A sparrow search algorithm-optimized LSTM framework with EMD denoising for rolling element bearing remaining useful life prediction

Abstract

Accurate remaining useful life (RUL) prediction of rolling element bearings is essential for implementing predictive maintenance strategies in rotating machinery. However, this task remains challenging due to severe signal noise and the inherently complex, nonlinear nature of bearing degradation processes. To address these limitations, this paper proposes a novel hybrid deep learning framework that synergistically integrates adaptive signal processing, intelligent optimization, and probabilistic sequence modeling. The proposed methodology comprises three key stages. First, Empirical Mode Decomposition (EMD) is applied to denoise raw vibration signals and extract physically meaningful degradation features by adaptively decomposing them into intrinsic mode functions (IMFs). Second, the Sparrow Search Algorithm (SSA) is employed to automatically optimize the critical hyperparameters of a Long Short-Term Memory (LSTM) network, thereby enhancing its capability to capture long-term temporal dependencies inherent in the degradation trajectory. Third, a first-passage-time model based on the inverse Gaussian distribution is introduced to provide probabilistic RUL predictions with quantified uncertainty, extending beyond conventional point estimates. The proposed EMD-SSA-LSTM framework is experimentally validated on the publicly available bearing accelerated life test dataset. Comparative results demonstrate that our approach significantly outperforms benchmark methods, including GA-LSTM and PSO-LSTM, achieving superior prediction accuracy, faster convergence, and enhanced robustness. This work provides a comprehensive and effective solution for data-driven bearing prognostics, contributing to reliable predictive maintenance in industrial applications.

Introduction

Rolling element bearings are fundamental components in rotating machinery, whose failure is a primary cause of unscheduled downtime and substantial economic losses in industrial systems¹. Predicting the Remaining Useful Life (RUL) of bearings is therefore a cornerstone of predictive maintenance strategies, aiming to enhance operational reliability and safety while reducing maintenance costs^2–4. However, achieving accurate RUL prediction remains challenging due to the complex, nonlinear, and stochastic nature of bearing degradation, often compounded by high levels of noise in sensor data collected from harsh industrial environments. Despite the widespread application of traditional signal processing techniques, their limitations in extracting weak fault features from noisy vibration signals have been increasingly recognized. Sun et al.⁵ recently demonstrated that conventional methods such as Empirical Mode Decomposition (EMD) or wavelet transform alone are often insufficient for identifying early-stage raceway damage in rolling bearings, particularly under varying operating conditions like cold start and thermal stabilization. Their proposed joint analysis method combining EMD and wavelet transform significantly improved fault detection accuracy, highlighting the necessity of hybrid approaches for robust diagnostic information extraction. This observation aligns with the multi-scale nature of bearing degradation, where damage initiates at the micro-level and progressively evolves to meso- and macro-scale failures, as systematically characterized by Maruschak et al.⁶ through indentation, profilometry, and fractographic analyses. These multi-scale damage mechanisms manifest as distinct frequency components in vibration signals, which can be effectively separated by adaptive decomposition methods such as EMD. Consequently, the selection of specific Intrinsic Mode Functions (IMFs)–particularly those correlated with characteristic fault frequencies–provides a physically interpretable basis for constructing Health Indicators (HIs) that capture the true degradation state of the bearing. Lourari et al.⁷ proposed a comprehensive methodology integrating Variational Mode Decomposition (VMD) with Particle Swarm Optimization (PSO) to determine the optimal number of IMFs for feature extraction. In addition, based on this direction, Lourari et al.⁸ introduced a method that combines fully integrated empirical mode decomposition with adaptive noise and sequential backward selection for bearing fault diagnosis under variable loads. Subsequently, these authors extended this framework to include both vibration and current signals⁹, enabling fault diagnosis under variable operating conditions.

