Engine residual life prediction method based on deep survival analysis

Abstract

The application discloses an engine residual life prediction method based on deep survival analysis and belongs to the technical field of industrial equipment monitoring and equipment health prediction management. By constructing an engine residual life prediction model based on deep survival analysis, the nonlinear relationship between multi-source sensor signals and the residual service life distribution of the engine is modeled, and in view of the complex situation that part of the signals are censored or lack of residual service life (RUL) labels, the survival analysis method is used to realize real-time online prediction of the engine RUL. The application can effectively process censored data, fully excavate potential information of the data by dynamically optimizing the prediction process, maximize the data utilization rate, and realize accurate RUL prediction and quantification of the uncertainty based on Weibull distribution modeling, so that the life distribution of the engine can be accurately predicted, and the prediction precision of the residual life of the engine is improved.

Description

Technical Field

[0001] This invention belongs to the field of industrial equipment monitoring and equipment health prediction management technology, and relates to online prediction technology for the remaining service life of engine equipment. Specifically, it relates to an online prediction method for the remaining service life of engines based on a Bayesian deep survival analysis network. By constructing a Bayesian deep survival analysis network model, signal fusion is performed on multi-source sensor degradation signals with right censoring, and the remaining service life of the engine is predicted online. Background Technology

[0002] In engineering applications, equipment operation status monitoring systems can monitor operational signals closely related to equipment health in real time. By establishing degradation models, they can accurately predict remaining lifespan to determine optimal maintenance times, demonstrating significant potential in preventing equipment failures and reducing maintenance costs. With the rapid development of sensor and information technologies, degradation signals collected during equipment operation can be transmitted in real time to cloud servers via wireless communication networks for in-depth analysis. This ease of data acquisition makes data-driven methods easier to implement and provides a data foundation for real-time predictive maintenance of equipment. Predicting the remaining useful life (RUL) of equipment based on its current health status specifically refers to predicting the time when a system or component will fail or cease to perform its intended function. Existing RUL prediction methods typically rely on degradation signals of the equipment and perform regression analysis on multi-source time series data. However, in practical applications, complete degradation signal data is often lacking during equipment operation. Many devices are still in service, and their exact failure time is unknown. We can consider this signal as right-censored data. For such devices, only partial degradation signals are available, and there is no remaining useful life label, making accurate prediction of remaining useful life complex. We call this type of data "time-to-event" (TTE) data, used to describe the duration from the starting point to the occurrence of a certain event (such as equipment failure or censoring). This type of data is widely present in many engineering and manufacturing scenarios, such as medical, aviation, and railway, and statistically, censored data often exceeds 95%. While the widespread use of sensor devices has improved data availability and facilitated the fusion of TTE data and high-dimensional degradation signals, existing RUL prediction methods rely solely on complete data for modeling, often neglecting the important information contained in censored signals. They also struggle to effectively handle censored data lacking regression labels, leading to biased results. This poses a significant challenge to achieving accurate prediction of remaining lifetime. Therefore, there is an urgent need for more effective methods to fully utilize censored data to improve prediction accuracy. Summary of the Invention

[0003] To overcome the shortcomings of the existing technologies, this invention proposes an online prediction method for engine remaining service life (RUL) based on a Bayesian deep survival analysis network. This invention constructs an engine RUL prediction model based on deep survival analysis (Bayesian LSTM-SURV), aiming to establish a nonlinear relationship between multi-source sensor signals with right censoring and the engine's RUL. For complex situations where some signals are censored or lack RUL labels, the invention achieves real-time online prediction of the engine's RUL through survival analysis.

[0004] Specifically, the online prediction model for engine remaining service life (RUL) proposed in this invention transfers information from training data to test data using a Bayesian method, fully utilizing the degradation information of censored and complete data. Simultaneously, considering the uncertainty of service life, it uses a Weibull distribution to model the service life distribution and innovatively designs a loss function based on survival analysis theory to capture the degradation information contained in the time series termination (TTE) data, thereby establishing a nonlinear relationship between the multidimensional time series degradation signal and RUL. This model overcomes the RUL prediction modeling problem caused by signal censoring and missing labels, fully mining the degradation information contained in censored data and significantly improving prediction accuracy. Furthermore, it enhances the ability to quantify uncertainty, providing a more robust solution for real-time RUL prediction.

[0005] The method of this invention comprises two stages: (a) an offline stage, in which multi-sensor degradation signals are modeled based on a Bayesian mixture effect model and a Bayesian LSTM-SURV model is trained to construct a nonlinear relationship between degradation signals and remaining lifetime; and (b) an online stage, in which dynamic Bayesian updates are performed on the degradation signals, and the engine's RUL is predicted in real time based on the trained Bayesian LSTM-SURV model. Specifically, in the offline stage, firstly, the degradation signals of the engine are fitted using a Bayesian mixture effect model to obtain model parameters to characterize the degradation trend of the signals; secondly, a Bayesian LSTM-SURV model is established, with the input being a labeled complete signal and an unlabeled censored signal. The output of the model does not focus on the specific value of RUL, but rather considers that the engine's lifetime follows a Weibull distribution, thereby capturing the probabilistic characteristics of the unit degradation process over time, such as the survival probability. Furthermore, this work introduces survival analysis to effectively process TTE data, which includes complete degradation data of faulty equipment and censored data of in-service equipment. Traditional methods often fail to fully utilize the potential information contained in censored data, resulting in limited model prediction performance. Therefore, this invention innovatively proposes a loss function for the Bayesian LSTM-SURV model, combining survival analysis theory. This allows the model to utilize not only complete data but also fully mine the degradation information contained in censored data, thereby improving the model's prediction accuracy. In the online phase, this invention proposes a signal update framework based on Bayesian methods to characterize degradation signals, remove noise, and transfer features from training data to test data to achieve signal updates for in-service engines. Subsequently, the Bayesian LSTM-SURV model uses the updated degradation signals to perform real-time RUL prediction. In summary, the signal update framework based on Bayesian methods proposed in this invention not only dynamically updates the signal but also quantifies the uncertainty in the degradation process, thereby significantly improving the accuracy of remaining service life prediction, reducing prediction bias, and enhancing the model's reliability and generalization ability in practical applications.

