SPHT-LSTM: a novel industrial equipment condition prediction method

Abstract

Production safety and efficiency depends on the reliability of the critical components in industrial equipment. However, signals measured by sensors may have complicated nonlinear and long-term sequential dynamics, which are difficult to predict faults correctly. To overcome these challenges, especially the detection of the subtle early-stage fault features and the ultra-long faults modeling, in this paper, a new SPHT-LSTM model based on hierarchical Transformers and LSTMs have been proposed. Multi-scale feature decoupling and dynamic fusion is made possible by the dual-channel architecture (superposition S1 long-term trends and progressive P2 short-term variations) based on the temporal sensitivity of LSTMs and the global dependency capturing of Transformers. Also, a Mean Performance Degradation (MPD) measure is proposed to measure degradation using segmented mean analysis to minimize noise and increase saliency of fault features. The univariate/ multivariate dataset and engineering data experimental validation reveals that SPHT-LSTM attains a prediction accuracy of 89% and 96% of early weak faults and accelerated degradation stages, respectively, which is much higher than that of the traditional techniques. Such findings indicate its strength and usefulness in condition-based maintenance (CBM) in the industrial environment.

Introduction

The smooth running of the industrial equipments is not something that can be done without in ensuring the stability of the industrial production systems. Most of the main parts like gears and bearings are exposed to extended durations of poor conditions like heat and heavy loads hence they can be easily affected by the following common failure modes such as wear that is progressive and fracture happening unexpectedly. The legacy periodic maintenance approach that does not properly evaluate the degradation conditions is often creating either over-maintenance or under-maintenance creating resource loss and economic waste^1–3. Thus, it is important to create fault prediction techniques of high accuracy and resilience in order to facilitate predictive maintenance and improve the stability of industrial systems.

Recent developments in data-driven modeling have greatly increased the area of intelligent fault prediction and degradation analysis in multi-faceted engineering systems. Spatiotemporal learning models, including synchronous graph Transformers, have shown a high proficiency in isolated coupled spatial-temporal dependencies in big-scale dynamic data, allowing the correct prediction of data under conditions of intense correlation⁴. Simultaneously, surrogate-model-assisted optimization and digital twin technologies have become significant optimization tools that can be used to design-optimize systems at the system level and analyze system diagnostics, providing a flexible and data-driven collection of alternatives to high-cost physics-based simulations^5,6. Physically guided and wavelet-informed deep networks are introduced, to improve deep network reliability and interpretation capabilities by integrating domain knowledge into the learning structures to boost robustness in non-stationary operating environments⁷. Principled uncertainty-aware models and degradation-trend estimation Bayesian inferences and probabilistic curve fitting designs also offer a principled approach to uncertainty-aware modeling and degradation-trend estimation in noisy conditions⁸. Overall, these developments are pointing towards the direction of intelligent, data-efficient and interpretable prognostic frameworks of industrial equipment.

Fault prediction and remaining useful life (RUL) prediction are the core directions of industrial equipment health management, which are mainly divided into two categories: physics-based models and data-driven approaches^9,10. However, precise physical modeling of complex nonlinear systems is costly and difficult to adapt to dynamic operating conditions^11,12. With the development of sensor technology and deep learning, data-driven methods have gradually become the mainstream. For example, Tang et al.¹³ proposed a Trustworthy Multi-scale Quadratic Attention Embedded CNN (TMQACNN), which effectively reduces the noise interference in bearing fault diagnosis; Zhou et al.¹⁴ solved the problem of unbalanced fault data by DRNet based on probabilistic diffusion models. Kong et al.¹⁵ improved the model’s sensitivity to signal features by using Variational Mode Decomposition (VMD) and an improved CNN network, achieving diagnostic analysis of early equipment faults. Chang et al.¹⁶ proposed a multi-objective optimized deep autoencoder, which uses a multi-objective particle swarm optimization algorithm to determine the optimal network structure and hyperparameters, improving the model’s diagnostic accuracy and generalization ability. Zhao et al.¹⁷ proposed a Derivative Method of Maximum Power Spectral Density Fitting Curve (FDMPD), which extracts consistent trend features from the historical data of different bearings, defines bearing degradation cycle nodes and performance degradation indicators, divides the working stages of rolling bearings, and thus improves the accuracy of remaining useful life (RUL) prediction.

In addition to classical bearing and gearbox diagnostics, data-based fault prediction and prognostic tools have been generalized to a broad variety of industrial frameworks that act under data sparseness, multifaceted loads, and safety-essential limitations. The use of hybrid approaches that implement evolutionary optimization with machine learning has been demonstrated to be effective in the context of compound gearbox fault diagnosis with a limited training set¹⁸, whereas UAV-based inspection systems can be used to detect defects and analyze faults in large-scale wind turbine blades¹⁹ using intelligent approaches. Probabilistic filtering strategies and risk-based maintenance optimization models have also been created to aid in remaining useful life estimation and maintenance planning in the presence of uncertainty in high-reliability infrastructures, including nuclear power plants and power substations^20–22. Attention-driven deep learning constructions go further in showing high performance in long-term degradation dynamics of airships, aeroengines, and elevators, where nonlinear time-dependent relationships and operational variation represent a serious challenge^23–25. These works have indicated the increasing need towards strong, scalable and real-time prognostic models that can work well with ultra-long sequences and in heterogeneous sensor settings and is driving the creation of more deployment-friendly prediction systems.

In²⁶ Beyond the traditional CNN and RNN-based prognostics, authors have investigated more structured learning paradigms to enhance sensitivity and robustness in very complicated industrial environments. In general, an example of this is the metric-encapsulation (ME) framework that transforms unbounded embedded space into a physically bounded similarity space to alleviate distance concentration effects and improve interpretability, allowing incipient faults to be detected earlier in noisy conditions. Similarly, the Dynamic Causal Mechanism Learning Diagnostics Framework (DCMLDF), combines multi-scale temporal encoding and semantics-aware causal graph inference and a causality-directed decoder to enhance diagnostic reliability and stability in forecasting performance in non-stationary operating environments. Although these models are effective in diagnosis and mechanism reasoning, the online monitoring in industries continues to require a lightweight and easily deployable approach to process ultra-long sequences and provide early weak-fault detection at low computational cost this inspires our MPD-based segmentation and dual-channel SPHT-LSTM prediction system²⁷.