In the current research of mechanical bearing fault prediction and RUL prediction^10–12, the mainstream methods can be roughly divided into three categories: one is the method based on physical mechanism modeling, which predicts the degradation process of bearing wear and fatigue by establishing physical models, but it relies on professional knowledge and has poor adaptability; Secondly, based on statistics and traditional machine learning methods, such as support vector machine (SVM)¹³, random forest (RF)¹⁴ and hidden markov model (HMM)¹⁵, it can identify the degradation trend to some extent, but it has limitations in feature extraction and time series processing. Thirdly, deep learning methods^16–18, such as convolutional neural network (CNN)¹⁹, self-encoder²⁰, Transformer²¹ and LSTM^22,23, have emerged in recent years. For example, in energy system control, a hybrid strategy that combines fuzzy logic has demonstrated excellent capabilities in regulating system state under dynamic operating conditions, as demonstrated in the voltage tracking of a vehicle mounted PEMFC system²⁴. Furthermore, Lourari et al.²⁵ proposed a hierarchical signal processing framework for rotating equipment fault diagnosis using Hilbert Huang transform (HHT) and singular value decomposition (SVD). By segmenting the vibration signal, extracting time-frequency information using HHT, reconstructing the signal from non-zero HHT components, and applying SVD for dimensionality reduction, their method achieved a diagnostic accuracy of using supervised learning. In particular, LSTM has gradually surpassed traditional methods in prediction accuracy and generalization ability because of its natural advantages in modeling time dynamic processes, and has become one of the mainstream choices. long short-term memory, LSTM network, as a cyclic neural network structure that can effectively process time series data, has been widely used in degradation modeling and remaining useful life prediction of mechanical bearings because of its excellent time series modeling ability²⁶. LSTM can retain the long-term dependence information through the gating mechanism, and avoid the problem of gradient disappearance, which makes it excellent in capturing the small but critical state changes during the operation of bearings, and is especially suitable for dealing with the common nonlinear and non-stationary degradation processes in mechanical systems^27–30. However, although LSTM has powerful modeling ability, its performance is still highly dependent on the network structure and the reasonable configuration of superparameters, such as the number of hidden layer units, learning rate, dropout ratio and so on. The traditional manual parameter adjustment method is not only inefficient, but also easy to fall into local optimum, which affects the final prediction effect of the model. Therefore, in recent years, more and more researches began to introduce intelligent optimization algorithms, such as genetic algorithm (GA)³¹, PSO³² and sparrow search algorithm (SSA)³³, to optimize the parameters of the LSTM model automatically, so as to improve the performance and robustness of the model.

In³⁴, Wang et al. developed a toll station congestion prediction model using an LSTM-PSO algorithm with average queue length, service time, and traffic volume as inputs, and established a lane scheduling optimization model under Level-of-Service (LOS) Grade II constraints. To improve the accuracy of degradation prediction for mechanical components such as rolling bearings, Hu et al. proposed a novel prediction method based on LSTM networks. The model parameters were optimized using an Improved Particle Swarm Optimization (IPSO) algorithm. The LSTM model with optimized parameters was then employed to forecast the degradation trend of bearing performance. Experimental results demonstrated that the proposed approach effectively identifies both the degradation states and performance trends³⁵. Lin proposed an intelligent control system based on LSTM networks optimized by GA, referred to as the GA-LSTM system, to determine optimal control parameters for semi-active variable stiffness isolation systems³⁶. Numerical simulations and shaking table tests demonstrated that the GA-LSTM system effectively reduces isolation displacement and superstructure acceleration under both near-fault and far-field earthquakes, outperforming conventional control methods and offering excellent seismic performance and adaptability. In¹⁷, Akpudo and Jang-Wook proposed a multi-sensor prediction approach that extracts highly predictable statistical features from vibration and pressure sensor signals using multi-level wavelet decomposition and then fuses them. Based on this, they developed a prognostic algorithm MOGA-LSTM, an LSTM model optimized by a multi-objective genetic algorithm–which takes the fused sensor features as input and outputs the RUL of the electromagnetic pump.

LSTM networks have been successfully applied to RUL prediction, the field of deep learning for time-series analysis is rapidly evolving. The Transformer architecture has emerged as a powerful alternative, particularly renowned for its self-attention mechanism that can theoretically capture long-range dependencies without the sequential processing constraints of RNNs. This has led to its success in various prognostic applications, such as the degradation prediction of proton exchange membrane fuel cell performance³⁷. However, the choice between LSTM and Transformer is often task-dependent. Transformers excel at modeling global contexts but typically require large datasets for training and can be computationally intensive. In contrast, LSTMs, with their inductive bias for sequential data, can often be effectively trained on smaller, domain-specific datasets like ours and provide a more computationally efficient solution. Given the proven efficacy of LSTM in capturing the medium-to-long-term degradation trends in mechanical systems and the manageable size of our training data, it was selected as the core network for this initial framework.