[0006] This invention, based on Bayesian methods and combining survival analysis theory with deep neural networks, proposes a Bayesian LSTM-SURV model. Assuming that engine lifespan follows a Weibull distribution, it constructs a nonlinear relationship between multi-sensor degradation signals and engine range-limited lifetime (RUL). In the offline phase, the degradation signals are first fitted using a Bayesian mixture-effects model to estimate the prior distribution of the parameters. Then, a Bayesian LSTM-SURV model is designed using survival analysis, enabling it to predict the lifespan distribution using both complete and censored data. In the online phase, the degradation signals are modeled and denoised using Bayesian methods, and features from the training data are transferred to the test data. By updating the posterior distribution of the signals in real time, accurate prediction of RUL is achieved.

[0007] The technical solution provided by this invention is as follows:

[0008] The engine remaining life prediction method based on deep survival analysis includes the following steps:

[0009] 1) Aircraft engine service signal collection and data representation: Collecting J-dimensional degradation signal data Y from I engines. i For each engine i, record the time T when the event occurs. i and indicator variable d i T i d represents the time of engine failure or deletion. i If the engine is faulty (1) or censored (0), then the complete observation data O for the i-th engine is... i ={Y i ,T i ,d i}

[0010] 2) Construct a Bayesian mixture effect model to model multivariate degradation signals:

[0011] 21) The degradation process of the engine is characterized by a Bayesian mixture effect model, and the training data is denoised.

[0012] 22) The two-stage method is used to estimate the model parameters.

[0013] 3) Constructing and training an engine remaining life prediction model based on deep survival analysis:

[0014] The engine remaining life prediction model based on deep survival analysis includes an input layer, a deep survival analysis (LSTM-SURV) network layer, and an output layer.

[0015] 31) Model input: Multi-sensor degradation signals are used as the input of the model, and sliding window technology is used to process the multi-sensor degradation signals (time series). In this process, the time series batches are normalized to a preset fixed length L to ensure that the multi-sensor degradation signal data maintain the same length L.

[0016] 32) Deep Survival Analysis Network Layer: This layer combines a Long Short-Term Memory (LSTM) network layer, a random deactivation layer, and two fully connected layers to learn the degradation information represented by multi-sensor degradation signals. The LTM network layer can process time-series data, extracting effective features from multi-sensor signals and capturing the time dependencies in the degradation process. This invention also incorporates a novel loss function designed based on survival analysis theory, enabling the LTM network layer to fully learn the degradation information contained in both complete and censored data. The random deactivation layer prevents overfitting by randomly discarding neurons, while the fully connected layers are used to further fuse and process the extracted features.

[0017] 33) Model output: Assuming that the lifetime of all units follows a Weibull distribution, the output of the model is set to the Weibull distribution parameters λ and k that the engine lifetime follows.

[0018] 34) Offline model training:

[0019] In the offline phase, the constructed Bayesian LSTM-SURV model is trained using the denoised training set data from step 2) to fully learn the degradation information contained in the complete and censored data, achieve accurate prediction of RUL, and update the model parameters of the neural network.

[0020] 4) Online parameter updates and remaining service life prediction:

[0021] After obtaining model training parameters using a grid search method in the offline phase, the remaining service life of in-service engine i can be predicted. At prediction time point t... * By combining historical data with individual information—specifically, utilizing a Bayesian mixture-effects model learned from the training dataset, using the model's fitted parameters as prior information, and combining this with sensor degradation signals from in-service engines as observed information—the posterior distribution of the parameters is calculated using Bayes' theorem to update the Bayesian mixture-effects model parameters, thereby achieving information transfer from training data to test data. The update process, based on the Bayesian method, uses the Bayesian mixture-effects model learned during training as prior information, combines it with sensor observation data (likelihood function) from in-service engines, and updates the posterior distribution of the model parameters using the Bayesian theorem. Specifically, based on the Bayesian method, the posterior distribution of sensor j for engine i... Follow the mean The sum and variance are The degradation signal of in-service engine i is updated using the posterior distribution of the multivariate normal distribution. Then, it is input into the Bayesian LSTM-SURV model to obtain the estimated parameters of the Weibull distribution that governs the lifespan of in-service engine i. and Therefore, calculate

[0022] 5) Construct model evaluation metrics to evaluate the model:

[0023] To evaluate the performance of the proposed Bayesian LSTM-SURV model, this invention uses the root mean square error (RMSE) as the model evaluation metric. RMSE quantifies the deviation between predicted and actual values, effectively assessing the model's prediction accuracy. Furthermore, this invention compares the proposed model with existing Time-dependent Survival Neural Network (TSNN) and DeepSurv models, comparing the prediction performance of the three models on the same dataset.

[0024] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0025] With the development of sensor technology, monitoring systems based on equipment operating status can monitor operating signals closely related to equipment health status in real time and perform predictive maintenance. However, existing RUL prediction methods are mostly based on regression analysis of degradation signals. These methods rely solely on complete data for modeling, neglecting the important information contained in censored data, thus affecting the accuracy of prediction. Therefore, these methods are insufficient to handle censored data lacking regression labels. This invention provides an online prediction model method for engine remaining service life based on a Bayesian deep survival analysis network. This method models the nonlinear relationship between multi-sensor degradation signals and engine RUL, effectively improving the accuracy and reliability of prediction. In the offline stage, the degradation signal is fitted using a Bayesian mixture effect model to estimate the prior distribution of parameters. Combined with survival analysis, a Bayesian LSTM-SURV engine remaining service life online prediction model is constructed, enabling it to predict the life distribution using both complete and censored data. In the online stage, a Bayesian method is used to model and denoise the degradation signal, and the features of the training data are transferred to the test data to update the posterior distribution of the signal in real time, thereby achieving real-time RUL prediction. The technical advantages of this invention include:

[0026] (1) By combining survival analysis and neural networks, a novel loss function was designed to capture degradation information in complete and censored data, thereby modeling the nonlinear relationship between degradation signal and remaining lifetime, and fully considering the censored signal and RUL tag missing problem;

[0027] (2) The present invention adopts the Bayesian method, extracts information from the training data and transfers it to the prediction process of the test data, dynamically optimizes the prediction results, fully explores the potential information of the data, and thus makes the most of the existing data.