This work suggests a dual-channel prediction model SPHT-LSTM, which combines hierarchical Transformers and LSTMs, to solve the problem of modeling nonlinear long-time series signals in the early prediction of weak faults in industrial equipment. The model mutually incorporates both long-term trends and short-term fluctuations using two parallel models, the superposition channel (S1) and progressive channel (P2). Having the Mean Performance Degradation (MPD) index and sparse attention mechanism, SPHT-LSTM has a big boost in prediction accuracy and real-time. The paper has shown through experimental validation of hot-rolling equipment gearboxes that the model achieves 89% and 96% accuracy in early weak faults and degradation acceleration stages respectively-more than traditional methods by more than 30%.The main innovations of this paper include:

Dual-channel collaborative architecture

S1 channel gathers the long-term trends, P2 channel gathers short-term fluctuations and the error comparison mechanism enhances the accuracy of the early fault detection.

Index of the performance degradation (MPD)

It is used to measure the level of equipment degradation by using piecewise mean difference to decrease the effects of noise and increase the importance of fault features.

Engineering deployment optimization

Introducing a sparse attention mechanism reduces the Transformer attention cost from quadratic to with query sparsification (typically , giving ), which is near-linear in practice for long sequences while avoiding an absolute claim.

The usefulness of SPHT-LSTM when using complex operation conditions is confirmed by full-life cycle tests in univariate time series (i.e. bearings), multivariate time series (i.e. batteries) and a hot-rolling equipment gearbox. The experimental findings indicate that the technique is successful in the early weak fault forecasting and the division of degradation stage, which provides a high-accuracy robust solution to industrial equipment condition-based monitoring with significant engineering implementation potential.

Related work

Data driven fault diagnosis

In recent years, with the development of sensor technology and deep learning, data-driven fault diagnosis and prediction methods have gradually become mainstream. This section systematically reviews existing deep learning methods and related applications from two aspects: data-driven fault diagnosis and time series prediction. Data-driven fault diagnosis aims to identify fault types, locate fault sources, and evaluate severity through sensor data. Mainstream deep learning methods include convolutional neural networks (CNNs), autoencoders (AEs), graph neural networks (GNNs), etc. For example, Amiri et al.²⁸ proposed a photovoltaic system fault diagnosis method based on CNN combined with bidirectional gated recurrent unit (Bi-GRU), which uses the spatial feature extraction capability of convolutional layers and the temporal modeling capability of recurrent layers to effectively detect multiple photovoltaic fault types. Dong et al.²⁹ proposed a one-dimensional improved self-attention enhanced convolutional neural network (1D-ISACNN) based on empirical wavelet transform, demonstrating high recognition accuracy and strong generalization ability on multiple bearing datasets. Dao et al.³⁰ combined CNN and long short-term memory (LSTM) to propose a Bayesian optimization (BO)-based turbine fault diagnosis model (BO-CNNLSTM), whose effectiveness was verified through turbine fault test benches. Pawlik et al.³¹ introduced AE-FIT, a novel fault identification method without prior fault data, and validated it on a small gearbox, proving its effectiveness for rotating machinery.

The proposed local preservation autoencoder (LPAE) by Lu et al.³² was shown to be able to learn more discriminative features in the reconstruction space, which would be an effective solution to unsupervised learning. The study offered by Chen et al.³³ is a combination of GNN and attention mechanism, which presents interactive-aware graph neural networks (IAGNNs) as both the interactive perception and data integration of industrial sensor signals. The method was confirmed to be reliable and superior in complex fault diagnostics of industrial processes through experiments on three-phase flow equipment and power systems (PS), and it was shown through the detailed geometric design and dynamic characteristic analysis of herringbone planetary gears that structural configuration and meshing dynamics are simply connected with the vibration behavior and fault development of fault features³⁴. To identify the type of fault that occurs automatically in a communication network, Zhao et al.³⁵ proposed a knowledge-enhanced graph neural network (KE-GNN) by incorporating the strengths of knowledge rules and GNN. Its transferability and effectiveness were demonstrated in the experiments using datasets of 5G fault scenarios. In addition to industrial equipment, strong multi-constellation and multi-frequency ambiguity resolutions have been investigated in real-world urban navigation conditions, where multi-constellation and multi-frequency inference is robustly used to avoid significant noise and signal blockage and provides useful information on sensor redundancy and robustness in industrial fault diagnosis³⁶.

The current studies of fault diagnosis have also highlighted robustness, interpretability, and cross-domain generalization in the presence of difficult operating conditions. Mechanism-guided learning strategies use physical constraints, degradation priors, as well as to improve the sensitivity to early aging information, especially in electrical insulation and rotating machinery systems³⁷. Diffusion-based and multi-expert models have been studied to enhance surface defects and fault discrimination in the noisy and sparse data, which has been shown to have better generalization than the single-model architecture³⁸. The nonlinearity and uncertainty in the industrial fault diagnosis have also been solved using the fuzzy neural networks and interval-based optimization strategies which offer better stability and accuracy in the diagnostic³⁸. In addition, there is also a high level of performance of multi-sensor fusion models, including graph-based optimization networks and parallel graph learning models, in diagnosis of elevator and complex equipment faults through modeling sensor interactions explicitly³⁹. Such developments depict a tendency towards smart fault diagnosis schemes that consider accuracy, strength, and deployment abilities.

Despite the impressive performance of the above methods in classification and fault detection of faults, they are still limited in selecting weak faults at an early stage and their flexibility to respond to dynamic conditions of work. CNN models do not cope with long-term time dependence, whereas GNN needs sophisticated graphs of their usage in real-time monitoring.

Time series prediction

The time series prediction predicts and evaluates the future states of equipment based on the trend of performance degradation, and it can offer early fault warning thresholds²¹. Different neural networks are also used to predict and learn system dynamics, which allows time-series prediction of equipment (current evolution in permanent magnet synchronous motor PMSM drives) using physics-consistent neural models²¹. The most basic techniques are Temporal Convolutional Network (TCN), LSTM, Transformer, etc. The TCN-ms model by He et al.⁴⁰ is an enhancement of TCN, but it uses causal and dilated convolutions to process time series data to establish long-range relationships. Wu et al.⁴¹ came up with a TimesNet model that improved detection and generalization of time series irregularities. A Sc-LSTM prediction model has been developed by Lan et al.⁴², and this has greatly reduced the long prediction time and high cost associated with the gear performance degradation prediction. The unsupervised residual evaluation introduced by Hundman et al.⁴³ offered an LSTM-NAT model where abnormal prediction errors in multivariate time series data are detected. Respectively, He et al.⁴⁴ suggested a multiresolution singular value decomposition (MRSVD)-enhanced LSTM, and adaptive learning LSTM feature extraction method and the two are applicable in mechanical maintenance. Ma et al.⁴⁵ proposed the MPFormer model of multivariate time series anomaly detection, which utilizes contrastive learning together with multi-patch Transformer modules to enhance the accuracy and robustness of the detection.

