ​​of the data in each stage can be expressed as follows:

[0149]

[0150] S1.2, residual control chart: Use the new observation data to establish the MSAR model to obtain the residual, which can be expressed as:

[0151]

[0152] Since the present invention divides the degradation process into three different stages, a three-stage residual control chart is proposed to monitor the degradation process; the residual obtained by establishing the MSAR model using new observation data, the residual control chart is drawn and the residual sequence is dynamically monitored according to the predefined control line; the control lines of the three stages are defined as:

[0153]

[0154] Where a2 and b2 are distributions between two adjacent distributions. Generally covers 99% of the distribution range, and in the residual control chart, L = 3; and in,

[0155] like Figure 2As shown in the figure, the first stage interval is (CL1, CL2), the second stage interval is (CL2, CL3), and the third stage interval is (CL3, CL4). Specifically, when the residual exceeds the upper control limit CL2, it marks the end of the first stage and the beginning of the second stage. When the residual exceeds the upper control limit CL3, it marks the end of the second stage and the beginning of the third stage. We can use the established residual control chart to segment the degradation stage of the new observation data online. The specific process is as follows: Figure 3 shown.

[0156] The process of online segmentation of the three degradation stages is described as follows: the residual of the degradation data is monitored and updated using formula (7); if the residual is within the range [CL1, CL2), the system is considered to be in the first degradation stage; after a period of operation, the residual will be at the mutation point x a The degradation from the new system to this critical point is regarded as the first degradation stage. As the system ages, when the residual falls into the range of [CL2, CL3), until the mutation point x b Jump to [CL3, CL4), this process system is considered to be in the second degenerate stage; when the residual is at the mutation point x b If it jumps to [CL3,CL4), the system is considered to be in the third degradation stage;

[0157] Therefore, three degradation stages separated by the residual control chart can be obtained:

[0158]

[0159] S2. Data preprocessing and feature extraction: Preprocess the multi-stage degradation data obtained in step S1, and then use the MSWCNN feature extractor to extract features from the preprocessed data, and input the data at different degradation stages into the MSWCNN at different stages respectively; input the extracted feature data into the regression prediction module to obtain the RUL prediction result; then minimize the covariance distance between the source domain and target domain features through CoMo-TL, align the feature distributions of the two domains, and realize accurate prediction of RUL in the target domain by the model under cross-working conditions.

[0160] Wherein, step S2 further includes the following steps:

[0161] S2.1. Data preprocessing: According to the mutation points obtained in step S1, the vibration data of the source domain and the target domain are divided into multi-stage vibration data, and the vibration data of each stage of different channels are normalized. The minimum-maximum normalization is used to scale them to [0, 1] to eliminate the dimensional differences between different channels. The process is defined by formula (9):

[0162]

[0163] In the formula, represents the normalized data of the zth channel in the i-th row, Represents the original data of the zth channel in the i-th row;

[0164] After normalizing the data, the RUL label is calculated; the ratio of the current RUL to the entire life cycle is used as the target label for RUL fitting. Assuming that the i-th sampling point is the i-th sample, there are a total of n sampling points, and each sampling point represents an operating cycle, that is, the bearing fails at the n-th sampling point, then the RUL label y of the i-th sample is i It can be defined by formula (10):

[0165]

[0166] S2.2, MSWCNN module design: The structure diagram of the MSWCNN module is shown in Figure 4 As shown in the figure, it consists of three parts: multi-scale convolution path, pooling weighting mechanism and feature fusion; the network processes the input vibration signal through convolution layers with convolution kernels of different sizes. The features captured by convolution kernels of different sizes can reflect the different time widths of the signal, so that the model can process global trends and local details at the same time; then, a pooling weighting mechanism is proposed, which obtains the channel weights of features of different scales through average pooling and maximum pooling, and uses a fully connected network to adaptively adjust the channel weights; finally, the fusion of output features of different convolution layers is realized through the concatenate method.