[0028] (3) Unlike traditional methods that directly predict RUL values, this invention assumes that engine life follows a Weibull distribution. By modeling the life distribution and survival function, it can not only calculate and predict RUL values, but also quantify the uncertainty of these prediction results. Attached Figure Description

[0029] Figure 1 This is a flowchart illustrating the model method provided by the present invention.

[0030] Figure 2 A framework diagram of the model method provided by this invention.

[0031] Figure 3 This is a schematic diagram of the Long Short-Term Memory (LSTM) network structure used in this invention.

[0032] Figure 4 This is a schematic diagram showing the comparison results of the root mean square error (RMSE) of the training data under different p1 (complete data ratio) and p2 (censoring time ratio) of Embodiment 1 of the present invention.

[0033] Figure 5 This is a schematic diagram of the Weibull distribution conditional probability density function predicted for 6 engines randomly selected from the test dataset in Embodiment 1 of the present invention.

[0034] Figure 6 This is a schematic diagram of the survival curves of engine c under actual and predicted conditions at times t0 = 40, 45, 50, and 55 in Embodiment 1 of the present invention.

[0035] Figure 7 This refers to Embodiment 1 of the present invention, under the conditions of p1 = 0.7 (complete data ratio) and p2 = 0.6 (censoring time ratio), P under different actual remaining useful lives levels. CI (Actual lifespan L) i A diagram showing the proportion of predictions falling within the 90% confidence interval.

[0036] Figure 8 This is a schematic diagram of partial signal Bayesian update of engine i in Embodiment 2 of the present invention.

[0037] Figure 9 This is a schematic diagram showing the RMSE comparison results of the training data under different p1 (complete data ratio) and p2 (censoring time ratio) in Embodiment 2 of the present invention.

[0038] Figure 10 This refers to Embodiment 2 of the present invention, under the conditions of p1 = 0.7 (complete data ratio) and p2 = 0.6 (censoring time ratio), P under different actual remaining useful lives levels. CI (Actual lifespan L)i A diagram showing the proportion of predictions falling within the 90% confidence interval. Detailed Implementation

[0039] The present invention will be further described below with reference to the accompanying drawings and embodiments, but the scope of the invention is not limited in any way.

[0040] This invention provides an online prediction model method for engine remaining service life (RUL) based on a Bayesian deep survival analysis network. It models the nonlinear relationship between multi-sensor degradation signals and engine RUL, effectively improving the accuracy and reliability of predictions. In the offline stage, a Bayesian mixture-effects model is used to fit the degradation signals, estimating the prior distribution of parameters. Combined with survival analysis, a Bayesian LSTM-SURV online prediction model for engine RUL is constructed, enabling it to predict the life distribution using both complete and censored data. In the online stage, Bayesian methods are used to model and denoise the degradation signals, and features from the training data are transferred to the test data. The posterior distribution of the signal is updated in real time, thereby achieving real-time RUL prediction. A model evaluation index is constructed for model evaluation. Figure 1 This is a flowchart illustrating the model method provided by the present invention.

[0041] The specific embodiments of the present invention are as follows:

[0042] 1) Signal collection and data representation during aircraft engine service:

[0043] This invention focuses on estimating the event occurrence time T for each engine i given a J-dimensional degradation signal. i The distribution of [the data]. Assume there are I engines, each engine i has J degradation signals, and each degradation signal has n [signals / values]. i The specific data for each observation is as follows:

[0044]

[0045] For ease of representation, this invention assumes that the J degradation signals for each engine are observed at the same time point. Therefore, Y consists of J-dimensional time series data, based on which the remaining service life is predicted. Given an engine i, the event occurrence time can be represented as:

[0046]

[0047] Among them, C R L represents the right-censored set. i C represents the time when the complete data failure occurred. i This indicates the endpoint of the censored data, and uses the indicator variable d. i To indicate whether the engine has been deleted, i.e.:

[0048]

[0049] Where, d i This indicates the survival state, i.e., if engine i is in L i If there is a fault, then d i The value is 1; if the engine is in C i If deleted, then d i The value is 0. Therefore, the complete observation data for the i-th engine can be represented as O. i ={Y i ,T i ,d i}

[0050] 2) Construct a Bayesian mixture effect model to model multivariate degradation signals:

[0051] 21) The Bayesian mixture effect model is used to characterize the engine degradation process, and the training data is denoised:

[0052] y i,j,t =X i,j,t β i,j +ε t (Equation 3) Among them, y i,j,t X represents the j-th signal value of the i-th engine at time t (j = 1, ..., J). i,j,t Yes (q) j +1)-dimensional polynomial basis function vector (qj represents the highest-order term of the signal-time function), denoted as β i,j Yes (q) j The regression parameter vector of dimension +1 follows a multivariate normal distribution, ε t This represents Gaussian noise, which follows a normal, independent, and identically distributed pattern. The order of the polynomial regression may differ for different sensors, and can be determined using quantitative model selection methods. This invention employs the Bayesian Information Criterion (BIC) to determine the optimal order of the polynomial form of the model.

[0053] For censored units, the limited information provided by the censoring signal is often insufficient to accurately predict lifetime. Therefore, this invention considers transferring features from the training data to the test data, which can be achieved through Bayesian methods. This method utilizes the similarity between units and group behavior in the data, effectively achieving information transfer and modeling the features of the data. Specifically, the training data provides prior information for the model parameters, which can be used to update the posterior distribution of the test data.

[0054] 22) The two-stage method is used to estimate the model parameters.

[0055] To perform Bayesian updates during the online phase, this invention designs a joint conjugate prior, namely the regression parameters. in and These are the mean vector and the covariance matrix, respectively, ε t Let Gaussian noise, following a normal independent and identically distributed pattern, be represented as: Its mean is 0 and its variance is Model parameters These parameters will be utilized through Bayesian updates during the online phase. The maximum likelihood estimation (MLE) method estimates ψ. (j) One natural way to achieve this is by maximizing the following marginal likelihood:

[0056]

[0057] in, Represents the signal vector of the engine. It is the parameter vector β i,j and variance The joint prior distribution is given. However, the aforementioned marginal likelihood is very complex and unsuitable for direct calculation, making parameter estimation via maximum likelihood estimation complicated. To improve parameter estimation, this invention employs a two-stage method to estimate the hyperparameter ψ. (j) This is a simpler and more efficient method. In this method, the first step is to use the MLE method to fit the degradation signal of each unit and estimate the parameters. The second step is to use... Estimating hyperparameter ψ (j) Although the two-phase method may introduce some bias, this bias is generally negligible in Bayesian online update processes. For parameters... Its maximum likelihood estimate is expressed as Equation 5:

[0058]

[0059] Where, n i X represents the number of signals collected by the i-th engine. i,j The design matrix of sensor j for engine i can be represented as follows:

[0060] Formula 6:

[0061]

[0062] Where, q j This represents the highest-order term of the signal-time function. For parameters... Its maximum likelihood estimate is expressed as Equation 7:

[0063]

[0064] Where I represents the number of engines. and These are estimates of the parameter vector and variance.