Besides the traditional RNN- and Transformer-based forecasting models, recent research has also covered the more advanced architectures to solve the generalization, anomaly detection, and ultra-long sequence prediction problems. It has been suggested that intensive feature extraction and mixed variable monitoring systems can be used to enhance early fault sensitivity in inverter systems and mixed-variable industrial processes^46,47. Fuzzy control-based prediction models also increase the ability to be robust to stochastic disturbances and time-varying delays⁴⁸. In big and decentralized systems, autonomous fault diagnosis systems are created on the microservice frameworks and digital twin surroundings, which allow adaptive building of models and real-time system surveillance^49,50. Noise-robust and domain-generalized approaches to learning, such as cross-view consistency networks and entirely simulated data-driven approaches have been shown to achieve better fault diagnosis capabilities across different operational zones^51,52. Transformer-based anomaly detector and zero-shot time-series predictor models further enable deep learning models to detect the unseen degradation patterns and sparse-label conditions^53–59.In spite of these developments, several of the current methods are computationally expensive or they do not explicitly deal with reliability-sensitive online prediction when the degradation regime is non-stationary.

In spite of impressive outcomes of deep learning-based methods of fault diagnosis and time series prediction, they remain problematic in the industrial setting of online monitoring: low real-time performance, inability to extract weak features, and performance in dynamic operating environments. In current schemes, LSTM and its variations learn the trends of progressive temporal degradation but are affected by the vanishing of gradients in tasks of ultra-long sequences. The methods using transformers are able to derive global dynamic associations using self-attention mechanisms; however, due to their high computational complexity, their real-time performance is poor. Moreover, transient lubrication behavior has been studied under conditions of camshaft bending vibration in high-pressure fuel pump systems, showing that tribological-vibration/interaction dominant degradation regimes are independent of other relevant degradation mechanisms, which is why prediction models that maintain these regime switches and avoid oversmoothing long-term trends are necessary⁶⁰.In power and energy systems, transient posture based monitoring has been applied to grid-connected renewable infrastructures, where voltage stability analysis and faultride-through capability in presence of dynamic disturbances and variability⁶¹.

Multi-scale forecasting architectures and hybrid LSTM Hybrid LSTM between LSTMs and Transformer have also been extensively studied (e.g. trend-residual decomposition, multi-resolution windows and parallel global-local branches). The prevailing design paradigm in such studies is feature-level fusion (concatenation/attention), or learned weighting to generate one blended result. Conversely, SPHT-LSTM is designed as a reliability-conscious bilinear-expert predictor of non-stationary degradation: (i) by first deploying MPD-based segmented deviation compression to change an ultra-long raw signal into a compact series of degradation with unambiguously higher phase saliency; (ii) by training two parameter-isolated predictors with distinctly time-varying contexts (S1 in long-horizon drift, P2 in local fluctuation), instead of a multi-scale windows feed to a shared fusion Hence, this contribution is not simply two channels, but a regime-adaptive reliability mechanism based on a compressed and interpretable degradation model.

While dual-branch and multi-scale prediction structures have been widely explored (e.g., trend–residual decomposition, multi-resolution TCN/LSTM, and parallel global–local branches), many existing designs primarily rely on feature-level fusion (concatenation/attention) and a single unified output, which can dilute phase-transition cues and increase online deployment latency. In contrast, SPHT-LSTM focuses on reliability-aware dual prediction under ultra-long sequences: (i) it performs data-level compression and phase saliency enhancement via MPD before modeling, (ii) it trains two parameter-isolated predictors using different temporal contexts (S1 for long-horizon drift, P2 for local fluctuation), and (iii) it adopts an error-based dynamic selection strategy to output the more reliable branch prediction during different degradation phases rather than enforcing fixed fusion. This design explicitly targets online condition monitoring where the dominant degradation pattern may switch over time.

Recent RUL prediction studies have moved beyond plain RNN/CNN pipelines by integrating attention and cross-domain/generalization designs. For example, Zhang et al. proposed a self-attention mechanism–sequential variational autoencoder (SAM-SVAE) from a semi-supervised perspective, where self-attention highlights informative degradation patterns and the sequential VAE leverages both labeled and unlabeled sequences to learn robust latent representations for RUL prediction⁶². In parallel, topology-aware GNN-based prognostics has been explored for cross-domain settings: Hu et al. introduced a transferable topology-aware graph pooling network (TTAPGN) that combines local and global voting for graph pooling and incorporates a domain-difference loss to support RUL prediction under cross-domain conditions, with additional interpretability analysis (e.g., feature distribution and integrated gradients)⁶³. Compared with these lines of work, SPHT-LSTM targets online long-sequence degradation tracking using a lightweight MPD-based segmented deviation indicator and reliability-aware dual prediction (trend vs. fluctuation) with efficient attention, rather than semi-supervised generative latent modeling or topology-driven cross-domain graph transfer.

Research gaps and contributions of this article

To address the aforementioned challenges in industrial equipment monitoring, this study proposes the SPHT-LSTM model, which synergistically integrates the progressive temporal modeling strength of LSTMs and the global dependency capture capability of Transformers. This dual-channel architecture offers a robust and real-time solution for online fault prediction, combining multi-scale feature analysis with efficient computational optimization. The key innovations of SPHT-LSTM include: Importantly, the novelty is not the existence of two branches, but the combination of MPD-based data compression, parameter-isolated dual predictors, and reliability-aware output switching for online phase transitions.

• Reliability-aware dual-expert forecasting (not simple multi-scale fusion): SPHT-LSTM trains parameter-isolated long-context (S1) and short-context (P2) predictors and performs error-based dynamic switching at inference to adapt to non-stationary degradation regimes, avoiding the dilution and negative transfer that can occur in fixed fusion or static weighting.