While several recent studies have explored deep learning-based approaches for bearing fault diagnosis and RUL prediction^38–40, most existing methods either rely on manual feature extraction or treat signal preprocessing and model optimization as separate stages. For instance, recent hybrid frameworks proposed for fault diagnosis in rotating machinery^41–44 effectively combine time-frequency analysis with convolutional neural networks or vision transformers, achieving high classification accuracy. However, these studies primarily focus on fault classification under limited labeled data or varying operating conditions, rather than on predicting the RUL, which requires modeling long-term degradation trends and quantifying prediction uncertainty. Moreover, they often overlook the physical interpretability of decomposed signal components and the need for systematic hyperparameter optimization under fair computational budgets. Recent surveys⁴⁵ have highlighted the need for integrated frameworks that jointly address signal denoising, hyperparameter tuning, and temporal dependency modeling. However, to the best of our knowledge, no prior work has systematically combined EMD, SSA, and LSTM networks within a single end-to-end prognostic framework. This paper differs from existing studies by: (1) proposing a novel hybrid architecture that synergistically integrates adaptive signal decomposition, intelligent optimization, and probabilistic sequence modeling; (2) demonstrating superior performance over benchmark methods (GA-LSTM, PSO-LSTM) on the XJTU-SY dataset with rigorous error analysis; and (3) explicitly discussing the generalization limitations and future pathways toward industrial deployment and standard compliance. By addressing these gaps, our work advances the state-of-the-art in data-driven bearing prognostics and provides a practical foundation for intelligent predictive maintenance.

Despite recent progress in hybridizing deep learning with optimization algorithms for hyperparameter tuning, three critical gaps persist: (i) EMD-based preprocessing remains largely decoupled from physical damage mechanisms; (ii) fair comparisons among optimizers under identical computational budgets are rarely conducted; and (iii) most models provide only point estimates, overlooking uncertainty quantification. To address these gaps, this paper proposes a physics-informed hybrid framework that integrates EMD-based feature extraction, SSA-optimized LSTM, and a first-passage-time model for probabilistic RUL prediction, with fair comparison to GA and PSO under unified conditions.

The remainder of this paper is organized as follows: Section "Core architecture of RUL model for rolling element bearing" details the core architecture, including the health indicator construction with EMD and the probabilistic RUL framework. Section "Bearing RUL prediction network model" elaborates on the LSTM network and the SSA-based optimization process. Section "Numerical simulation" presents experimental results and a comparative analysis. Finally, Section "Conclusion" concludes the paper and suggests future research directions.

Core architecture of RUL model for rolling element bearing

According to the transmission medium, Rolling bearing can be divided into ball bearings, cylindrical bearings, and tapered bearings. Rolling element bearings are critical components in mechanical systems, and their operational status directly affects the reliability of the system⁴⁶. Due to the fact that bearings typically operate under high loads, high speeds, and complex working conditions, predicting their RUL in advance can help reduce the risk of sudden failures, achieve predictive maintenance, extend system lifespan, and reduce maintenance costs⁴⁷.

The degradation process of rolling bearings exhibits significant nonlinear and non monotonic characteristics, which are not only random noise but often contain physically meaningful patterns. One of them is transient ’self recovery’ behavior, manifested as temporary improvement in health indicators. This can be attributed to mechanisms such as stress release after local micro peeling or re lubrication of the contact surface, which temporarily alleviate the degradation rate. It is interesting that this degradation recovery coupling mechanism is a common phenomenon in different engineering systems. For example, in proton exchange membrane fuel cells, highly nonlinear reversible voltage loss recovery phenomena occur due to catalyst hydration kinetics and operational interventions, which have a profound impact on life prediction⁴⁸.

Exponential degradation model

A rolling element bearing fundamentally comprises four integral components: an inner ring affixed to the rotating shaft, an outer ring secured in a stationary housing, rolling elements (such as balls or rollers) that carry the load between these rings, and a cage which spaces and guides these elements. Lubricant and seals are typically included to minimize wear and contamination, as shown in Fig. 1.

Fig. 1.

The structural diagram of rolling element bearing.

The precise interaction and material integrity of these parts–particularly the raceways and rolling elements–directly govern its operational vibration and degradation patterns, forming the physical basis for the health indicators and predictive models discussed.

The exponential degradation mathematical model is

1

where, represents the material degradation coefficient, calibrated through accelerated testing and measured in . Among the extracted IMFs, is empirically/analytically identified as the most sensitive component to the bearing’s characteristic failure frequency, and is therefore selected for constructing the Health Indicator.

The Health Indicator (HI) is defined by normalizing the cumulative vibration energy to the [0, 1] range, where HI = 0 indicates a brand-new bearing and HI = 1 signifies complete failure.