[0065] 3) Constructing and training an engine remaining life prediction model based on deep survival analysis:

[0066] The engine remaining life prediction model based on deep survival analysis includes an input layer, a deep survival analysis network layer, and an output layer. Figure 2 Model framework diagram:

[0067] 31) Model input: Multi-sensor degradation signals are used as the input of the model, and a sliding window technique is used to process the time series. In this process, the time series batches are normalized to a preset fixed length L to ensure that the multi-sensor degradation signal data maintain the same length L.

[0068] 32) Deep Survival Analysis Network Layer: This layer combines a Long Short-Term Memory (LSTM) network layer, a Dropout layer, and two fully connected layers to learn the degradation information represented by multi-sensor degradation signals. The LTM network layer can process time-series data, extracting effective features from multi-sensor signals and capturing the time dependencies in the degradation process. This invention also incorporates a novel loss function designed based on survival analysis theory, enabling the LTM network layer to fully learn the degradation information contained in both complete and censored data. The Dropout layer prevents overfitting by randomly discarding neurons, while the fully connected layers are used to further fuse and process the extracted features.

[0069] 33) Model Output: Assuming the lifetime of all units follows a Weibull distribution, the model output is set to the Weibull distribution parameters λ and k that the engine lifetime follows. Specifically, as shown below:

[0070] In the survival analysis of the engine, the engine's survival function S i (t) represents the probability that unit i survives after a given time t, defined as:

[0071] S i (t)=P(L i >t)=1-F i (t) (Equation 8)

[0072] Among them, S i (0) = 1, Let f(t) represent the probability of failure, and f(t) represent the probability density function (PDF) of unit i. Assuming engine i has survived to time t0, the conditional survival function S... i (t∣t0) is represented as:

[0073]

[0074] The remaining lifespan can be predicted as follows:

[0075]

[0076] in, This invention employs the Weibull distribution to characterize the probability density function f(t) of lifetime. The Weibull distribution is commonly used for lifetime modeling and is an effective tool for analyzing system reliability. It has several advantages, including flexibility, wide applicability, intuitive parameter interpretation, and adaptability. In survival analysis, the Weibull distribution is widely used due to its effectiveness in modeling survival time and failure rate distributions.

[0077] Fault occurrence time L i This refers to the specific moment from when the engine starts running until it fails, typically representing the engine's lifespan. Its probability density function is:

[0078]

[0079] Here, λ>0 and k>0 represent the scale and shape parameters of the Weibull distribution, respectively, which control the location and scale of the distribution and determine its shape. The survival function S(t) can be calculated using the probability density function f(t) as follows:

[0080]

[0081] Among them, S(L i ) represents the survival function; L i The time of the failure;

[0082] Based on Equations (9) and (10), the remaining lifetime can be calculated as follows:

[0083]

[0084] Among them, RUL i (t0) represents the remaining lifespan of engine i at time t0; t0 represents the current time point, i.e., the starting time point for predicting the remaining lifespan. Clearly, the distribution of the predicted lifespan or remaining lifespan is equivalent to the two parameters λ and k of the predicted Weibull distribution.

[0085] Neural network methods are widely used in data fusion due to their flexible structure and ability to effectively approximate arbitrary nonlinear functions. Among them, Long Short-Term Memory (LSTM) networks can effectively capture important information in sequence data and solve long-term dependency problems, which makes them perform well in handling various complex sequence tasks.

[0086] This invention uses the denoised multi-sensor degradation signals as input to a Bayesian LSTM-SURV model, and employs a sliding window technique to manage the length of the time series. During this process, time series batches are normalized to a preset fixed length L to ensure that the multi-sensor degradation signal data maintains the same length L and has S feature dimensions. The goal of the Bayesian LSTM-SURV model is to approximate g to achieve data fusion and engine remaining life prediction, denoted as [λ]. i ,k i ] = g(y i,1 ,y i,2,…, y i,L ), where the slice signal y i,L =[y i,1,L ,y i,2,L ,…,y i,J,L The BayesianLSTM-SURV model combines long short-term memory network layers, random deactivation layers, and fully connected layers to fuse degradation signals from multiple sensors and output two parameters λ. i and k i Used for lifetime distribution prediction. Figure 3 The structure of an LSTM is shown, which includes three special gating mechanisms designed to control the flow of information in the LSTM network: the forget gate, the input gate, and the output gate. Their formulas are as follows:

[0087]

[0088] Among them, y i It is a vector of multidimensional signals, i t It is an input gate, f t It's the Gate of Oblivion, o t It's an output gate, g t It is a candidate state, W i W f W o and W g These are the weights of the input gate, forget gate, output gate, and candidate state, respectively. i b f b o and b gHere, σ represents the biases of the input gate, forget gate, output gate, and candidate state, respectively; σ is the sigmoid activation function; f is the activation function tanh; and c is the bias of the candidate state. t It is a unit state, c t-1 It represents the cell state at the previous time step, h. t This is the hidden state output at time t. The neural network output is processed by the Softplus activation function to ensure that the output is non-negative, and the output engine i is at the prediction point t. i Two distribution parameters: λ i and k i , representing the scale parameter and shape parameter of the Weibull distribution, respectively. To prevent overfitting, this invention applies random dropout to the output layer.