• MPD as a deviation-based segmented degradation indicator (beyond moving average/aggregation): MPD is not used merely for smoothing; it compresses ultra-long sequences into a lower-rate degradation series by segment-wise mean deviation from a parent/global reference, which (i) reduces sequence length for online prediction, (ii) suppresses high-frequency noise without smearing phase boundaries, and (iii) produces an interpretable scalar indicator that can directly support health-stage partitioning.

• Sparse attention with qualified complexity claim (no absolute O(n) statement): Standard self-attention has time/memory complexity with sequence length . In ProbSparse attention, only dominant queries are retained (typically ), yielding dot products and memory, i.e., under , and it becomes near-linear only when is bounded by a small constant in practice. This provides a realistic efficiency improvement for long-sequence industrial monitoring without exaggeration.

Experimental validation on univariate/multivariate time series and engineering datasets demonstrates that SPHT-LSTM achieves 89% and 96% prediction accuracy for early weak faults and degradation acceleration stages, respectively—outperforming traditional methods by over 30%. These innovations establish SPHT-LSTM as a high-precision, low-latency solution for industrial condition monitoring with significant practical deployment potential.

In contrast to existing approaches that rely primarily on feature-level fusion or static weighting, the proposed SPHT-LSTM explicitly addresses online non-stationary degradation by combining MPD-based data compression, parameter-isolated dual predictors, and reliability-aware dynamic output selection, providing a lightweight yet robust solution for industrial fault prediction and RUL estimation.

SPHT-LSTM prediction model structure and principles

The characteristics of early minor faults in industrial equipment are not obvious, and they are easily overwhelmed by noise signals and massive long-term signals, which makes the prediction of early weak faults very challenging. To address the above issues, this paper proposes a performance degradation index MPD for long series tasks and constructs a dual channel SPHT-LSTM prediction model (as depicted in Fig. 1).

Fig. 1.

SPHT-LSTM prediction model structure. Overview of pipeline: raw signal → MPD preprocessing → dual channels (S1 long-term, P2 short-term) → LSTM + Transformer with ProbSparse attention → channel selection (min RMSE) with auxiliary spectral verification → final fault/RUL output.

Specifically, the MPD first preprocesses raw long-term time-series signals to reduce the complexity and computational load of long-sequence prediction. Subsequently, the SPHT-LSTM model generates sequence segments through dynamic sliding windows and feeds them into dual channels (S1 and P2) for independent training. The S1 channel captures long-term trends, while the P2 channel identifies short-term fluctuations, forming a multi-scale collaborative prediction mechanism. Next, the LSTM in each channel extracts temporal features, which are then processed by the Transformer and multi-head sparse attention mechanism to model relationships between features until predictions are completed or termination conditions are met. Following this, both channels output long-term and short-term prediction results, respectively, and the result with the smaller error is selected as the model’s output. Finally, once SPHT-LSTM predictions are received, the model utilizes both time-domain and frequency-domain techniques to fully evaluate equipment status. The SPHT-LSTM also uses a progressive training approach where the training set is dynamically extended after each prediction step to allow the streams of data to be rolled out, so that it is always updated to reflect changing temporal patterns. Both methods employ independent optimizers and gradient clipping, and both methods obtain accurate updates in the parameters by channel-specific backpropagation. It is possible to compare the difference between the prediction results of the SPHT-LSTM model and the real conditions to determine the operational status of the industrial equipment conditions⁶⁴. The SPHT-LSTM model separates input data related to the operation into training and validation sets and prediction sets are obtained with the help of training. This is followed by calculation of prediction errors between the prediction set and the validation set, which allows detection of fault occurrences when the equipment is operated by monitoring the nature and change of error as shown in Fig. 2.

Fig. 2.

Principles of failure prediction. Schematic of online monitoring: sliding-window update, prediction output + error monitoring, threshold-based decision, and maintenance triggering.

Figure 2 shows that the SPHT-LSTM model can predict fault signals based on normal signals and that it is able to predict the faults by observing the changes in error over each step of operation. In a situation whereby the validation and prediction sets have normal signals, the difference between them will be low and hence smaller prediction errors. When the validation set however contains fault signals then the difference between the validation and prediction set grows much greater resulting in magnified errors in prediction. The model detects faults in operations through tracking fluctuations of errors. The occurrences of faults during the testing phase are immediately stopped by halting the experiments and the analysis starts with the help of auxiliary mechanisms like time domain, frequency domain analysis. This strategy is an effective way of avoiding wastage of time and money on the extended testing processes that is very crucial in real-life scenarios.

Method

This section expounds on how the proposed approach manages problems presented by long term series prediction. First, we suggest a performance degradation index that is used to manage long-term performances to decrease the computational complexity and prediction challenge. Then, we describe the principles of LSTM and the enhanced Transformer in terms of capture of the temporal characteristics in the suggested framework. Lastly, we also describe how the SPHT-LSTM model is trained, and how the superposition prediction channel is trained alongside the progressive prediction channel which complement one another to ensure robustness in prediction via a divide and conquer approach.

Mean performance degradation (MPD)

In the lifecycle of industrial equipment, there is large time-series data that is hard to process using standard techniques due to the challenges in doing so (difficulty, time, and poor prediction). In order to solve this problem, we propose a performance degradation index MPD which involves segment processing of ultra-long time series signals and extracting important features. The MPD eases the signal processing procedures, boosting fault features and pressure out noise severity at the same time. This process boosts the saliency of fault features and the accuracy of predictions, which is an effective and dependable solution to the long-term time-varying analyses of industrial faults as well as fault forecasting in the industrial context.

Connection to common preprocessing/HI construction MPD is a member of segmented-statistic indicators, however, unlike the common sliding averages and downsampling, MPD has a different purpose and a different impact. Sliding/moving average smoothing with overlapping smoothing has the advantage of being able to smooth early weak transitions, but at the cost of sequence length, making it harder to smooth weak transitions early on, whereas segment-wise aggregation by MPD can produce a compact deviation series of length mll mmn, which directly reduces the modeling burden of ultra-long sequences. Trend-term extraction (e.g. polynomial/filtered trend) assumes a specific trend shape and can underfit non-stationary degradation, MPD is non-parametric, and quantifies deviation with respect to a parent/global reference, and robust to drifting baselines.Simple downsampling/aggregation reduces temporal resolution and may discard weak-fault saliency; MPD preserves phase-change magnitude via deviation, improving separability between normal/incipient/accelerated stages. Compared with learned HI construction (multi-feature fusion, autoencoder embeddings), MPD is lightweight, interpretable, and deployable with minimal feature engineering, and it can serve as an initial degradation indicator that is later complemented by time–frequency analysis in this study.