Remark 1

This exponential model provides a physics-inspired foundation for defining the Health Indicator. However, to capture the complex and stochastic nature of actual bearing degradation, the functional relationship between the historical HI sequence and the RUL is learned directly from data using the LSTM network. The inverse Gaussian distribution of the RUL not only provides a point estimate (the mean) but also quantifies the prediction uncertainty (the variance). This allows for riskinformed maintenance decisions, where actions can be scheduled based on a desired confidence level (e.g., 95% probability that failure will occur after time T).

RUL probability density function (first passage time)

In the first-passage-time framework, the failure threshold of the health indicator is set to , based on engineering experience. The RUL is defined as the earliest time at which the HI reaches or exceeds the failure threshold, given the initial condition :

2

Assuming the degradation process follows a drift-diffusion model:

3

where denotes standard Brownian motion, is the mean degradation rate, and represents the stochastic fluctuation intensity, the first-passage time to the threshold is inverse Gaussian (IG) distributed.

The closed-form probability density function (pdf) is given by:

4

where is the remaining health margin.

By matching with the standard IG parameterization , the equivalence is:

5

This yields the expected RUL and variance as:

6

From an engineering perspective, the average RUL is proportional to the remaining health margin and inversely proportional to the mean degradation rate, while the uncertainty in RUL increases with higher noise levels and decreases with faster degradation rates.

The cumulative distribution function (CDF) of the IG distribution is:

7

where is the standard normal CDF. The survival function gives the probability that the bearing will survive beyond time t, and the hazard rate can be used for risk-based maintenance scheduling.

Remark 2

For parameter estimation, historical full-life data can be used to construct the HI curve and estimate and via regression or maximum likelihood methods. The drift can be estimated from the slope of the HI in a quasi-linear degradation stage, while is estimated from the variance of residuals or increments. In online monitoring, Bayesian filtering or Kalman filtering can be applied to recursively update the parameters and refresh the IG distribution, using the current HI value to recalculate .

Remark 3

Practical considerations include the sensitivity of results to the chosen failure threshold, the possibility of negative drift rates (indicating no approach to failure), and the non-monotonicity of early-stage degradation data, which may require segmenting the process into quasi-monotonic phases before fitting. If is uncertain, a prior distribution can be assigned to it, and the resulting RUL distribution becomes a mixture of IG distributions. Confidence intervals for RUL can be obtained from IG quantiles, enabling maintenance schedules to be planned with explicit reliability targets.

Assumption 1

Linear cumulative damage criterion: The health index increment satisfies, which is applicable to high cycle fatigue scenarios dominated by rolling contact fatigue (material stress below of yield strength) and conforms to the description of bearing failure mechanisms in ISO 15243 standard;

Model input data processing

In bearing RUL prediction, the network input usually comes from vibration signals (acceleration, velocity, displacement, etc.), but these signals often have problems such as strong signal noise, weak degradation characteristics, and multi frequency component aliasing. EMD was proposed by Huang et al. for decomposing complex signals into several IMFs and a residual term, each IMF being a locally stationary narrowband signal^49–51. In bearing RUL prediction, EMD can separate low-frequency trend terms (representing long-term changes in health indicators) from high-frequency components (including fault impact information), select IMF related to bearing characteristic frequencies as network inputs, and improve feature quality before the input layer to reduce network learning difficulty.

EMD sifting process

The EMD algorithm decomposes a signal through an iterative sifting process, which extracts oscillatory modes at different time scales. Given the original signal x(t), the sifting procedure is carried out as follows:

1. Identify local extrema - Locate all local maxima and minima of x(t).

2. Construct envelopes - Generate the upper envelope and lower envelope by cubic spline interpolation of the maxima and minima, respectively.

3. Compute the local mean - The mean of the two envelopes is calculated as

   8

   which represents the local average of the signal at each time instant.
4. Extract the detail component – Subtract the local mean from the original signal to obtain

   9

   where is intended to be the first IMF.
5. Check IMF criteria - Evaluate whether satisfies the two necessary conditions for an IMF: (i) the number of extrema and zero-crossings differ by at most one, and (ii) the local mean is zero.

