Residual life prediction method based on domain invariance and consistent ordinal number representation learning

Abstract

A residual life prediction method based on domain invariance and consistent ordinal number representation learning comprises the following steps: firstly, calculating an MMD distance between different bearing domains in a training set, and minimizing the distance through gradient descent to enable feature distribution of each source domain to be globally aligned, extracting domain-invariant and ordered degeneration features, introducing a Gaussian mixture prior coding-decoding structure, and finally obtaining a residual life prediction result; the features are mapped to a potential space constrained by Gaussian mixture distribution through a variational auto-encoder, and the inadaptability of traditional single Gaussian distribution to nonlinear degradation is reduced; and in combination with MMD alignment, ordinal number loss and local consistency constraints, the dominant situation of a single constraint is avoided, the feature space has domain invariance and continuous orderliness at the same time through joint optimization, and the generalization ability of the model in a cross-domain task is remarkably improved.

Description

Technical Field

[0001] This invention belongs to the field of equipment testing and relates to a method for predicting remaining lifetime based on domain invariance and consistent ordinal representation learning. Background Technology

[0002] Predictive Health Management (PHM) technology assesses reliability and predicts Remaining Useful Life (RUL) by monitoring performance degradation throughout the entire lifecycle of equipment in real time, thereby optimizing equipment maintenance strategies and reducing operational risks. Current RUL prediction methods can be categorized into three types: those based on empirical knowledge, physical models, and data-driven approaches. Among these, deep learning (DL) technology has attracted significant attention in bearing RUL prediction due to its powerful nonlinear fitting capabilities. However, traditional DL methods require that the distributions of training and test data be identical or similar. However, due to differences in service environments, operating conditions (OCs), and failure modes, the degradation processes of different bearings are significantly inconsistent. To address this issue, transfer learning (TL) technology has been introduced to estimate RUL, which can be further subdivided into domain adaptation (DA) and domain generalization (DG) methods. While DA and DG techniques are widely used, directly aligning feature distributions can lead to overfitting and lacks constraints on the orderliness of the feature space. Summary of the Invention

[0003] To overcome the shortcomings of existing technologies, this invention provides a Remaining Life (RUL) prediction method based on domain invariance and consistent ordinal representation learning. A domain generalization framework is proposed, which alleviates overfitting optimization problems by aligning cross-domain feature distributions and introducing Gaussian mixture priors through an encoder-decoder structure. Specifically, maximum mean difference (MMD) is used to align multiple source domain feature distributions, cross-domain MMD losses are calculated for different domain feature distributions, and Gaussian mixture distributions are used as the prior constraint latent space. This relaxes rigid constraints while preserving key discriminative information, avoiding overfitting. Simultaneously, to address the problem of modeling degradation time-series relationships, a triplet loss function is constructed to enhance the similarity of features in adjacent degradation states. Local consistency loss constraints are also introduced to ensure the stability of short-term degradation trends and the invariance of local structures, thereby strengthening the ordered representation of the global degradation path. Furthermore, a joint loss function is constructed to integrate domain invariance constraints, distribution prior constraints, ordered feature constraints, and local consistency constraints into a unified framework to improve the generalization ability of RUL prediction.

[0004] The technical solution adopted by this invention to solve its technical problem is:

[0005] A remaining lifetime prediction method based on field invariance and consistent ordinal representation learning, the method comprising the following steps:

[0006] Step 1: Data Preprocessing: Vibration data was acquired using two accelerometers to collect horizontal vibration signals. Each sample group had a time-domain vibration signal length of 2560 points (sampling frequency 25.6 kHz, duration 0.1 seconds). All vibration signal samples were Z-score normalized. μ represents the characteristic mean, and σ represents the standard deviation, eliminating sensor dimensional differences and enhancing data distribution consistency; the RUL labels for the entire bearing life cycle are linearly normalized, mapping the actual RUL values ​​to... The input is a standardized vibration signal. Temporal features are extracted using a stacked residual convolutional module, outputting a 256-dimensional feature vector. A distributed encoder-decoder uses a three-layer fully connected network to compress the features into the latent space. The decoder reconstructs the original features using a symmetric structure, with the latent space constrained by a Gaussian mixture distribution. The regressor consists of three fully connected layers, each followed by a ReLU activation function. The final output is a normalized RUL prediction value.