[0089] 34) Offline model training:

[0090] In the offline phase, the Bayesian LSTM-SURV model constructed in step 3) is trained using the denoised training set data from step 2) to fully learn the degradation information contained in both the complete and censored data, achieving accurate prediction of RUL and updating the model parameters of the neural network. The details are as follows:

[0091] Let all model parameters be denoted as Where J = 0, 1, 2, ..., J, and in these parameters, {θ, α} represent the model parameters of the neural network, which are the same for all units; parameters The variability of each sensor across different units is captured and can be estimated based on (Equation 5) and (Equation 6).

[0092] During model training, a loss function is designed based on the maximum likelihood estimation of the Weibull distribution, and the distribution parameter k is constrained through joint optimization. i .

[0093] The first part of the loss function is a likelihood function based on survival analysis, which considers the probability of the event occurring, including the censored data at time point C. i Survival probability and complete data at time point L i The probability of failure.

[0094] ①The first part of the loss function is shown below:

[0095] With complete data, we obtain complete degradation data and information on the exact time of failure. Where C R Denotes the right-censored set, and uses f(t) i The probability distribution of faults is described by T. For censored data, T... i ∈CR Using S(t) i The probability distribution of censoring events is described by f(t) of the complete data. The joint likelihood function of the complete data and the censored data is obtained by expressing the probability distribution of the censored data as f(t). i ) and right-censored data S(t) i The likelihood function is estimated by multiplying the two numbers together. Therefore, the likelihood function can be expressed as:

[0096]

[0097] Where θ represents the unknown parameters of the neural network, n represents the number of engines, and L i C represents the time when the complete data failure occurred. i d represents the endpoint of the censored data. i The survival state is defined as follows: if engine i is in L... i If there is a fault, then d i The value is 1; if the engine is in C i If deleted, then d i The value is 0. Based on (Equation 11) and (Equation 12), (Equation 17) can be further expressed as:

[0098]

[0099] Where, λ i and k i These represent the scale parameter and shape parameter of the Weibull distribution output by the neural network, respectively.

[0100] Taking the negative logarithm of (Equation 18), we obtain the first part ζ1 of the loss function:

[0101]

[0102] ②The second part of the loss function is the shape parameter k i The lower bound of k ensures i Greater than 1, that is:

[0103] Here, α is a negative constant representing the penalty term coefficient. Based on the previous introduction to the Weibull distribution, it is constrained by k>1 during the degradation process. The final loss function L is obtained by linearly combining the two loss functions ζ1 and ζ2. net :

[0104]

[0105] Where n is the number of engines; L i C represents the time when the complete data failure occurred. i d represents the endpoint of the censored data.i The survival state is defined as follows: if engine i is in L... i If there is a fault, then d i The value is 1; if the engine is in C i If censored, then d i The value is 0, λ i and k i , where are the scale parameter and shape parameter of the Weibull distribution, respectively, and α is the penalty term coefficient.

[0106] 4) Online parameter updates and remaining service life prediction:

[0107] After obtaining training parameters using a grid search method in the offline phase, the remaining service life of in-service engines can be predicted. Assuming the in-service unit is represented by i, its available degradation signal is denoted as... Where j = 1, 2, 3, ..., J, t * This indicates the time point for prediction. To achieve accurate RUL prediction, historical data and individual information are combined. Specifically, a Bayesian mixture effect model learned from the training dataset is used, with the model fitting parameters as prior information and the sensor degradation signals of the in-service engine as observational information. The posterior distribution of the parameters is calculated using the Bayesian formula to update the Bayesian mixture effect model parameters, thereby achieving information transfer from training data to test data. In the proposed signal update framework based on the Bayesian method, for sensor j of engine i, its posterior distribution... It is a value with mean and variance The multivariate normal distribution is calculated as follows:

[0108]

[0109] in, For sensor j at time t * The previous design matrix, Let the variance of the measurement error of sensor j be the estimated variance. and β is obtained based on the degradation signal data of the training engine. i,j The mean and covariance matrix of the prior distribution. This represents the degradation signal of the in-service engine. The updated degradation signal of engine i can be calculated using the mean of the posterior distribution according to Equation 3. After obtaining the updated degradation signal, it is then input into the Bayesian LSTM-SURV model to obtain the estimated parameters of the Weibull distribution that the in-service engine i follows during its lifetime. and Thus, RUL is calculated. i The estimated value The specific formula is as follows:

[0110]

[0111] Among them, t * This indicates the point in time when the prediction is made.

[0112] 5) Construct model evaluation metrics to evaluate the model:

[0113] To evaluate the performance of the proposed Bayesian LSTM-SURV model, we selected the Root Mean Square Error (RMSE) as the model evaluation metric. RMSE quantifies the deviation between predicted and actual values, effectively assessing the model's prediction accuracy. Furthermore, this invention compares the proposed model with existing Time-dependent Survival Neural Networks (TSNNs) and DeepSurv models, comparing the prediction performance of the three models on the same dataset. Examples are provided below to further illustrate this invention.

[0114] Example 1: Simulation Verification

[0115] For the simulation data, this invention assumes that the degradation signal is collected from two sensors, and the actual degradation signal path of unit i has a polynomial trajectory with personalized random effects. Specifically:

[0116]

[0117] Among them, X i,j,t =[1,t,t 2 ], j∈{1,2}, regression parameters in and These are the mean vector and the covariance matrix, respectively, ε j Let Gaussian noise, following a normal independent and identically distributed pattern, be represented as: Its mean is 0 and its variance is The parameters used in (Equation 25) are listed in Table 1. This invention assumes that the actual lifespan of the engine follows a Weibull distribution, with a scale parameter λ = 20 and a shape parameter k = 80. The failure time Li of each unit is then generated by uniform sampling.

[0118] Table 1. Parameter settings for degradation signals

[0119]

[0120] To explore and compare the impact of the proportion of censored data on the accuracy of remaining lifetime prediction, this invention defines two parameters: the proportion of complete data p1 and the proportion of censored time p2. By setting different values ​​of p1 and p2, this invention can construct training sets with different degrees of censoring. The two parameters are defined as follows:

[0121]

[0122]

[0123] Where, N F C is the number of complete lifetime units (i.e., complete data), N is the total number of training data, and C is the total number of training data. i It is the end point of the censored data, L i This represents the time of failure before censoring. In practical applications, p2 may vary for different units; however, for simplicity, we apply the same p2 to all units. In this invention, p1 is set to 0.3, 0.5, and 0.7; p2 is set to 0.2, 0.4, 0.6, 0.8, and 0 to compare the impact of different proportions of censored data on prediction accuracy. Specifically, p2 = 0 indicates no censored data. Based on this, we can generate training sets with different levels of censoring.