   • If meets these conditions, it is designated as the first IMF, .
   • If not, the sifting process is repeated: replace x(t) with and return to step1. This iterative sifting continues until the resulting component fulfills the IMF criteria.
6. Compute the residual - After extracting a valid IMF, the residual signal is obtained by

   10
7. Decompose the residual - The entire sifting procedure (steps a-f) is applied to the residual to extract subsequent IMFs. In general, for the k-th stage,

   11

   The decomposition terminates when the final residual becomes a monotonic function or contains no oscillatory components. Upon completion, the original signal can be expressed as

   12

   where denotes the i-th IMF, capturing oscillations at a specific time scale, and represents the overall trend of the signal. This adaptive decomposition isolates physically meaningful components and effectively separates noise from degradation-related features in bearing vibration signals.

Physical interpretability analysis of EMD components

To rigorously justify the selection of EMD components and verify their physical interpretability, a two-step quantitative analysis was conducted, addressing the correlation with overall degradation and linkage to specific fault mechanisms.

First, a quantitative correlation analysis was performed to identify which IMF carries the most significant information about the bearing’s health degradation. The energy of each IMF was calculated for every time window, and the Pearson correlation coefficient between these energy sequences and the constructed HI was computed. As summarized in Table 1, IMF3 exhibits the strongest negative correlation with the , indicating that its energy increases most consistently as the bearing degrades (since HI approaches 1 at failure). This strong correlation establishes IMF3 as the primary carrier of the degradation trend.

Table 1.

Correlation coefficients between IMF energies and the Health Indicator.

IMF Component	Correlation with HI
IMF1	− 0.45
IMF2	− 0.71
IMF3	− 0.89
IMF4	− 0.62
IMF5 (Residual)	0.28

Second, to link IMF3 to specific physical degradation mechanisms, envelope spectrum analysis was applied to a segment of the IMF3 signal from the final stage of the bearing’s life. The envelope spectrum is particularly effective in identifying the periodic impact signals generated by faults like raceway spalling. The theoretical fault characteristic frequencies for the test bearing-Ball Pass Frequency Outer Race and Ball Pass Frequency Inner Race were calculated based on the bearing geometry and rotational speed.

Therefore, the selection of IMF3 for constructing the HI in Eq. (1) is justified both by its high statistical correlation with the degradation trend and its direct connection to a known physical degradation mechanism.

Bearing RUL prediction network model

The HI of bearings during operation evolves over time, forming a time-dependent degradation curve. The essence of RUL prediction is:

13

Predicting the time from the future to failure based on the state of the past period of time.

Principle and structure of LSTM

Early weak degradation features may affect the overall lifespan trend, requiring the model to remember information over a longer time span. Ordinary RNNs are prone to gradient vanishing or exploding⁵², and cannot effectively capture long-term dependencies, while LSTM can avoid this problem through gating mechanisms. HI is often nonlinear (affected by load, temperature, speed, etc.) and has high noise. LSTM is an improved RNN proposed by Hochreiter&Schmidhuber (1997), which controls the flow of information in the cell state through input gates, forget gates, and output gates, achieving long-term dependency modeling^53–55.

LSTM gating mechanism

The LSTM gating mechanism controls information flow through three interacting gates, as shown in Fig. 2. The forget gate determines how much of the previous cell state to retain. The input gate and candidate state jointly update the cell state via . Finally, the output gate controls the hidden state output . This structure enables long-term dependency modeling essential for RUL prediction.

Fig. 2.

The schematic diagram of LSTM.

The Forget Gate in an LSTM determines the proportion of information from the previous cell state to retain or discard. It computes a gating vector:

14

where the sigmoid activation produces values between 0 (completely forget) and 1 (fully keep), thus enabling the model to filter out irrelevant or outdated information from long-term memory.

The input gate decides how much new information to write into the cell state, and the candidate state contains the actual new content.

15

16

The Input Gate uses weight matrices and bias vectors to determine how much new information should be written to the cell state. It generates a gating vector via the sigmoid function , which controls the write proportion, and computes the candidate cell state through the hyperbolic tangent , which scales values to .

• Update Cell State

  17

  The cell state update in LSTM combines retained past memory with new candidate content through element-wise multiplication, enabling the model to preserve information over long time spans, mitigate vanishing gradients.
• Output Gate:

  18
• Hidden State:

  19

  The Output Gate in LSTM, parameterized by and , controls which parts of the updated cell state are revealed through , producing the hidden state that serves as short-term memory for the next time step or prediction layer.

LSTM parameter optimization

The main concepts of SSA

This work focuses on high-precision parameter estimation for LSTM hyperparameter tuning–a challenge shared across engineering fields, such as parameter identification in proton exchange membrane fuel cell models⁵⁶. Inspired by methods like those in⁵⁶, future work may incorporate competing objectives–such as predictive accuracy. Notably, SSA has been successfully implemented to optimize LSTM networks in various time-series forecasting applications, such as short-term load forecasting⁵⁷ and bearing RUL prediction.