[0167] The structure of the multi-scale convolution path is as follows Figure 5 As shown in Figure 2, the multi-scale convolution path applies convolution kernels of different sizes to different convolution branches to extract spatial features of different scales from multi-channel signals. For different branches, the multi-scale convolution path introduces a pooling weighting mechanism to select important spatial features on the channel according to their relevance to the prediction task, and finally performs special fusion. In the multi-scale convolution path, for the i-th scale, the size of its receptive field is determined by The width of the convolution kernel used in is determined by, The output of can be defined by formula (11):

[0168]

[0169] In the formula, represents the convolution calculation of the i-th scale in the first convolutional layer, ReLU represents the rectified linear unit activation function, and F BN represents the batch normalization operation, X*W1 represents the convolution operation, W1 represents the weight of the first convolutional layer, and b i Indicates bias, represents multi-stage data, b, c, l represent the batch size, the number of channels and the length of each channel respectively; the output feature S of the i-th scale i It can be defined by formula (12):

[0170]

[0171] Pooling weighting mechanism such as Figure 6 As shown in the figure, the pooling weighting mechanism is used to assign weights to multi-scale features in order to better capture the spatial and temporal characteristics of the input signal. The parameters of each convolution kernel channel in this mechanism represent the importance of the feature, and these parameters are adaptively updated according to the relevance of the data of different input channels to the prediction task, that is, features with strong relevance will have larger weights, and features with weak relevance will have smaller weights;

[0172] First, average pooling and maximum pooling operations are performed on the input features to generate two different spatial descriptors W avg and W max , respectively represent the average pooling feature and the maximum pooling feature, and the weight obtained by the pooling weighting mechanism for:

[0173]

[0174] In the formula, F FC represents the fully connected layer function, Softmax(·) represents the activation function to ensure that the total weight is 1, Indicates a connection operation; and They represent the average pooling feature and the maximum pooling feature of the i-th scale respectively, which can be defined by equations (14) and (15):

[0175]

[0176]

[0177] The feature weights calculated by formula (13) are used to calculate the output features S of the multi-scale convolution path. i The weighting process can be defined by formula (16):

[0178]

[0179] In the formula, Represents the weighted output features.

[0180] The output features of each scale after pooling weights are fused with the output features of different convolutional layers through the concatenate method. This process can be defined by formula (17):

[0181]

[0182] In the formula, S + represents the features fused by the concatenate method, F concat Represents the calculation function of the concatenate method; the features fused by the concatenate method are further extracted by the MSWCNN module and the final feature X′ is output:

[0183] X′=Conv 2 (S + ), (18);

[0184] In the formula, Conv 2 (·) represents the convolution calculation of the second convolutional layer, which can be defined by formula (19):

[0185] Conv 2 (S + )=ReLU(F BN (S + *W2+b i )), (19);

[0186] Where W2 represents the weight of the second convolutional layer;

[0187] Based on the above feature extraction steps, the feature extraction process of the MSWCNN module can be summarized as follows:

[0188]

[0189] S2.3, CoMo-TL domain adaptation method: Combining correlation alignment (Coral) and model transfer, a deep domain adaptation framework based on CoMo-TL (Coral-model transfer) is established to achieve accurate prediction of RUL.

[0190] Correlation Alignment: Correlation Alignment (Coral) is a method applied to cross-modal information retrieval. Its main idea is to learn the conversion relationship between two different modalities by introducing an adaptive layer, thereby achieving cross-modal information retrieval. Coral extracts features from the input data through two branches, and then maps image features and text features into a common subspace. In the common subspace, Coral uses an adaptive layer to learn the conversion relationship between two different modalities.