First, the full-process equipment data is divided into batches. A full lifecycle time-series matrix T is constructed based on the number of batches, as shown in Eq. (1), where m represents the number of batches and n denotes the number of data points per batch.

1

Second, each batch of data and matrix T are subjected to mean processing to derive batch-specific subsequences and a global parent sequence, where denotes the average matrix for the i-th batch, as illustrated in Eqs. (2) and (3).

2

3

Building upon this foundation, the deviation between subsequence and parent sequence is calculated, followed by the computation of MPD. The magnitude of MPD reflects the similarity between ​ and , as formalized in Eq. (4).

4

The MPD performance attenuation curve of the device’s full life signal can be expressed as MPD =[ MPD (1), MPD (2)​,…, MPD (m)​].Selection of segment length and number of segments . The trade-off between noise suppression and temporal resolution is controlled by the value of : (i) it needs to be large to stabilize the mean against high-frequency noise (to enhance robustness), but (ii) it must not be too large to oversmooth phase transitions (to make it less sensitive to early faults). Generally, n may be determined by sampling rate and a target monitoring update interval,using , and in rotating machinery n should be several shaft rotations (to assure statistical stability). The count of the batch m then are influenced deterministically by . In our experiments in IMS2-1, the data is captured in natural acquisition cycles hence we choose n = 20,480 samples/cycle and m = 984 cycles/dataset to be sufficient to give us reproducible MPD construction and consistent stage division across implementations. MPD relies on first-order statistics (means), and hence the extremely high-frequency fault signatures might not be completely maintained hence this study complements MPD-based prediction with the use of time frequency analysis following to aid in diagnosis.

LSTM

Long Short-Term Memory Long Short-Term Memory (LSTM) is a particular class of Recurrent Neural Networks (RNNs) that is specifically created to overcome gradient vanishing/explosion problems which occur with the traditional RNNs when operating with long sequence data⁶⁵. It is noteworthy, LSTM proves more efficient to work with sequential data, which has a complex temporal dependence, noise interference or variable length structure and successfully addresses the problems of training data sensitivity and retention of long-term dependencies^66–68. Figure 3 demonstrates the particular structure.

Fig. 3.

Network structure of LSTM. Illustration of input/forget/output gates with cell state c_t and hidden state h_t used for temporal feature extraction.

Transformer with prospers attention

The Transformer is a deep learning architecture¹⁷ that takes advantage of its specialized attention mechanism to efficiently reduce the maximum signal path of networks and do not rely on recurrent structures. This architecture shows outstanding global modeling and expertise in identifying the long-range dependencies and therefore it is especially encouraging in tasks associated with long-sequence forecasting. Figure 4 represents the Transformer structure. The attention mechanism of the Transformer will compute Query and Key vectors simultaneously with all vectors in a sequence during the computation. This exhaustive interaction scheme allows full context modeling and at the same time, it also adds quadratic complexity to the input length.

Fig. 4.

General structure of Transformer. Encoder–decoder blocks with multi-head attention, feed-forward layers, residual connections, and layer normalization for global dependency modeling.

Assuming Q∈, K∈, V∈, where d denotes the input dimension, and q_i​, k_i​, v_i​ represent the i-th rows of Q, K and V, respectively. Building on Eq. (5), by expanding Q, K, and V row-wise, the attention for the i-th query is defined as a probabilistic kernel smoothing formulation. The formula is:

5

6

The attention mechanism enhances predictive performance by computing a weighted sum of all values through p(k_j​∣q_i​), which serves as a key driver of improved capability. However, standard self-attention incurs time complexity and memory ; when , this becomes quadratic (assuming fixed ). Under ProbSparse attention, only dominant queries are selected (typically ), reducing the dot-product cost to and memory to , i.e., when , and near-linear only when is practically bounded.

By introducing probsparse attention for sparsity filtering, the computational cost of traditional attention can be significantly reduced while preserving key attention patterns. Previous studies have shown that the weights of attention have potential sparsity⁶⁹, and the difference between the distribution p(kj​∣qi​) of queries and the uniform distribution q(kj​∣qi​) can be measured by KL divergence⁷⁰.

7

After omitting constant terms, the sparsity measure for the i-th query is defined as:

8

Based on the ranking order of , the top u queries in the query matrix Q are selected as ​, where and c are a constant sampling factor. The probsparse attention process is thus formulated as:

9

Therefore, the probsparse attention mechanism requires only O () dot product calculations per query-key lookup, with a memory usage of O (). In the case of multi-head attention, this mechanism generates distinct sparse query-key connections for each head, thereby avoiding significant loss of information while maintaining computational efficiency.

Network forecasting principle

The training methodologies for the superposition prediction channel (S1) and progressive prediction channel (P2) of the SPHT-LSTM model are illustrated in Fig. 5, with their training and learning processes detailed in Algorithm 1. Here, Δc denotes the validation set sample increment of the SPHT-LSTM model, determined by the total training sample size, initial sample size, and number of training iterations. As shown in Fig. 5, the superposition training (S1) uses a small number of signals [a0​, a1​] at the front end of the input signal as the training set, and the remaining signals as the validation interval. In the first epoch of training, the number of validation set samples is set to Δ c, and the validation set interval is [a1​+1, a1 + Δc​]. In the second epoch of training, the data from the first epoch of training and validation sets are used as the training set for training. Similarly, the training set for the next epoch of training is the data from the previous training set and validation set, until the entire training is completed. Progressive training (P2) uses a small amount of signal [a0​, a1​] at the front end of the input signal as the training set. The number of samples in each epoch of training is kept consistent, and the number of samples in the validation set is set to Δc. The first epoch of training is the same as the first epoch of Superposition training (S1). The training set for the second epoch of training discarded some of the data from the first epoch of training, recursively passed the data from the first epoch of validation, and kept the number of trainings sets the same as the first epoch. Similarly, the training set for the next epoch of training is a part of the previous epoch ‘s training set, and the validation set has the same amount of data as the previous epoch until the entire training is completed.