The SSA is a nature-inspired metaheuristic optimization algorithm proposed by Xue and Shen⁵⁸, which simulates the foraging, vigilance, and anti-predation behaviors of sparrow populations. As a gradient-free optimization method, SSA exhibits remarkable flexibility in handling continuous, discrete, or mixed-variable optimization problems, making it particularly suitable for non-convex, high-dimensional, and nonlinear search spaces. The algorithm divides the population into three functional roles: producers that explore promising regions by guiding the population toward optimal areas, scroungers that follow producers to exploit discovered food sources, and warners that detect potential dangers and alert the population to avoid predators, thereby achieving an effective balance between global exploration and local exploitation⁵⁹.

SSA principle and formulation

Remark 4

LSTM networks have multiple hyperparameters such as learning rate, number of hidden units, sequence length, and dropout rate–that have a substantial impact on prediction accuracy. Manually tuning these parameters is challenging because the search space is nonlinear and high-dimensional, the relationship between hyperparameters and model performance is typically non-convex, and gradient-based methods are often ineffective since the loss function is non-differentiable with respect to hyperparameters. The SSA addresses these challenges effectively. Its global search capability enables efficient exploration of the entire parameter space, reducing the risk of being trapped in local minima. Its balanced exploration–exploitation mechanism, driven by the producer–scrounger dynamics, allows simultaneous discovery of new promising regions and refinement of known good solutions. Moreover, its random vigilance and alert strategies enhance adaptability to complex fitness landscapes, preventing premature convergence. Consequently, SSA can automatically discover near-optimal LSTM hyperparameters, thereby improving the accuracy and generalization capability of RUL prediction models. The schematic diagram of SSA is shown in Fig. 3.

Fig. 3.

The schematic diagram of SSA .

Let:

• number of sparrows in the population;

• dimension of the search space (number of variables to optimize);

• position matrix, where each row represents one sparrow’s candidate solution.

  1. Role assignment
     • Producers ( of population): actively search for food, explore globally.
     • Scroungers: follow producers, focus on local exploitation.
     • Warners ( of population): detect danger and adjust population positions.
  2. Producer position update
     For producer i at iteration t :

     20

     In this formulation, is the control parameter, denotes the maximum number of iterations, ( 0,1 ) is the safety threshold, is a random number, Q is a random value drawn from a normal distribution, and L is a vector of ones. When , indicating no perceived danger, producers explore in the vicinity of their current positions; otherwise, they adopt more aggressive movements to search for better solutions.
  3. Scrounger position update
     Scrounger i updates position as:

     21

     where, represents the position of the best producer at iteration denotes the position of the worst producer, is a random value drawn from a normal distribution, and L is a vector of ones. Scroungers positioned near the end of the ranking tend to focus on local exploitation around promising areas, while those ranked higher may explore farther from the current best position to discover new potential solutions.
  4. Warner position update
     Warner i modifies position:

     22

     where, is a random step drawn from a normal distribution. Warners update their positions by moving toward the current best location while remaining vigilant, enabling them to avoid potential threats such as predators or, in optimization terms, premature convergence to local optima.

Remark 5

SSA is a powerful metaheuristic algorithm that mimics the foraging and alerting behavior of sparrows, effectively exploring and utilizing the solution space. When applied to LSTM hyperparameter adjustment for azimuth RUL prediction, SSA helps overcome manual trial and error and gradient limitations, resulting in better convergence and prediction accuracy. It was found that this relatively small configuration (population size 20, maximum iterations 50) is sufficient to complete the current low-dimensional hyperparameter optimization task, ensuring fast convergence and high computational efficiency, while still achieving significant performance improvements compared to the baseline model.

Control objective

The core control objective of this study is to construct a high-precision and highly robust framework for predicting the RUL of rolling bearings, in order to serve accurate predictive maintenance decisions. Specifically, the aim is to achieve the following goals through systematic technological integration: 1) effectively suppress noise interference and extract key features that can truly reflect the degradation state of bearings from the original vibration signals; 2) Accurately modeling complex degradation dynamics, capturing the nonlinear and long-term dependent temporal patterns during the performance degradation process of bearings; 3) Realize efficient model optimization, automatically determine the optimal hyperparameter configuration for deep learning models, and improve model performance and training efficiency; 4) Provide uncertainty quantification, go beyond single point prediction, and output probabilistic RUL interval estimates, thereby providing richer and more reliable information ort for risk assessment and maintenance planning. The overall architecture of the proposed EMD-SSA-LSTM hybrid model is shown in Fig. 4.