[0191] Define the source domain training samples as Label L s =y i ,i∈{1,2,...,L}; the unlabeled target domain data is Where d represents the number of output neurons in the network; the number of samples in the source domain and the target domain are ns and n t ; and represents the j-th dimension feature of the i-th sample in the source domain and the target domain respectively, C s and C t Represent the feature covariance matrices of the source domain and the target domain respectively; the Coral loss is defined as the covariance distance between the source domain and the target domain features, expressed by formula (21):

[0192]

[0193] in Represents the F2 norm. According to the covariance distance, the covariance calculation formula between the source domain and the target domain can be obtained:

[0194]

[0195] After obtaining the covariance between the source and target domains, the chain rule is used to calculate the gradient relative to the input features:

[0196]

[0197] The final Coral loss function minimizes the covariance matrix between the target domain and the source domain, thereby reducing the data distribution difference between the two domains. The Coral loss function can be defined by formula (24):

[0198]

[0199] Where t represents the number of Coral loss layers in the deep network, and λ represents the weight for weighing the adaptation and classification accuracy in the source;

[0200] Deep domain adaptation framework based on CoMo-TL: This paper combines Coral and model migration and proposes a CoMo-TL (Coral-model migration) method to solve the problem of accurately predicting the RUL of rolling bearings when the data and target domain data distribution are inconsistent and the target domain is not labeled; the structure of CoMo-TL is as follows Figure 7 shown.

[0201] CoMo-TL consists of Coral-based deep domain adaptation and model-based migration. In the first part, the preprocessed state monitoring data in the source domain and target domain are simultaneously input into the feature extraction module, and Coral domain adaptation is performed after passing through the multi-scale weighted convolution layer to obtain the Coral loss. By minimizing the Coral loss of the source domain and target domain features, the feature distributions of the two domains are automatically aligned, thereby achieving domain adaptation. In the second part, the network parameters of the feature extraction module of the model trained in the source domain data are migrated to the target domain data to achieve accurate prediction of RUL.

[0202] like Figure 7 As shown, the feature extraction module input comes from the data Then the input features are mapped to the subspace Where Z s =φ1(D s ); at the same time, the target domain data Input feature extraction module and map to the same subspace Where Z t =φ1(D t ); After feature extraction, the source domain labeled data Z s Input the regression prediction module, map it to the subspace where the RUL result is located, and output the prediction result in By minimizing Z s and Z t Coral loss and the average loss error between the source domain predicted RUL and the true RUL, update the overall trainable parameters of the source domain model;

[0203] Model optimization goal: The loss function of the model in the present invention is composed of the loss function L of the multi-stage RUL prediction module pred And the Coral loss function L between the source domain and the target domain Coral Composition, among which L Coral It has been defined by formula (24), and the loss function L of the multi-stage RUL prediction module can be calculated by MAE pred :

[0204]

[0205] In the formula, n represents the number of samples, y i represents the true RUL of the i-th sample, represents the predicted RUL of the i-th sample;

[0206] Therefore, the final loss function L of the model proposed in this invention is final It can be defined by formula (26):

[0207] L final =L pred +L coral , (26).

[0208] Example 2

[0209] This embodiment uses actual experiments to verify the method involved in the present invention; this embodiment uses the bearing data set of Xi'an Jiaotong University and Changxing Shengyang Technology Co., Ltd. (XJTU-SY) to evaluate the performance of the proposed method; the experimental device is as follows Figure 8As shown in FIG. 1 , the experimental design includes an AC motor, a motor speed regulator, a rotating shaft, a supporting bearing, a hydraulic loading system and a bearing under test. The specific bearing model tested is LDK UER204. The experimental design simulates three different working conditions, as shown in Table 1.

[0210] Table 1 Detailed information on the operating conditions of the accelerated life test

[0211] Working conditions Condition 1 Condition 2 Condition 3 Speed / (rpm) 2100 2250 2400 Radial force / (kN) 12 11 10

[0212] In addition, durability tests were performed on five bearings under each operating condition. During the experiment, vibration data were collected using a DT9837 portable dynamic signal collector and two PCB 352C33 unidirectional accelerometers, which were fixed on magnetic bases in the horizontal and vertical directions respectively. The sampling parameters were set to a frequency of 25.6 kHz, an interval of 1 minute, and each sampling lasted 1.28 seconds. The specific details of each tested bearing, such as operating status, total number of data samples, actual life, and fault location, are shown in Table 2.