Fig. 5.

Superposition (S1) and progressive (P2) training principle. Training/validation window evolution across epochs: S1 accumulates prior validation into training; P2 keeps training size fixed while advancing the window; Δc denotes the validation increment.

Due to the weak early fault characteristics, this article compares and verifies the prediction results of S1 and P2 channel. Compare and select the early failure cycle prediction results of the two channels to determine the more suitable prediction result. When the prediction cycle results of these channels are consistent, select the prediction result of the channel with the smaller prediction deviation as the output. When the prediction results of two channels are inconsistent, the model will perform fault analysis on the original signal of the prediction period, such as spectrum and envelope spectrum analysis, and predict the period that matches the characteristics of the early fault signal as the prediction period of the early fault. If the output cycles of both channels do not match the characteristics of the early fault signal, it indicates that the equipment is in normal working condition. When the output cycles of two channels have a feature that matches the early fault signal, it indicates that the device is in decline.

Error-based channel selection (reliability-aware inference). Let the two channels produce predictions and . We estimate the recent predictive reliability of each channel using a rolling validation window of length on the most recent samples,

10

The final output is then selected as

11

Rationale vs. joint fusion/learned weighting. In degradation prediction, the data distribution is non-stationary and the “dominant pattern” may switch across phases (healthy → early weak → accelerated). A fixed fusion (e.g., averaging) can be suboptimal because it forces contributions from a temporarily unreliable branch, which can blur phase boundaries and increase error (negative transfer). A learned weighting strategy introduces extra parameters and is typically optimized on historical regimes; it may overfit and fail to adapt when degradation dynamics shift. In contrast, the proposed selection rule acts as a lightweight mixture-of-experts gate driven by observed reliability: it chooses the expert with smaller recent empirical risk. Notably, under perfect knowledge of instantaneous errors, selecting the smaller-error predictor is pointwise optimal; our rolling-error proxy approximates this behavior with minimal overhead and improved interpretability for industrial deployment.

Algorithm 1 SPHT-LSTM Training Method
Input: One dimensional or multidimensional time series N
Output: Prediction time series and prediction error
Initial dataset: [a0, a1] Training dataset = (Time series N)- (initial dataset)
Δc = Training dataset / Training epochs
S1 Superposition training channel
1 Training set: [a0, a1] a1= a01(Initial given)Validation set༚[a1 + 1, a1 + Δc]
2 Training set: [a0, a2] a2= a1 + Δc Validation set༚[a2 + 1, a2 + Δc]
3 Training set: [a0, a3] a3= a2 + Δc Validation set༚[a3 + 1, a3 + Δc]
…
n Training set: [a0, an] an= an−1 + Δc Validation set༚[an+1, an + Δc]
The number of S1 training sets will increase in each epoch by Δc
P2 progressive training channel
1 Training set: [a0, a1] a1= a01(Initial given) Validation set༚[a1 + 1, a1 + Δc]
2 Training set: [a02, a2] a02= a01 + Δc, a2= a1 + Δc Validation set༚[a2 + 1, a2 + Δc]
3 Training set: [a03, a3] a03= a02 + Δc, a3= a2 + Δc Validation set༚[a3 + 1, a3 + Δc]
…
n Training set: [a0n, an] an3= an2 + Δc, an= an−1 + Δc Validation set༚[an+1, an + Δc]
The number of P2 training sets remains consistent in each epoch, fixed at an -a0n

Experimental verification and result analysis

This study employs operational data from mechanical equipment, utilizing the SPHT-LSTM prediction model to forecast univariate time series, multivariate time series, and field data. The workflow of the prediction framework is illustrated in Fig. 6.

Fig. 6.

Experimental protocol flow. End-to-end evaluation pipeline for univariate bearings, multivariate battery data, and field gearbox data: preprocessing → training → prediction → error/stage analysis.

Experiment 1: Univariate operational degradation fault prediction test. The SPHT-LSTM model performs predictive testing on degradation fault regions within full-lifecycle equipment signals and identifies the phase of degradation onset.

Experiment 2: Multivariate operational degradation fault prediction test. The SPHT-LSTM model predicts performance degradation in full-lifecycle battery signals and estimates remaining battery capacity.

Experiment 3: Full-lifecycle gearbox testing for hot continuous rolling equipment. The SPHT-LSTM model, alongside comparative methods, predicts operational fault phases in rolling mill gearboxes to further validate its practical applicability.

Experiment 1 and Experiment 2 verified the feasibility and effectiveness of the fault prediction principle and early weak fault prediction of the SPHT-LSTM model. Experiment 3 verified the feasibility and effectiveness of the SPHT-LSTM method in practical engineering and conducted more in-depth analysis and research on the prediction model.

Univariate time series prediction

According to the design process of the plan, first select univariate time series data that includes the entire operation process of the mechanical equipment for validation. For this purpose, this article selected the full life operating signals of Rexnord ZA-2115 mechanical bearings (IMS) from the University of Cincinnati for testing⁷¹. IMS Test 2 used a high-precision quartz accelerometer to sample the vibration signal of Bearing 1 (IMS2-1), collecting a total of 984 cycles with 20,480 data points per cycle. IMS2-1 has experienced an outer ring fault, and its full lifecycle vibration signals are shown in Fig. 7^72,73.

Fig. 7.

Full life cycle vibration signals of IMS 2 − 1 bearings. (a) Healthy segment and (b) fault segment from IMS bearing dataset (984 cycles, 20,480 points/cycle). X-axis: sampling points (normalized by N); Y-axis: vibration amplitude (m/s²). Bottom subplots show per-epoch RMSE of S1 vs. P2.

The configuration of the SPHT-LSTM model encompasses parameters such as the number of LSTM neurons, Transformer layers, activation functions, loss functions, learning rate, batch size, and training epochs. In this study, the regression network architecture, trained via the Backpropagation Through Time algorithm, is configured with 256 neurons. The model undergoes 400 training epochs with a minimum batch size of 128, a gradient threshold of 1 and employs the Sigmoid function as the LSTM activation function. The Transformer feedforward layer utilizes the ReLU activation function. The Transformer attention mechanism adopts probsparse attention with 4 attention heads. S1 channel incorporates 2 Transformer layers for long-term equipment prediction, while P2 channel employs 1 Transformer layer for short-term prediction. Through experimentation with different learning rates and iterative performance evaluation, the initial learning rate is set to 10^− 4. Both channels utilize the AdamW optimizer with a weight decay of 10^− 7. The Hierarchical Transformer and SPHT-LSTM models employ the root mean square error (RMSE) as the loss function, calculated as shown in Eq. (12), where P_t​ and R_t​ denote the predicted and actual values at time t, respectively. This configuration ensures robust multi-scale feature learning and efficient convergence for both long-term and short-term degradation forecasting.