[0124] Table 2 details the data generation process. The training dataset consists of 500 units, i.e., N = 500, and the test dataset consists of 100 units, i.e., N s =100, each unit corresponds to one time period. Following the steps provided in Table 2, the training dataset O of this invention can be generated. i , where i = 1, 2, ..., 500.

[0125] Table 2. Description of the data generation process

[0126]

[0127] To determine the optimal structure of the Bayesian LSTM-SURV model and prevent overfitting, this invention employs five-fold cross-validation on the training dataset, reserving 20% ​​of the units for validation. The model with the smallest loss function is ultimately selected. During cross-validation, grid search is used to determine the hyperparameters of the neural network, including the number of hidden nodes in the Long Short-Term Memory (LSTM) layer, the number of hidden layers, the number of training epochs, the learning rate, and the penalty term coefficient α of the loss function. These hyperparameters vary with the proportion of complete data (p1) and the proportion of censoring time (p2). Furthermore, specific values ​​are set for the randomly deactivated layers and fully connected layers (FC) of the Bayesian LSTM-SURV model: 64 hidden nodes for FC1 and 32 hidden nodes for FC2. The dropout rate for the randomly deactivated layers is 0.2. For the activation function, both FC1 and FC2 use the Sigmoid activation function, and the output layer uses the Softplus function to ensure a positive output value, satisfying the constraint that the parameters λ and k of the Weibull distribution must be positive.

[0128] During training, all input data needs to be normalized and denoised. The data with the smallest loss function is selected as the training result, and this neural network structure and parameters are saved, thus obtaining the trained neural network. After the model training is completed, this invention evaluates the model's prediction accuracy by calculating the root mean square error (RMSE), defined as:

[0129]

[0130] Where N is the number of engines. and y i These are the engine i-predicted and the actual RUL.

[0131] Figure 4 The paper presents a comparison of the RMSE results of the training data under different p1 = 0.7, 0.5, 0.3 (proportion of complete data) and p2 = 0, 0.2, 0.4, 0.6, 0.8 (proportion of censored time). By observing the trend of RMSE changing with p2 in the graph, it is clear that the model's RMSE decreases as p2 increases. This phenomenon indicates that, with a fixed proportion of complete data p1, the increase in degradation signals in the censored data improves the accuracy of the model's predictions. Next, this invention considers the influence of different p1 values, predicting that a larger p1 value results in a lower RMSE, which is consistent with... Figure 4The results are consistent with those in the previous example. This is because a higher proportion of complete data (p1) provides a more complete degradation signal, i.e., more degradation states and remaining service life prediction information. As the equipment operates for longer periods, more degradation signals are collected, and the degradation trend becomes more significant. The RMSE of the model training corresponding to different p1 and p2 are shown in Table 3.

[0132] Table 3. Root mean square error of model training for different p1 and p2 values

[0133]

[0134] During training, the model takes multi-sensor degradation signals as input and predicts the lifetime distribution of each unit, following a Weibull distribution, by outputting parameters λ and k. Compared to traditional point estimation methods, the model proposed in this invention predicts the distribution of remaining useful life by outputting parameters of the lifetime distribution, thus capturing lifetime uncertainty more comprehensively. This distribution-based approach improves the accuracy and reliability of remaining useful life prediction by providing detailed information about lifetime (including probability density function, confidence interval, survival function, etc.). The prediction results are then presented from the perspective of distribution prediction.

[0135] Figure 5 The plot shows the conditional probability density function (PDF) predicted for six randomly selected cells (#42, #21, #57, #17, #81, #95) in the test dataset. The PDF is plotted based on the λ (scale) and k (shape) parameters of the Weibull distribution predicted for each cell using a Bayesian LSTM-SURV model. As can be seen from the plot, the predicted lifetimes (solid line) are very close to the actual lifetimes (dashed line), mainly concentrated near the peak of the PDF, i.e., where the failure probability is highest. Furthermore, all actual lifetimes fall within the gray shaded area, which represents the 90% confidence interval (CI) of the cell lifetime calculated based on the Weibull distribution. This indicates that the model effectively captures the variability in the degradation process of different cells and provides lifetime distribution predictions with high confidence.

[0136] Figure 6This paper compares the actual and predicted conditional survival curves of unit c at four different prediction time points (t0 = 40, 45, 50, 55). The largest discrepancy exists between the predicted and actual values ​​at the early prediction time t0 = 40. However, as more degradation signal information is collected, the predicted survival curve at t0 = 55 becomes increasingly closer to the actual curve. The discrepancy in the early predictions is due to their high reliance on the accuracy of prior information obtained from historical data. Since less observational data was available for unit c, the conditional survival curves in the early stages primarily reflect the overall trend rather than the individual-specific characteristics of unit c. As more degradation signal data is collected, the estimated survival curves become increasingly able to capture individual degradation trends, leading to more accurate predictions of survival and remaining lifespan.

[0137] To further evaluate the accuracy of the confidence intervals (CIs) predicted by the model, this invention defines P. CI As an indicator, P CI Indicates actual lifespan L i It falls within the predicted 90% CI[A] i B i The proportion within ] is based on the prediction parameter λ of the Weibull distribution. i and k i It is calculated. The formula is as follows:

[0138]

[0139] Where, N s L is the total number of test units. i A is the actual lifetime of the i-th unit. i and B i These represent the lower and upper bounds of the 90% confidence interval for the prediction, respectively. It is an indicator function, if the actual lifetime L i If the value falls within the 90% confidence interval, it equals 1; otherwise, it equals 0.