Fig. 4.

The overall architecture of the proposed EMD-SSA-LSTM hybrid model.

Numerical simulation

In order to better verify the effectiveness of the proposed EMD-SSA-LSTM model in predicting the remaining life of mechanical bearings, we selected two prediction models based on GA-LSTM and PSO-LSTM for detailed comparison. The parameter settings of the two comparison prediction models can be found in references⁶⁰ and⁶¹.

Experimental setup and dataset description

To validate the effectiveness of the proposed EMD-SSA-LSTM model, experiments were conducted on the publicly available and widely recognized XJTU-SY rolling bearing accelerated life test dataset. This dataset was established by a research institute in collaboration with our unit, providing comprehensive operational fault data of rolling bearings under three different operating conditions. The test bearings are LDK UER204 deep groove ball bearings, with geometric parameters summarized in Table 2 . The experiments were performed under three operating conditions: Condition 1 (35 Hz rotational speed, 12 kN radial load), Condition 2 (37.5 Hz, 11 kN), and Condition 3 (40 Hz, 10 kN) . For each condition, five bearings were tested to failure, resulting in 15 run-to-failure datasets. Vibration signals were monitored through vertically and horizontally mounted accelerometers with a sampling frequency of 25.6 kHz, sampling interval of 1 minute, and each sample duration of 1.28 seconds. The failure locations recorded in the dataset include outer race failures, inner race failures, cage failures, and combined failures involving multiple components.

Table 2.

Specifications of the LDK UER204 rolling bearing 510.

Parameter	Value
Inner diameter	20 mm
Outer diameter	47 mm
Width	14 mm
Ball diameter	7.92 mm
Pitch diameter	34.55 mm
Number of rolling elements	8
Contact angle

Parameter configuration and fair comparison setup

The proposed EMD-SSA-LSTM model involves hyperparameters for both the LSTM network and the SSA optimizer. For the LSTM, a sliding window of length 5 is used to predict the next Health Indicator value, i.e., the input sequence length is 5 and the prediction horizon is 1. The network is trained with the Adam optimizer, where the initial learning rate is subject to optimization, and L2 regularization () together with gradient clipping (threshold 1.0) is applied to ensure stable training.

The SSA is employed to automatically tune three critical LSTM hyperparameters: the number of hidden units, the initial learning rate, and the number of training epochs. Their search ranges are set empirically based on preliminary experiments (e.g., hidden units in [50,200], learning rate in [0.001,0.01], epochs in [100,300])). The SSA is configured with a population size of 20 and a maximum of 50 iterations, which is sufficient for this low-dimensional hyperparameter optimization problem.

To guarantee a fair comparison among SSA, genetic algorithm (GA), and particle swarm optimization (PSO), a controlled experimental design is adopted. All optimizers are allocated an identical computational budget measured by the total number of fitness evaluations (i.e., the number of times the LSTM model is trained and validated). Accordingly, the population size and maximum iterations are unified to 20 and 50 respectively for all algorithms, resulting in 1000 evaluations each. Algorithm-specific parameters are set following common practices in the literature: for SSA, the producer ratio is and the warner ratio ³³; for GA, crossover probability 0.8 and mutation probability 0.1³¹; for PSO, cognitive and social acceleration constants with inertia weight linearly decreasing from 0.9 to 0.4³². This unified setup ensures that performance differences stem solely from the optimizers search capabilities rather than from unequal computational resources.

Simulation demonstration

This section presents the experimental results of the proposed EMD-SSA-LSTM framework for rolling element bearing RUL prediction. The experiments were conducted on the XJTU-SY bearing dataset, and the performance of the proposed method is compared with two benchmark models: GA-LSTM and PSO-LSTM. All models were trained on the first of the lifetime data and tested on the remaining . The following figures illustrate the decomposition of the raw vibration signal, the prediction accuracy on both training and test sets, the error distributions, and various quantitative metrics.