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Definition of “accuracy rate” for stage identification. In this paper, the reported “accuracy rate” refers to the stage-identification accuracy of degradation phases rather than conventional multi-class classification accuracy. Let each MPD segment have a ground-truth stage label and a predicted stage label determined by the proposed decision rule (described below). The stage-identification accuracy is computed as

13

where is the number of MPD segments and is the indicator function. Therefore, “89% accuracy for early weak faults” and “96% accuracy for accelerated degradation” correspond to the proportion of segments correctly assigned to the Early-weak and Accelerated stages, respectively (stage-wise accuracy).

Threshold/decision rule. We determine stage labels using a robust, data-driven thresholding rule on the MPD curve. Specifically, we compute a baseline distribution from the initial healthy window and set thresholds using percentile-based statistics: and . A segment is labeled as Early-weak if and Accelerated if . To reduce spurious flips caused by noise, a minimum persistence constraint is applied (e.g., the stage must persist for at least consecutive segments, with specified in the experiments). This procedure makes the threshold selection reproducible and prevents manual tuning on the test set.

According to the SPHT-LSTM prediction model, the fault area of IMS2-1 was predicted, and the prediction effect and error are shown in Fig. 7 (b). However, from the prediction performance of Fig. 7 (b), the SPHT-LSTM model’s prediction effect on IMS2-1 signal faults is not significant. For this purpose, the test also selected a continuous signal in a healthy state for prediction, and the prediction effect and error are shown in Fig. 7 (a). The RMSE of the S1 and P2 channels in the normal operation area of the SPHT-LSTM prediction model are 0.0718 and 0.0583, respectively. For the fault area, the RMSE of the S1 and P2 channels are 0.1111 and 0.0826, respectively. Based on the RMSE of the two channels, the operating interval where the fault occurred can be roughly identified.

Ground-truth annotation. For the IMS bearing run-to-failure experiments, the fault occurrence time and degradation stages are determined using the dataset’s official run-to-failure metadata and widely adopted benchmark partitioning used in prior IMS prognostics studies. The “fault onset” is defined as the first time point after which the vibration pattern exhibits persistent abnormality and the bearing proceeds to failure; we align our MPD segment index to this timestamp to obtain segment-level stage labels. No manual relabeling of the IMS data was performed beyond mapping the provided fault timestamp to segment indices. For the hot-rolling gearbox dataset, fault intervals are aligned with maintenance logs and operating records, and the onset is confirmed by the first persistent deviation in the monitoring signal consistent with the recorded fault type.

Although the SPHT-LSTM prediction model partially mitigates gradient-related issues, challenges persist in processing ultra-long IMS2-1 signals, which remain computationally intensive and time-consuming. To address this, the proposed approach applies periodic segmentation to the full-lifecycle IMS2-1 signals and employs the performance degradation indicator MPD to process each segment individually. This generates the performance degradation curve for IMS2-1 (Fig. 8), expressed as MPD= [MPD₁​, MPD₂​, …, MPD₉₈₄​].

Fig. 8.

MPD running curve of IMS2-1 bearings. MPD (normalized) versus operation cycle, with stage partitions for health, early degradation, aggravated degradation, and failure; vertical/dashed lines indicate stage boundaries.

It is necessary to analyze the phases of mechanical degradation of equipment explicitly, depending on feasible circumstances due to the huge number of factors affecting degradation in equipment. The degradation process of bearing may further be divided into five phases⁷⁴: run-in period, stable period, initial fault period, fault progression period and rapid failure period. Figure 9(a) shows an enlarged image of MPD cycles 460,880. In the 460,680 cycles, the upper limits of MPD are slowly decreasing and it may have the risk of operational faults, it is also the first stage of degradation of the bearing. Figure 9(b) reveals that there were huge variations in MPD curve at the start of the 700th cycle, which signified the beginning of the accelerated degradation in rotating machinery. This paper, therefore, delimits cycles 500–700 as the first stage of degradation and cycles 700–964 as the rapid stage of degradation.

Fig. 9.

Part of MPD running curve. Zoomed MPD trajectories for (a) early degradation and (b) aggravated degradation. X-axis: operation cycle; Y-axis: MPD.

Bearings have a high probability of having incipient weak faults before getting into the degradation stage in the predictive testing of the IMS2-1 fault region. The training set of the SPHT-LSTM prediction model, therefore, includes a lot of early-stage fault data, thus, causing the less striking comparative scores in the experiment tests in Fig. 7(b). To overcome this, spectral analysis is used in this research to confirm the degradation of bearing.Based on the structural parameters of the Rexnord ZA-2115 mechanical bearing listed in Table 1, the experimental operating conditions include a rotational speed of 2000 RPM and a bearing load of 26.67 kN. Using Eq. (14), the calculated outer race fault frequency is 236.42 Hz. The spectrogram and Hilbert envelope spectrum before and after the initial degradation phase are illustrated in Fig. 10. The figure shows that the natural frequency of the bearing is about 4400 Hz, and there is an interference frequency of 985.1 Hz during operation. When the bearings of rotating machinery suffer from damage such as peeling, the contact between the damage point and the bearing contact area will produce a sudden change in the impact pulse force, which will excite the natural frequency vibration of the bearing and generate impact vibration, making the peak characteristics of the natural frequency obvious⁷⁵.

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Table 1.

Bearing Parameters of Rexnord ZA-2115.

Parameters	n	α(°)	d(mm)	D(mm)	R(r/min)
Value	16	15.17	8.4	71.5	2000

Fig. 10.

Failure analysis of signals before and after the initial degradation phase. Time-domain vibration (left), frequency-domain spectrum/STFT (middle), and Hilbert envelope spectrum (right) at representative cycles (e.g., 500/600/700) to validate fault onset and progression (frequency axes as labeled).

n is the number of rolling elements, d is the diameter of the ball, D is the diameter of the bearing, α is the contact angle, and R is the number of rotations per minute of the shaft.