[0140] Figure 7 This demonstrates the actual lifetime L at different remaining lifetime levels, with a complete data proportion p1 = 0.7 (corresponding to p2 = 0.2, 0.4, 0.6, 0.8) and a censoring time proportion p2 = 0.6 (corresponding to p1 = 0.7, 0.5, 0.3). i The proportion P that falls within the 90% confidence interval of the prediction CIIn the figure, "All" represents all engines to be predicted, and "TL" represents engines with a remaining life of less than or equal to L. Overall, an upward trend is observed in both figures because as engines approach failure, more observational data is collected, making the model's inferences about the degradation process more accurate, resulting in more actual lifespans falling within the predicted 90% confidence interval. In the left figure, when p1 = 0.7, the model maintains strong predictive performance at p2 = 0.6, similar to the result at p2 = 0.8; in the right figure, when p2 = 0.6, the result of the solid line (p1 = 0.7) is significantly better than the result of the dashed line (p1 = 0.5). Therefore, p1 = 0.7 and p2 = 0.6 is a relatively "cost-effective" combination, achieving satisfactory prediction results with relatively less data.

[0141] To verify the effectiveness of the proposed model, it is compared with the Time-dependent Survival Neural Network (TSNN) and the DeepSurv model. TSNN is a non-parametric model with a single hidden layer. This method uses multi-sensor data as input and discretizes future time intervals into a series of different time points β1 < β2 ... < β... k Each time interval is marked with a 1 or a 0 to indicate the time β. k To determine whether a fault has occurred, this method effectively structures the TSNN as a combination of multiple binary classifiers. To incorporate historical data, an exponential decay ratio is used to enhance fault risk estimation based on degraded data. The DeepSurv model combines the Cox proportional hazards (Cox PH) model with a deep multilayer perceptron. Although it does not explicitly consider the temporal dependence of covariates on event occurrence, it exhibits high adaptability when combined with neural networks, making it an effective Cox PH model. This invention uses these two methods as benchmarks for model comparison. In the experimental parameter settings, the number of hidden layer neurons in the TSNN is set to 20, and the DeepSurv model has two hidden layers, each containing 64 neurons.

[0142] Table 4 presents the RMSE comparison results of the Bayesian LSTM-SURV model, TSNN, and DeepSurv. The results show that the RMSE of all three methods improves with increasing p1 and p2, which is expected given the increasing dataset size. However, the proposed model (Bayesian LSTM-SURV) consistently outperforms the baseline method across all scales. Bayesian LSTM-SURV achieves the best prediction result with an RMSE of 5.500 when p1 = 0.7 and p2 = 0.8. In contrast, the DeepSurv model consistently underperforms the other models, primarily due to its reliance on the latest observations and the overall level obtained during training, failing to effectively utilize historical time-series data.

[0143] Table 4. Comparison of root mean square error of different models

[0144]

[0145] Example 2: C-MAPSS Turbofan Engine Dataset Case Study

[0146] Specifically, the engine degradation simulation data was generated using C-MAPSS software developed by NASA. This software is widely used to simulate the engine health monitoring of large commercial turbofan engines. Users can obtain signal data under different flight conditions by adjusting different flight parameters. The engine simulated by C-MAPSS has a total of 21 output signals, as shown in Table 5.

[0147] Table 5.21 Description of Output Signals

[0148]

[0149]

[0150] Continued from Table 5.21, which introduces the output signals.

[0151]

[0152] This dataset comprises 100 training units with 20,631 observation periods and 100 test units with 13,096 observation periods. In the training dataset, the engine continuously degrades until failure. To simulate censored data, we set the proportion of complete data p1 = 0.7, 0.5, 0.3 and the proportion of censored time p2 = 0, 0.2, 0.4, 0.6, 0.8, consistent with the simulation verification. In the test dataset, the data is cut off at a certain moment before failure, but its remaining lifetime is known. This invention uses the training data to estimate the prior distribution parameters of the signal and train the neural network, and uses the test data for prediction and to verify the model's prediction performance. The specific implementation is as follows:

[0153] (1) Variable selection and data preprocessing:

[0154] In the dataset, the operating parameters and sensor variables numbered S1, S5, S6, S10, S16, S18, and S19 showed no significant changes and were therefore removed during the modeling process. To maintain data consistency and quality, min-max normalization was employed to improve prediction performance. Subsequently, this invention uses a Bayesian method for signal denoising. The core idea of ​​the Bayesian method is to utilize prior knowledge and observed data to estimate the true signal, thereby reducing the impact of noise. Specifically, the training data is fitted using a Bayesian mixture-effects model to model potential degradation trends; for the test data, the signal is corrected by applying Bayesian updates in conjunction with prior knowledge, thereby effectively reducing noise and improving prediction accuracy. In the Bayesian method, the specification of the prior typically depends on the availability of relevant historical data or expert knowledge. In this invention, since the test unit i is similar to the training data, we choose a prior distribution based on the training data, thereby transferring information from the training data to the test data. Figure 8 The diagram illustrates the Bayesian update process for a portion of the degraded signal from a randomly selected unit i, including the original measurements (fluctuating solid line), the directly fitted signal (smooth solid line), and the signal updated by Bayesian analysis (dashed line). Clearly, the dashed line, incorporating prior information, captures the degradation trend more accurately than the smooth solid line, which only fits the observations. Therefore, the Bayesian method achieves more accurate and effective denoising.

[0155] (2) Display of lifespan prediction results:

[0156] During training, this invention employs five-fold cross-validation and uses a grid search method to estimate the hyperparameters of the BayesianLSTM-SURV model, such as the number of hidden layers, the number of neurons, and the number of iterations. The model parameters are as follows: the first fully connected layer (FC1) has 64 hidden nodes, the second fully connected layer (FC2) has 32 hidden nodes, and a dropout rate of 0.2 is used to mitigate overfitting. Both FC1 and FC2 layers use the ReLU activation function, while the output layer uses the Softplus activation function. Finally, the model with the minimum loss is selected. After training, the model is applied to 100 test units for performance evaluation.

[0157] Figure 9 Table 6 shows the RMSE comparison results of the training data at p1 = 0.7, 0.5, 0.3 (proportion of complete data) and p2 = 0, 0.2, 0.4, 0.6, 0.8 (proportion of censored time), demonstrating the model's performance under different proportions of censored data. The results are consistent with the simulation validation; as p1 and p2 increase, the model's RMSE decreases accordingly. This is obvious because larger p1 and p2 indicate a higher proportion of complete data and censored time. More degraded data provides the model with richer information, enabling it to more accurately monitor the engine's degradation trajectory and improve the accuracy of RUL prediction.