[0007] Step 2: Domain-Invariant Feature Learning: Different bearings exhibit inconsistent feature distributions due to variations in operating conditions, materials, or degradation paths. Traditional models experience a sharp performance drop in the test domain. Domain-invariant features can be addressed by minimizing the MMD distance between domains. MMD measures the difference between two distributions in the regeneration kernel Hilbert space (RKHS). The difference in feature distributions across different domains in the training set is calculated using the multi-kernel maximum mean difference, serving as a regularization term for model optimization. This ensures that the degradation feature distributions of different bearings are globally consistent and forced to align to a unified feature space. Simple feature alignment can lead to overfitting in a given domain, resulting in poor generalization performance in unknown domains. Prior distribution constraints are imposed on the domain-invariant representations within the feature space to alleviate overfitting. A variational distribution encoder-decoder structure is designed to map the aligned high-dimensional features to a latent space constrained by the prior distribution. The latent space distribution is constrained to a multimodal Gaussian mixture model (GMM), with each Gaussian component corresponding to a different degradation stage. The variational posterior distribution q is optimized using KL divergence. φ Matching (h|z) with the prior distribution p(h).

[0008] Step 3: Continuous Ordered Feature Learning: Most TL methods focus on classification problems where labels are discrete, while RUL prediction is used as a regression problem where labels are continuous. Therefore, ordered regression is introduced to capture the sequential relationship in the feature space. Triple sample inputs are constructed, and the loss function forces the feature distance and label difference to maintain an ordinal relationship. This constraint ensures the continuity and orderliness of the degraded feature space. A small Gaussian noise perturbation is added to the features to constrain the similarity of the prediction results before and after the perturbation. This loss strengthens the short-term stability of the degradation process, avoids feature abrupt changes caused by high-frequency noise, and thus maintains the mapping consistency within the local neighborhood. In RUL prediction, the label space used to run to the failed bearing is continuous and uniformly distributed. When training the network, it is challenging to guarantee the degradation continuity of the feature space using only label constraints. This is because there is high-frequency noise interference inside the vibration signal, making it impossible to form a continuous feature representation related to the label. Moreover, the bearing degradation process represented by the data is not absolutely monotonic.

[0009] Step 4: Construct the joint loss function: This is done by using the weight coefficients λ corresponding to each loss term, namely MMD loss, variational inference loss, ordinal loss, and local consistency loss. m , λ p , λ o , λ l By balancing the various optimization objectives, this joint optimization ensures that the feature space possesses both domain invariance and continuous order, significantly improving the model's generalization ability in cross-domain tasks. The joint loss function formula is as follows:

[0010]

[0011] in This represents the mean squared error (MSE) loss between the true ground value and the estimated RUL. Indicates MMD loss. Let represent variational inference loss, and represent ordinal loss. This represents the loss of local consistency.

[0012] Furthermore, in step 2, the domain-invariant feature learning process is as follows:

[0013] 2.1) The difference in feature distribution across different domains in the training set is calculated using the multi-core maximum mean difference, and used as a regularization term for model optimization. This ensures that the degradation feature distributions of different bearings are globally consistent and are forced to align to a unified feature space. The formula is defined as follows:

[0014]

[0015] Where S m ,S nLet M and N represent the samples drawn from two domains, respectively, and let N be the number of samples in each domain. The nonlinear function φ() can map the original input to a vector in RKHS. The kernel function is represented by the radial basis function (RBF) kernel. The MMD regularization term for multiple source DG tasks is represented as:

[0016]

[0017] K represents the number of source domains;

[0018] 2.2) After aligning the feature distributions, an encoder-decoder framework is used to map the features into a latent space constrained by the prior distribution. This framework includes a distributed encoder E. φ (·) and distributed decoder D θ (·), where φ and θ represent network parameters from two modules. A Gaussian mixture distribution is chosen as the prior, which theoretically can fit any distribution, thus learning complex degenerate representations. For ease of distinction, the low-dimensional embedding in the latent space is denoted by h, and a variational posterior q(z|x) is constructed to approximate the true posterior p(z|x). p(h) is the Gaussian mixture prior, represented as:

[0019]

[0020] Where π i Let μ represent the coefficients of the i-th Gaussian component, where the sum of all coefficients equals 1. Let I denote the identity matrix. i , Let represent the mean and variance of the i-th Gaussian component, respectively;

[0021] Variational posterior q φ (h|z) is represented as:

[0022]

[0023] Where ξ i Let μ represent the weight of the i-th Gaussian component, where μ is the mean and variance of the i-th Gaussian component. i , express;