[0213] Table 2 Information of the tested bearings

[0214]

[0215]

[0216] This embodiment mainly studies the multi-level RUL prediction of bearings under different working conditions. Therefore, it is necessary to divide the data sets under different working conditions into source domain and target domain according to the requirements, and verify the effectiveness of the proposed method. Three experimental tasks are carried out in this embodiment, as shown in Table 3. Task 1 involves the transfer from working condition 2 to working condition 1. Task 2 involves the transfer from working condition 3 to working condition 2. Task 3 involves the transfer from working condition 1 to working condition 3. For the target domain, the data is divided into 3:7, that is, 30% of the data is selected as the training set and the rest of the data is used as the test set. It should be noted that all source domain data and the training set of the target domain are used for training, and only the test set of the target domain is used for testing. Therefore, the prediction model can effectively learn the invariant features of different fields.

[0217] Table 3. Description of TL tasks under cross-working conditions

[0218] Prediction Task Source domain / training set Target domain / test set Task 1 Bearing 2_1; Bearing 2_2; Bearing 2_3 Bearing 1_4; Bearing 1_5 Task 2 Bearing 3_1; Bearing 3_2; Bearing 3_4 Bearing 2_4; Bearing 2_5 Task 3 Bearing 1_1; Bearing 1_2; Bearing 1_3 Bearing 3_3; Bearing 3_5

[0219] In this embodiment, multi-stage signals are used as input to the MSWCNN model to achieve complex feature extraction; the model uses three different kernel sizes (3, 5, and 9) in the multi-scale convolution layer in each selective convolution recursive block to capture features of different scales, thereby enhancing the model's ability to handle different feature dimensions and complexities.

[0220] After the convolutional layer, the selective mechanism uses adaptive pooling and a series of fully connected layers for feature recalibration, allowing the model to prioritize features with more information; the transformation within the first selective convolutional recursive block maintains 64 channels, while the second block reduces the channel depth from 64 to 32, further compressing the spatial dimension of the feature map.

[0221] In terms of RUL prediction, the architecture adopts a "64-128-1" structure, using a fully connected layer to gradually refine the features into a single output prediction, which is very suitable for regression tasks such as RUL prediction. Table 4 lists the detailed specifications of the proposed model. The program runs on a 12th generation Core TM i5-12600KF CPU and On a personal computer with a GeForce RTX 3060OC graphics card, the GPU-accelerated Pytorch platform is used.

[0222] Table 4 Structure and parameters of the proposed method

[0223]

[0224] In order to determine the prediction accuracy of the method, the mean absolute error (MAE) and root mean square error (RMSE) are used in this embodiment to evaluate the difference between the actual results and the predicted results. The calculation formula is as follows:

[0225]

[0226] In the formula, and i Represent the predicted results and actual results respectively, and n represents the data length.

[0227] Experimental results analysis:

[0228] For the above test bearings, three residual control charts were created, such as Figure 9-14 As shown; for bearing 1_5, from Fig.10 a It can be seen that when the residual exceeds the control limit CL_2 = 1.09 at point 38, it moves from the first stage to the second stage; Fig.10 b shows that when the residual exceeds the control limit CL_3=3.32 at the 46th point, it jumps to the third degradation stage.

[0229] In addition, the comparison between the RUL prediction results and the actual life of the above-mentioned test bearings is also given, such as Figure 15-20As shown, by analyzing the comparison results between the RUL prediction results and the actual life of each test bearing, the accuracy and effectiveness of the bearing RUL prediction method involved in this embodiment can be verified; in order to further verify the effectiveness of the model, Figure 21-26 The comparison of the proposed model with the mainstream models ResNet (Deep Residual Network), CNN (Convolutional Neural Network) and LSTM (Long Short-Term Memory Network) is shown; Fig.21 As shown, the proposed model shows lower prediction errors in all three stages of bearings 1_4 compared with the comparative model, highlighting its superiority.