[0158] Table 6. Root mean square error of model training for different p1 and p2

[0159]

[0160] Figure 10 This shows the proportion of actual lifetime falling within the 90% confidence interval of the prediction at different remaining useful life levels, i.e., P. CI The left-hand figure shows the P values ​​for different censoring time proportions p2 (0.2, 0.4, 0.6, 0.8) with a fixed p1 = 0.7. CI The right-hand plot shows the impact of different p1 values ​​(0.7, 0.5, 0.3) with a fixed p2 = 0.6. Overall, an upward trend was observed in both plots. As the equipment approaches failure, the accumulation of more degradation data makes the model's estimation of the degradation trend more accurate, resulting in more actual lifespans falling within the predicted 90% confidence interval.

[0161] In the case study section, further comparisons with benchmark methods (particularly TSNN and DeepSurv) are presented, as shown in Table 7, validating the performance of the proposed Bayesian LSTM-SURV model. The results demonstrate that the proposed model (Bayesian LSTM-SURV) outperforms the benchmark methods in all tests, particularly when p1 = 0.7 and p2 = 0.8, where it achieves the best prediction performance with an RMSE of 7.696. Furthermore, the DeepSurv model consistently underperforms the other models because it relies solely on the latest observations and the overall level obtained during training, without considering historical time-series data.

[0162] Table 7. Comparison of root mean square error of different models

[0163]

[0164] It should be noted that the purpose of disclosing the embodiments is to help further understand the present invention. However, those skilled in the art will understand that various substitutions and modifications are possible without departing from the scope of the present invention and the appended claims. Therefore, the present invention should not be limited to the content disclosed in the embodiments, and the scope of protection of the present invention is defined by the scope of the claims.

Claims

1. A method for predicting the remaining life of an engine based on deep survival analysis, characterized by, A remaining useful life prediction model based on deep survival analysis is constructed, a nonlinear relationship between multi-source sensor signals and a remaining useful life distribution of an engine is modeled, and through survival analysis, the remaining useful life of the engine is effectively predicted online in the presence of signal deletion or lack of remaining useful life RUL labels; The method comprises the following steps: 1) collecting complete observation data of the engine, including: engine degradation signal data, engine failure occurrence time or deletion time, engine failure or deletion indicator variable; 2) modeling the multi-sensor degradation signal, establishing a Bayesian mixed effect model representing the degradation process of the engine and performing model parameter estimation; 3) constructing an engine remaining useful life prediction model based on deep survival analysis and performing model training; By constructing an engine remaining useful life prediction model based on deep survival analysis, i.e., a Bayesian LSTM-SURV model, a relationship between the degradation signal and the remaining useful life is established; the engine remaining useful life prediction model comprises an input layer, a deep survival analysis network layer, i.e., an LSTM-SURV layer, and an output layer; The multi-sensor degradation signal is used as the input of the model; The LSTM-SURV layer is used to learn the degradation information represented by the multi-sensor degradation signal; the structure of the LSTM in the LSTM-SURV layer comprises a forgetting gate, an input gate and an output gate, which are used to control the flow of information in the LSTM; The parameter of the distribution to which the engine life is subjected is used as the output of the model; In the offline stage, the denoised multi-sensor degradation signal is input into the engine remaining useful life prediction model based on deep survival analysis constructed in step 3), and based on survival analysis, the loss function of the model is designed to comprise a first part and a second part, which are used to sufficiently learn the degradation information contained in the complete data and the deletion data, and to realize the prediction of the remaining useful life; The first part of the loss function comprises the survival probability of the deletion data at the deletion time and the failure probability of the complete data at the failure occurrence time; the second part of the loss function is the lower bound of the shape parameter, which is greater than 1; 4) performing prior distribution preprocessing based on the training data on the engine data to be predicted, migrating the processed information from the training data to the test data, dynamically updating the model parameters using the test set data, inputting the updated data into the trained engine remaining useful life prediction model based on deep survival analysis, and performing online parameter estimation and remaining useful life prediction; 5) constructing a model evaluation index to evaluate the constructed model; Through the above steps, online prediction of the remaining useful life of the engine based on the Bayesian deep survival analysis network is realized.

2. The method of claim 1, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, In step 4), the Weibull distribution is used to represent the probability density function of the remaining useful life of the engine. Survival function S(L i ) is calculated by the probability density function f(t) and is expressed as: where λ > 0 and k > 0 represent the scale parameter and shape parameter of the Weibull distribution, respectively; S(L i ) denotes the survival function; L i is the time to failure; The calculation of the remaining useful life is represented as: wherein RUL i (t0) represents the remaining useful life of engine i at time t0; t0 represents the current time point, i.e. the starting time point for the prediction of the remaining useful life; the prediction of the distribution of the engine life or the remaining useful life, i.e. the prediction of the two parameters of the distribution.

3. The method of claim 2, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, The loss function of the model training is represented as: where ζ1 is the first part of the loss function; ζ2 is the second part of the loss function; n is the number of engines; L i represents the failure time of complete data, C i represents the end point of censored data, d i is the survival status, i.e. if engine i fails at L i , the value of d i is 1; if the engine is censored at C i , the value of d i is 0, λ i and k i are the scale parameter and shape parameter of Weibull distribution respectively, and α is the penalty term coefficient.

4. The method of claim 3, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, In step 2), the Bayesian mixed effect model is used to represent the degradation process of the engine.

5. The method of claim 4, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, The two-stage method is used to estimate the model parameters.

6. The method of claim 5, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, The method for calculating an estimate of the engine remaining useful life RUL i is represented as:​ wherein, and are Weibull distribution estimated parameters subject to the in-service engine i lifetime; t * denotes the time point at which the prediction is made.

7. The method of claim 1, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, The output of the engine remaining useful life prediction model based on deep survival analysis constructed in step 3) is processed by the Softplus activation function.

8. The method of claim 1, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, Step 1) Collecting the complete observation data of the engine is represented as: O i = {Y i , T i , d i}, where Y i represents the data of the J-dimensional degenerate signal of the engine, T i represents the fault or missing time of the engine, and d i represents the engine fault or missing, with the distribution value being 1 or 0.

9. The method of claim 1, wherein the engine remaining life prediction based on the deep survival analysis is characterized by, In step 5), the root mean square error (RMSE) is used as the model evaluation index to effectively evaluate the prediction accuracy of the model by quantifying the deviation between the predicted value and the actual value.

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