[0024] 2.3) The optimization objective is equivalent to maximizing the lower bound of evidence (ELBO), written as:

[0025]

[0026] Where p(h) is the Gaussian mixture prior, D KL (||) indicates KL divergence calculation:

[0027]

[0028] 2.4) In actual training, the theoretical formula ELBO cannot be directly manipulated. Therefore, it is negativeized and decomposed into specific calculation terms. The prior constraints of the encoding / decoding framework on the features are as follows:

[0029]

[0030] Where z represents the reconstructed feature, Replacing the log probability of reconstruction with L2 loss directly achieves the same effect, but is easier to calculate. The latter term, KL divergence, directly minimizes the difference between the encoder output distribution and the prior (Gaussian mixture).

[0031] Furthermore, in step 3, the continuous ordered feature learning process is as follows:

[0032] 3.1) Construct feature pairs with similar labels and negative samples with dissimilar labels, ensuring that the feature distance is consistent with the label difference. For samples with corresponding labels (y... u ,y v ,y w The triple input (x) u ,x v ,x w The extracted features can be derived from (z) u ,z u ,z w ) indicates; when |y u -y v |less than|y u -y w The time series number constraint aims to maintain the following relationship:

[0033] d(z u ,z v ) <d(z u ,z w );

[0034] Where d() is used to calculate the Euclidean distance between two vectors;

[0035] 3.2) For the set of triples Ω={u,v,w||y u -y v |<|y u -y w |}

[0036] Ordinal constraints are expressed as:

[0037]

[0038] Where δ represents the margin to avoid the impact of fluctuations;

[0039] 3.3) Degradation processes typically exhibit local consistency, meaning the degradation state will remain unchanged or change only slightly in a short period. Imposing local consistency constraints on features and adding noise perturbations to them ensures similar prediction results for neighboring samples, enhancing the local stability of the degradation process. This is achieved by projecting the neighborhood of sample features onto points surrounding the corresponding labeled neighborhood and adding a small perturbation to the features to realize local consistency loss.

[0040]

[0041] Here, ε follows a standard Gaussian distribution, γ noise refers to the coefficient that controls the magnitude of the disturbance, and R() represents the regression network. By combining ordinal constraints and loss terms of local consistency, it can learn locally stable and ordered manifold characteristics.

[0042] The technical concept of this invention is as follows: First, the MMD distance between different bearing domains in the training set is calculated, and this distance is minimized through gradient descent, so that the feature distributions of each source domain are globally aligned. Domain-invariant and ordered degenerate features are extracted, and a Gaussian mixture prior encoder-decoder structure is introduced. The features are mapped to a latent space constrained by a Gaussian mixture distribution through a variational autoencoder, which alleviates the inadequacy of the traditional single Gaussian distribution to nonlinear degradation. By combining MMD alignment, ordinal loss, and local consistency constraints, a single constraint is avoided from dominating. Joint optimization makes the feature space simultaneously possess domain invariance and continuous order, which significantly improves the model's generalization ability in cross-domain tasks.

[0043] The beneficial effects of this invention are as follows:

[0044] (1) By using the maximum mean difference (MMD) to align the feature distributions of different domains, the problem of bearing degradation data distribution shift under different working conditions is solved, the model's adaptability to data from different sources is enhanced, and the generalization performance can be improved without training on target domain data.

[0045] (2) Using Gaussian mixture distribution (GMM) as a prior, the multi-stage characteristics of bearing degradation are captured, the KL divergence between the encoding distribution and the prior distribution is minimized, and the feature reconstruction error is constrained. The latent space of the feature distribution is constrained by the encoding-decoding structure to prevent the model from overfitting in the training domain.

[0046] (3) Based on triplet loss, anchor samples, positive samples (with similar labels), and negative samples (with distant labels) are selected, and the feature distances are constrained to satisfy the ordinal relationship. The ordinal relationship is forced to be maintained in the feature space. After adding a small perturbation to the features, the corresponding label prediction is required to remain unchanged. This ensures that the degradation state at nearby time points in the feature space is mapped to the nearby labels, capturing the local stability of degradation.

[0047] (4)λ m , λ p , λo , λ l To balance the various loss terms and avoid a single constraint from dominating, joint optimization enables the feature space to simultaneously possess domain invariance and continuous order, significantly improving the model's generalization ability in cross-domain tasks. Attached Figure Description

[0048] Figure 1 This is a framework diagram of the online prediction principle of the domain-invariant and consistent ordinal representation learning method.