[0230] In addition, in order to verify the effectiveness and superiority of the model, we used RMSE (root mean square error), PER (error rate of the last degradation point), R 2 The prediction performance of the proposed model and the comparison model are compared by using six evaluation indicators: mean absolute deviation (MAD), mean absolute percentage error (MAPE), and mean absolute error (MAE). The results are shown in Figure 2. Figure 27-32 As shown; the values ​​described therein are obtained by calculating the true RUL and predicted RUL of each bearing at the 10%, 50% and 100% stages, and the data are shown in Table 5; then, the average values ​​of various evaluation indicators of all bearings in each stage are calculated; Fig. 27 It shows that in bearing 1_4, the proposed model has the lowest RMSE, MAE, MAPE, MAD and PER values, R 2 The highest value indicates that its prediction performance is superior; Figure 28-32 The results and Fig. 27 Consistent.

[0231] Table 5. Overall performance evaluation of RUL prediction using different methods

[0232]

[0233] In summary, the present invention proposes a RUL prediction method for rolling bearings with multi-stage degradation under cross-working conditions. The method can divide the degradation stages, and input the data in different stages into the feature extraction module of MSWCNN in the corresponding stage for feature extraction. The CoMo-TL method is used to minimize the Coral distance between the source domain and target domain features, align the feature distributions of the two domains, and achieve deep domain adaptation. The method effectively solves the problems of decreased RUL prediction accuracy of rolling bearings due to differences in different degradation stages and poor model generalization performance due to large differences in the distribution of degradation data under different working conditions, and can obtain multi-stage RUL prediction results with higher accuracy under cross-working conditions.

Claims

1. A method for predicting the remaining useful life of a rolling bearing with multi-stage degradation under cross-operating conditions, comprising the following steps: S1. Identify the degradation stage of the bearing: Establish a degradation stage division method based on the multivariate time series autoregressive model and residual control chart, collect the degradation data of the equipment and establish the autoregressive model, and calculate the residual of the overall data; identify the degradation mutation point based on the residual, identify the online observation data based on the obtained mutation point, realize the division of the degradation process stage, and obtain the multi-stage degradation data of the test bearing; S2. Data preprocessing and feature extraction: Preprocess the multi-stage degradation data obtained in step S1, and then use the multi-scale convolutional neural network feature extractor to extract features from the preprocessed data, and input the data at different degradation stages into the multi-scale convolutional neural network at different stages respectively; input the extracted feature data into the regression prediction module to obtain the remaining service life prediction result; then minimize the covariance distance between the source domain and target domain features through CoMo-TL, align the feature distributions of the two domains, and realize accurate prediction of the remaining service life in the target domain by the model under cross-working conditions.

2. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 1 is characterized in that: The step S1 of identifying the bearing degradation stage comprises the following steps: S1.

1. Establishing a multivariate time series autoregressive model: Collect historical multivariate monitoring data of rolling bearings, and select relatively stable data to establish an autoregressive model; S1.

2. Residual control chart: Use new observation data to establish a multivariate time series autoregressive model to obtain residuals, draw a residual control chart, and dynamically monitor the residual series through predefined control lines to identify mutation points and obtain multiple degradation stages divided by the residual control chart.

3. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 2 is characterized in that: The step S2 of data preprocessing and feature extraction comprises the following steps: S2.1, data preprocessing: according to the mutation points obtained in step S1, the vibration data of the source domain and the target domain are divided into multi-stage vibration data, and the vibration data of each stage of different channels are normalized, and the minimum-maximum normalization is used to scale them to [0,1] to eliminate the dimensional differences between different channels; after completing the data normalization, the remaining service life label is calculated; S2.

2. Multi-scale convolutional neural network module design: A multi-scale convolutional neural network module is constructed based on the multi-scale convolution path, pooling weighting mechanism and feature fusion, and the fusion of output features of different convolutional layers is achieved through the connection function method; S2.

3. Correlation alignment-model transfer domain adaptation method: Combining correlation alignment and model transfer, a deep domain adaptation framework based on correlation alignment-model transfer is established to achieve accurate prediction of remaining useful life.

4. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 3 is characterized in that: The process of establishing a multivariate time series autoregressive model in step S1.1 includes: Collected historical multivariate monitoring data of rolling bearings: X = x 1 ,x 2 ,...,x M , where M represents the number of degraded data items, and relatively stable data are selected and expressed as Where m represents the number of sample points; definition and is the mutation point between the three degradation stages, and the data sequences of the three degradation stages are represented as and According to the data series The first-stage multivariate time series autoregressive model established is as follows: In the formula, is the mean of the data in the stationary part, p 1 represents the order of the model, represents the autocorrelation coefficient, is subject to normal distribution N 1 (0,C 1 ) error term; All parameters in the model are unknown and parameter estimation is required; select the data sequence Rewrite equation (1) into 1 =V 1 A 1 +E 1 ;in, In the formula, M represents the number of degradation data items, p represents the order of the model, and x a Indicates the first mutation point, x c Represents the upper limit of the selected data sequence, where c <a; According to the least squares method, we can get A 1 and C 1 Estimated value of and Defined by: The order of the model is p 1 Estimated value of Defined by: In the formula, represents the number of feasible data points; According to the above steps, the parameters of the multivariate time series autoregression model in the first stage can be estimated Repeat the above steps to estimate the model parameters of the second and third stages. and Substituting the model parameters of the three stages into formula (1) in turn, the predicted values ​​of the data in each stage are expressed as follows:

5. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 4 is characterized in that: The process of the residual control chart in step S1.2 includes: The residual is expressed as: In the formula, represents the residual value of the nth degradation data in the i-th stage, represents the true value of the nth degradation data in the i-th stage, represents the predicted value of the nth degradation data in the i-th stage; The residuals obtained by establishing the multivariate time series autoregressive model using the new observation data are plotted, and the residual control chart is dynamically monitored according to the predefined control lines; the control lines for the three stages are defined as: Where a2 and b2 are distributions between two adjacent distributions. Generally covers 99% of the distribution range, and in the residual control chart, L = 3; and in, The intervals of the three stages are defined as (CL1, CL2), (CL2, CL3) and (CL3, CL4) respectively; the three degradation stages of the new observation data are segmented online using the established residual control chart; Formula (7) is used to monitor and update the residual of the degradation data. If the residual is within the range of [CL1, CL2), the system is considered to be in the first degradation stage. After a period of operation, the residual will be at the mutation point x a The degradation from the new system to this critical point is regarded as the first degradation stage. As the system ages, when the residual falls into the range of [CL2, CL3), until the mutation point x b Jump to [CL3, CL4), this process system is considered to be in the second degenerate stage; when the residual is at the mutation point x b If it jumps to [CL3,CL4), the system is considered to be in the third degradation stage; Therefore, we get three degradation stages separated by the residual control chart:

6. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 5 is characterized in that: The data preprocessing process in step S2.1 includes: The process of dividing the vibration data into multiple stages, normalizing the vibration data, and scaling it to [0, 1] to eliminate the dimension difference is defined by equation (9): In the formula, represents the normalized data of the zth channel in the i-th row, Represents the original data of the zth channel in the i-th row; After normalizing the data, the remaining service life label is calculated; the ratio of the current remaining service life to the entire life cycle is used as the target label for the remaining service life fitting. Assuming that the i-th sampling point is the i-th sample, there are a total of n sampling points, and each sampling point represents an operating cycle, that is, the bearing fails at the n-th sampling point, then the remaining service life label y of the i-th sample is i Defined by formula (10):

7. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 6 is characterized in that: The multi-scale convolution path in step S2.2 includes: In the multi-scale convolution path, for the i-th scale, the size of its receptive field is calculated by The width of the convolution kernel used in The output of is defined by equation (11): In the formula, represents the convolution calculation of the i-th scale in the first convolutional layer, ReLU represents the rectified linear unit activation function, and F BN represents the batch normalization operation, X*W1 represents the convolution operation, W1 represents the weight of the first convolutional layer, and b i Indicates bias, represents multi-stage data, b, c, l represent the batch size, the number of channels and the length of each channel respectively; the output feature S of the i-th scale i Defined by formula (12):

8. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 7 is characterized in that: The pooling weighting mechanism in step S2.2 includes: Perform average pooling and maximum pooling operations on the input features to generate two different spatial descriptors W avg and W max , respectively represent the average pooling feature and the maximum pooling feature, and the weight obtained by the pooling weighting mechanism for: In the formula, F FC represents the fully connected layer function, Softmax(·) represents the activation function to ensure that the total weight is 1, Indicates a connection operation; and They represent the average pooling feature and the maximum pooling feature of the i-th scale respectively, and are defined by equations (14) and (15): The feature weights calculated by formula (13) are used to calculate the output features S of the multi-scale convolution path. i The weighting process is defined by equation (16): In the formula, Represents the weighted output features.

9. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 8, characterized in that: The process of feature fusion in step S2.2 includes: The output features of each scale after pooling weights are fused with the output features of different convolutional layers through the connection function method. The process is defined by formula (17): In the formula, S + represents the features fused by the connection function method, F concat The calculation function representing the connection function method; The features fused by the connection function method are further extracted by the multi-scale convolutional neural network module and the final feature X′ is output: X′=Conv 2 (S + ), (18); In the formula, Conv 2 (·) represents the convolution calculation of the second convolutional layer, which is defined by formula (19): Call 2 (S + )=ReLU(F BN (S + *W2+b i )), (19); Where W2 represents the weight of the second convolutional layer; Based on the above feature extraction steps, the feature extraction process of the multi-scale convolutional neural network module is summarized as follows:

10. The method for predicting the remaining useful life of a rolling bearing facing multi-stage degradation under cross-operating conditions according to claim 9, characterized in that: The process of the related alignment-model transfer domain adaptation method in step S2.3 includes: Through correlation alignment, the input data is passed through two branches for feature extraction, and then the image features and text features are mapped into a common subspace; in the common subspace, correlation alignment uses an adaptive layer to learn the conversion relationship between two different modalities; Define the source domain training sample as Label L s =y i ,i∈{1,2,...,L}; the unlabeled target domain data is Where d represents the number of output neurons in the network; the number of samples in the source domain and the target domain are n s and n t ; and represents the j-th dimension feature of the i-th sample in the source domain and the target domain respectively, C s and C t denote the feature covariance matrices of the source domain and the target domain respectively; the loss of the related alignment is defined as the covariance distance between the source domain and the target domain features, which is expressed by formula (21): in Represents the F2 norm, and the covariance calculation formula between the source domain and the target domain is obtained according to the covariance distance: After obtaining the covariance between the source and target domains, the chain rule is used to calculate the gradient relative to the input features: The final correlation alignment loss function minimizes the covariance matrix between the target domain and the source domain, thereby narrowing the data distribution difference between the two domains; the correlation alignment loss function is defined by formula (24): Where t represents the number of related alignment loss layers in the deep network, and λ represents the weight for weighing adaptation and classification accuracy on the source domain; In the feature extraction module, the input comes from the data Then the input features are mapped to the subspace Where Z s =φ1(D s ); at the same time, the target domain data Input feature extraction module and map to the same subspace Where Z t =φ1(D t ); After feature extraction, the source domain labeled data Z s Input the regression prediction module, map it to the subspace where the remaining useful life results are located, and output the prediction results in By minimizing Z s and Z t Update the overall training parameters of the source domain model based on the correlation alignment loss and the average loss error between the source domain predicted remaining useful life and the true remaining useful life; The loss function L of the multi-stage remaining useful life prediction module is calculated by the mean absolute error pred : In the formula, n represents the number of samples, y i represents the true remaining useful life of the i-th sample, represents the predicted remaining useful life of the i-th sample; Therefore, the final loss function L final Defined by formula (26): L final =L pred +L coral , (26)

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