Figure 5 displays the EMD of a representative bearing vibration signal with a modal number . The original non-stationary signal is adaptively decomposed into five IMFs and a residual trend (here the residual is negligible and not shown separately). Each IMF captures oscillations at a distinct time scale, progressing from high-frequency components (IMF1) to low-frequency trends (IMF5). This decomposition isolates the physically meaningful degradation information from noise and random fluctuations. In this study, IMF3 was empirically identified as the most sensitive to the bearing’s characteristic fault frequencies and was therefore used to construct the HI. Figure 6 compares the RUL predictions of the three models on the training set. The black solid line represents the target RUL (ground truth), while the coloured dashed/dotted lines correspond to GA-LSTM (blue dashed), PSO-LSTM (green dash-dot), and EMD-SSA-LSTM (red dotted). It is evident that the proposed EMD-SSA-LSTM closely follows the target trajectory throughout the entire degradation process. In contrast, GA-LSTM exhibits moderate deviations, especially in the early and middle stages, and PSO-LSTM shows larger fluctuations, indicating inferior learning of the degradation dynamics. The corresponding training errors are plotted in Fig. 7. The error of EMD-SSA-LSTM remains tightly bounded within units (the same unit as the RUL), while GA-LSTM and PSO-LSTM produce errors that frequently exceed and units, respectively. This confirms the superior fitting capability and stability of the proposed hybrid model.

Fig. 5.

The decomposition diagram of EMD with modal number .

Fig. 6.

Training set RUL predictions.

Fig. 7.

Training set prediction errors.

The generalisation ability of the models is evaluated on the unseen test set. Figures 8, 9 display the prediction and its errors for the three models. Again, EMD-SSA-LSTM achieves the lowest and most stable error, with its error band remaining below units. GA-LSTM and PSO-LSTM exhibit larger error magnitudes and more erratic behaviour, particularly at sample indices where the degradation trend changes abruptly. Figure 10 presents scatter plots of the predicted versus actual RUL on the test set, together with the coefficient of determination . The diagonal dashed line represents perfect prediction. The EMD-SSA-LSTM model achieves the highest value, indicating a stronger linear correlation between predictions and ground truth. GA-LSTM yields an of 0.167, while PSO-LSTM produces a negative , meaning its predictions are worse than simply using the mean of the target values. This clearly demonstrates that the proposed method captures the underlying degradation trend more effectively than the benchmark optimisers.

Fig. 8.

Test set RUL predictions.

Fig. 9.

Test set prediction errors.

Fig. 10.

Scatter plots of predicted versus actual RUL (test set).

To further quantify prediction accuracy, Figure 11 compares RMSE and MAE for training and test sets across GA-LSTM, PSO-LSTM, and EMD-SSA-LSTM. The proposed method achieves the lowest errors on both datasets, especially on the test set (e.g., test RMSE is about lower than GA-LSTM and lower than PSO-LSTM), confirming enhanced generalization from EMD denoising and SSA-based optimization. Figure 12 shows boxplots of absolute prediction errors for all six cases. EMD-SSA-LSTM exhibits the smallest median error and narrowest interquartile range on both sets, indicating higher accuracy and lower uncertainty. In contrast, GA-LSTM and PSO-LSTM display wider error distributions and several outliers, reflecting less reliable predictions.

Fig. 11.

Error metrics comparison.

Fig. 12.

Boxplot of absolute prediction errors.

Conclusion

This paper proposes an EMD-SSA-LSTM hybrid framework for rolling bearing RUL prediction, which integrates adaptive signal denoising, automated hyperparameter optimization, and probabilistic sequence modeling. Experimental validation on the XJTU-SY dataset demonstrates that the proposed method outperforms GA-LSTM and PSO-LSTM benchmarks, achieving RMSE and MAE improvements of and , respectively, while enabling damage stage identification under varying operating conditions. However, this work has limitations: validation is confined to a single bearing type and dataset, field tests have not been conducted, and the computational overhead of SSA optimization hinders real-time deployment. Future work will focus on cross-domain generalization, lightweight model design for edge deployment, and field validation through industry collaboration. Among these, developing lightweight models suitable for edge deployment represents the most immediate direction toward enabling real-time industrial prognostics.

Author contributions

Qin Li and Bo Zhang conceived the experiment(s), Qin Li. and Xinxiang Fang. conducted the experiment(s), Qin Li. analysed the results. Xinxiang Fang prepared the simulation part and complete the revised draft of the paper. All authors reviewed the manuscript.

Data availability

The bearing vibration datasets analyzed during the current study are proprietary and were obtained under a specific license agreement with our industrial collaborator for use in this research project only. Therefore, they are not publicly available. However, the datasets can be made available from the corresponding author upon reasonable request and with the permission of the collaborating entity.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-41852-1.