[0049] Figure 2 This is a diagram of the network structure for learning domain-invariant and ordered features. Detailed Implementation

[0050] The present invention will be further described below.

[0051] Reference Figure 1 and Figure 2 A remaining lifetime prediction method based on field invariance and consistent ordinal representation learning includes the following steps:

[0052] Step 1: Data Preprocessing: Vibration data was acquired using two accelerometers to collect horizontal vibration signals. Each sample group had a time-domain vibration signal length of 2560 points (sampling frequency 25.6 kHz, duration 0.1 seconds). All vibration signal samples were Z-score normalized. μ represents the characteristic mean, and σ represents the standard deviation, eliminating sensor dimensional differences and enhancing data distribution consistency; the RUL labels for the entire bearing life cycle are linearly normalized, mapping the actual RUL values ​​to... The input is a standardized vibration signal. Temporal features are extracted through a stacked residual convolution module, and a 256-dimensional feature vector is output. The distributed encoder-decoder compresses the features into the latent space through a 3-layer fully connected network. The decoder reconstructs the original features through a symmetrical structure, and the latent space is constrained by a Gaussian mixture distribution. The regressor contains 3 fully connected layers, each followed by a ReLU activation function, and the final output is a normalized RUL prediction value.

[0053] Step 2, Domain Invariant Feature Learning: Due to differences in operating conditions, materials, or degradation paths, the feature distributions of bearings can be inconsistent, causing traditional models to experience a sharp performance drop in the test domain. Domain invariant features can be addressed by minimizing the MMD distance between domains. MMD measures the difference between two distributions in the regeneration kernel Hilbert space (RKHS). The difference in feature distributions across different domains in the training set is calculated using the multi-kernel maximum mean difference, serving as a regularization term for model optimization. This ensures that the degradation feature distributions of different bearings are globally consistent and are forced to align to a unified feature space.

[0054] The formula is defined as:

[0055]

[0056] Where S m ,S n Let M and N represent the samples drawn from two domains, respectively, and let N be the number of samples in each domain. The nonlinear function φ() can map the original input to a vector in RKHS. The kernel function is represented by the radial basis function (RBF) kernel. The MMD regularization term for multiple source DG tasks is represented as:

[0057]

[0058] K represents the number of source domains;

[0059] After aligning the feature distributions, an encoder-decoder framework is used to map the features into a latent space constrained by the prior distribution. This framework includes a distributed encoder E. φ (·) and distributed decoder D θ (·). The symbols φ and θ represent the network parameters from the two modules. A Gaussian mixture distribution is chosen as the prior, which can theoretically fit any distribution, thus learning complex degenerate representations. For ease of distinction, the low-dimensional embedding in the latent space is denoted by h, and a variational posterior q(z|x) is constructed to approximate the true posterior p(z|x). p(h) is the Gaussian mixture prior represented as:

[0060]

[0061] Where π i Let I represent the coefficients of the i-th Gaussian component, with the sum of all coefficients equal to 1. Let I represent the identity matrix, and μ represent the coefficients of the i-th Gaussian component. i , Let represent the mean and variance of the i-th Gaussian component, respectively;

[0062] Variational posterior q φ (h|z) is represented as:

[0063]

[0064] Where ξ i Let μ represent the weight of the i-th Gaussian component, where μ is the mean and variance of the i-th Gaussian component. i , express;

[0065] The optimization objective is equivalent to maximizing the lower bound of evidence (ELBO), written as:

[0066]

[0067] Where p(h) is the Gaussian mixture prior, D KL (||) indicates KL divergence calculation:

[0068]

[0069] In actual training, the theoretical ELBO formula cannot be directly manipulated. Therefore, it is negativeized and decomposed into specific calculation terms. The encoding and decoding framework imposes prior constraints on the features:

[0070]

[0071] Where z represents the reconstructed feature, Replacing the log probability of reconstruction with L2 loss directly achieves the same effect but is easier to compute; the latter term, KL divergence, directly minimizes the difference between the encoder output distribution and the prior (Gaussian mixture).

[0072] Step 3: Learning Continuous and Ordered Features:

[0073] Most time-trading (TL) methods focus on classification problems where labels are discrete, while RUL prediction is used as a regression problem where labels are continuous. Therefore, we introduce ordered regression to capture the ordinal relationships in the feature space. We construct triplet input samples and force an ordinal relationship between feature distance and label difference through a loss function.

[0074] Construct feature pairs with similar labels and negative samples with dissimilar labels, ensuring that the feature distance matches the label difference. For samples with corresponding labels (y... u ,y v ,y w The triple input (x) u ,x v ,x w The extracted features can be derived from (z) u ,z v ,z w ) indicates. When |y u -y v |less than|y u -y w The time series number constraint aims to maintain the following relationship:

[0075] d(z u ,z v ) <d(z u ,z w );

[0076] Where d() is used to calculate the Euclidean distance between two vectors.

[0077] For the set of triples Ω = {u, v, w || y} u -y v |<|y u -y w |}

[0078] Ordinal constraints are expressed as:

[0079]

[0080] Where δ represents the margin to avoid the impact of fluctuations;

[0081] This constraint ensures the continuity and order of the degenerate feature space. A small Gaussian noise perturbation is added to the features, constraining the similarity of the prediction results before and after the perturbation.

[0082] Degradation processes typically exhibit local consistency, meaning the degradation state remains unchanged or changes only slightly over short periods. Therefore, we impose local consistency constraints on features by adding noise perturbations to ensure similar predictions from neighboring samples, thus enhancing the local stability of the degradation process. This is achieved by projecting the neighborhood of sample features onto points surrounding the corresponding labeled neighborhoods and adding a small perturbation to the features to implement local consistency loss.

[0083]

[0084] Here, ε follows a standard Gaussian distribution, and γ (noise) refers to the coefficient controlling the magnitude of the perturbation. R() represents the regression network. By combining ordinal constraints and loss terms for local consistency, it is possible to learn locally stable and ordered manifold features. This loss enhances the short-term stability of the degradation process, avoids feature abrupt changes caused by high-frequency noise, and thus maintains the mapping consistency within the local neighborhood.

[0085] Step 4: Construct the joint loss function: This involves using the weight coefficients λ corresponding to the MMD loss, variational inference loss, ordinal loss, and local consistency loss term. m , λ p , λ o , λ l By balancing the various optimization objectives, this joint optimization ensures that the feature space possesses both domain invariance and continuous order, significantly improving the model's generalization ability in cross-domain tasks. The joint loss function formula is as follows:

[0086]

[0087] in This represents the mean squared error (MSE) loss between the true ground value and the estimated RUL. Indicates MMD loss. Let represent variational inference loss, and represent ordinal loss. This represents the loss of local consistency.

[0088] The embodiments described in this specification are merely examples of implementations of the inventive concept and are for illustrative purposes only. The scope of protection of this invention should not be considered limited to the specific forms described in these embodiments; rather, it extends to equivalent technical means conceived by those skilled in the art based on the inventive concept.

Claims

1. A method for predicting remaining lifetime based on field invariance and consistent ordinal representation learning, characterized in that, The method includes the following steps: Step 1: Data Preprocessing: Vibration data is collected using two accelerometers to acquire horizontal vibration signals. The temporal vibration signal length for each sample group is set, and all vibration signal samples are Z-score standardized. The RUL labels for the entire bearing life cycle are linearly normalized. The standardized vibration signals are input, and temporal features are extracted using a stacked residual convolution module. The distributed encoding-decoding uses a 3-layer fully connected network to compress the features into the latent space. The decoder reconstructs the original features using a symmetrical structure. The latent space is constrained by a Gaussian mixture distribution. The regressor contains 3 fully connected layers, each followed by a ReLU activation function. The final output is the normalized RUL prediction value. Step 2, Domain Invariant Feature Learning: Domain invariant features are learned by minimizing the MMD distance between domains. MMD measures the difference between two distributions in the Hilbert space of the regeneration kernel. The difference in feature distributions of different domains in the training set is calculated by multi-kernel maximum mean difference and used as a regularization term for model optimization, making the degradation feature distributions of different bearings globally consistent and forcing alignment to a unified feature space. Prior distribution constraints are applied to the domain invariant representations in the feature space. A variational distribution encoder-decoder structure is designed to map the aligned high-dimensional features to the latent space constrained by the prior distribution. The distribution of the latent space is constrained to a multi-peak Gaussian mixture model, with each Gaussian component corresponding to a different degradation stage. The variational posterior distribution q is optimized by KL divergence. φ The matching of (h|z) with the prior distribution p(h); Step 3, Continuous Ordered Feature Learning: Introducing ordered regression to capture the sequential relationships in the feature space, constructing triplet sample inputs, and forcing the feature distance and label difference to maintain a consistent ordinal relationship through the loss function; adding small Gaussian noise perturbations to the features to constrain the similarity of the prediction results of the features before and after the perturbation; in RUL prediction, the label space used to run to the failed bearing is continuous and uniformly distributed. It is challenging to ensure the degenerate continuity of the feature space using only label constraints when training the network. Step 4: Construct the joint loss function: This is done by using the weight coefficients λ corresponding to each loss term, namely MMD loss, variational inference loss, ordinal loss, and local consistency loss. m , λ p , λ o , λ l By balancing the various optimization objectives, joint optimization ensures that the feature space simultaneously possesses domain invariance and continuous order. The joint loss function formula is as follows: in The mean squared error (MSE) loss represents the difference between the true ground value and the estimated RUL. Indicates MMD loss. Let represent variational inference loss, and represent ordinal loss. This represents the loss of local consistency.

2. The remaining lifetime prediction method based on field invariance and consistent ordinal representation learning as described in claim 1, characterized in that, In step 2, the domain-invariant feature learning process is as follows: 2.1) The difference in feature distribution across different domains in the training set is calculated using the multi-core maximum mean difference, and used as a regularization term for model optimization. This ensures that the degradation feature distributions of different bearings are globally consistent and are forced to align to a unified feature space. The formula is defined as follows: Where S m ,S n Let M and N represent the samples drawn from two domains, respectively, and let N be the number of samples in each domain. The nonlinear function φ() can map the original input to a vector in RKHS. The kernel function, selected as the radial basis function kernel, is used for the MMD regularization term in multiple source DG tasks, and is represented as follows: K represents the number of source domains; 2.2) After aligning the feature distributions, an encoder-decoder framework is used to map the features into a latent space constrained by the prior distribution. This framework includes a distributed encoder E. φ (·) and distributed decoder D θ (·), where φ and θ represent network parameters from the two modules. A Gaussian mixture distribution is chosen as the prior, and the low-dimensional embedding in the latent space is denoted by h. A variational posterior q(z|x) is constructed to approximate the true posterior p(z|x), and p(h) is the Gaussian mixture prior expressed as: Where π i Let I represent the coefficients of the i-th Gaussian component, with the sum of all coefficients equal to 1. Let I represent the identity matrix, and μ represent the coefficients of the i-th Gaussian component. i , Let represent the mean and variance of the i-th Gaussian component, respectively; Variational posterior q φ (h|z) is represented as: Where ξ i Let μ represent the weight of the i-th Gaussian component, where μ is the mean and variance of the i-th Gaussian component. i , express.

3. The remaining lifetime prediction method based on field invariance and consistent ordinal representation learning as described in claim 2, characterized in that, In step 2, the domain-invariant feature learning process further includes: 2.3) The optimization objective is equivalent to maximizing the lower bound of evidence, written as: Where p(h) is the Gaussian mixture prior, D KL (||) indicates KL divergence calculation: 2.4) Prior constraints on features by the encoding / decoding framework: Where z represents the reconstructed feature, The log probability of reconstruction is directly replaced by L2 loss, and the latter term KL divergence directly minimizes the difference between the encoder output distribution and the prior.

4. The remaining lifetime prediction method based on field invariance and consistent ordinal representation learning as described in claim 3, characterized in that, In step 3, the continuous ordered feature learning process is as follows: 3.1) Construct feature pairs with similar labels and negative samples with dissimilar labels, ensuring that the feature distance is consistent with the label difference. For samples with corresponding labels (y... u ,y v ,y w The triple input (x) u ,x v ,x w The extracted features can be derived from (z) u ,z u ,z w ) indicates; when |y u -y v |less than|y u -y w The time series number constraint aims to maintain the following relationship: d(z u ,z v )<d(z u ,z w ); Where d() is used to calculate the Euclidean distance between two vectors; 3.2) For the set of triples Ω={u,v,w||y u -y v |<|y u -y w |} Ordinal constraints are expressed as: Where δ represents the margin to avoid the impact of fluctuations; 3.3) Apply local consistency constraints to features and add noise perturbations to the features to ensure that the prediction results of neighboring samples are similar, thereby enhancing the local stability of the degradation process. Project the neighborhood of the sample feature onto the points around the corresponding labeled neighborhood, and add a small perturbation to the feature to achieve local consistency loss. Here, ε follows a standard Gaussian distribution, γ noise refers to the coefficient that controls the magnitude of the disturbance, and R() represents the regression network. By combining ordinal constraints and loss terms of local consistency, it can learn locally stable and ordered manifold characteristics.

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