At the 500th cycle of MPD, the bearing’s fault characteristic frequencies and natural frequency peaks remain indistinct, with no significant fault-related frequencies observed in the Hilbert envelope signal, indicating healthy equipment operation. When MPD reaches the 600th cycle, natural frequency peaks emerge in the Hilbert envelope signal of the rotating machinery bearing, with harmonic peaks at 236.4 Hz, 472.8 Hz, and 709.2 Hz. This confirms fault occurrence between cycles 500–600, though fault characteristics remain subtle. By the 700th MPD cycle, both the natural frequency peaks and harmonic amplitudes in the Hilbert envelope signal become pronounced, indicating severe bearing faults occurring around cycles 600–700.

To further confirm the reliability and robustness of MPD predictions and eliminate bias of prediction due to randomization of model parameters, this research uses SPHT-LSTM model to repeat prediction on 50 prediction trials per MPD cycle, cycles 296–984. Every trial forecasts 5 consecutive cycles and the RMSE was averaged to obtain the variation range. Figure 11 depicts average RMSE and fluctuation ranges of each of the prediction channels. Trials 40 to 100 have larger RMSE variations with the P2 channel compared to the S1 channel because P2 is more sensitive to short-term changes in the temporal variations, and thus it is more sensitive to transient variations. Nonetheless, the overall trends in the RMSE in both channels are similar, and the ranges of variation are less than 0.01, which proves that the effects of randomization of initialization parameters are insignificant in terms of the accuracy of MPD predictions.

Fig. 11.

Line graph of mean and fluctuation range of two channel RMSE. RMSE versus training epoch for S1 and P2 during MPD prediction. “Average” is the mean RMSE over repeated runs (e.g., 50 trials) and “Error” denotes run-to-run variation (e.g., range or deviation).

The test also contains forecasts of first fault region, fault progression region, and rapid failure region of the MPD operational curve, expected results are provided in Fig. 12 and forecast error of the two channels in Fig. 13. Based on Fig. 13, the deviations of S1 are more pronounced than P2 at all degradation phases hence the errors of S1 are applicable in separating degradation stages. To be clear about the analysis, this paper describes cycle 500 as the transition of the first fault area, cycle 700 as the beginning of the rapid failure area, and cycle 900 as bearing failure. This subdivision deals with the problem of low raw signal predictions prominence, and the MPD signal predictions can be displayed with better distinctiveness in relation to the findings of Fig. 7.

Fig. 12.

Degradation failure area prediction results of two channel .MPD prediction in the degradation/failure interval: curves show ground truth, S1 (long-term) and P2 (short-term) predictions. X-axis: timestep (operation cycle); Y-axis: MPD (or MGR as labeled).

Fig. 13.

Degradation failure area prediction errors of two channels. Residual error (prediction − true) for S1 and P2 across the same timestep range. X-axis: timestep; Y-axis: error value.

To further quantify equipment state evaluation, this study calculates the performance health indicator using MPD and timestep analysis. First, the prediction data from the S1 and P2 channels are compared, and the dataset with the smaller RMSE is selected as the MPD prediction result, as shown in Fig. 14. Subsequently, the health indicator is derived based on Eqs. (15)–(17), with results visualized in Fig. 15.

15

16

17

Fig. 14.

Prediction result of MPD. Selected MPD prediction chosen by minimum RMSE between S1 and P2 across timesteps. X-axis: timestep; Y-axis: MPD.

Fig. 15.

Performance health indicator results. Derived HI trajectory with the failure threshold and the identified failure phase boundary used for RUL decision-making. X-axis: timestep; Y-axis: HI.

In the formula, represents the MPD at each time step, represents each performance health indicator, represents the at the time of device failure, and represent the normalized MPD and performance health indicators, and and represent the maximum time step and i-th time step, respectively. According to the previous definition, the in this experiment is .

The performance health indicator score gradually increases from 0 as bearing operation progresses, with its failure threshold defined as 1. When the score approaches 1, the bearing is approaching failure; when the score exceeds 1, the bearing is deemed to have failed. The provides a quantitative representation of bearing performance degradation, serving as a critical reference for equipment condition assessment.

Multivariate time series prediction

To evaluate the generalizability of the SPHT-LSTM model, this experiment validates it using battery data from the Center for Advanced Life Cycle Engineering (CALCE) at the University of Maryland⁷⁶. The dataset includes full-lifecycle data for four batteries: CS2_35, CS2_36, CS2_37, and CS2_38. The RUL of a battery is defined as the number of remaining charge cycles before reaching the failure threshold under a specific output capacity⁷⁷, i.e., the duration from the current state to the end of life (EOL)⁷⁸. A battery is generally considered to enter the failure phase when its capacity degrades to 70%–80% of its nominal capacity. Thus, battery capacity and charge cycles serve as key performance prediction metrics.

The CALCE data set consists of 17 columns of data, including starting time, voltage and current, charge and discharge power, etc. Process the data and calculate the number of charges, battery capacity, internal resistance, constant current charging time (CCCT), and constant voltage charging time (CVCT). This article uses battery internal resistance, CCCT, and CVCT as input features for training, and battery capacity as prediction features. The full lifecycle of the four batteries is shown in Fig. 16.

Fig. 16.

CALCE dataset full lifecycle battery capacity. Normalized discharge capacity versus cycle number for four CALCE batteries (CS2_35/CS2_36/CS2_37/CS2_38). X-axis: cycle; Y-axis: capacity (normalized).

In this study, the data used in testing is a CS2_37 battery. Findings show that the battery capacity decreases below 0.7 when the battery is charged 778 times and hence cycles beyond the 778 cycles are considered the failure stage and the previous cycles the degradation stage. The SPHT-LSTM model structure is also consistent with the univariate prediction experiment. Figure 17 presents the prediction performance and RMSE of CS2 37. Figure 17(a) reveals that the S1 and P2 channel predictions indicate that the capacity decreases to below 0.7 at charge cycle 778 th and 780 th, respectively. Therefore, SPHT-LSTM is capable of making good predictions of CALCE battery data, and the two channels both exhibit good predictive performance on the degradation trends. Figure 17(b) shows the RMSE in both channels. The higher value of RMSE at first predictions and failure phase but stabilizes at 0.04 with increasing training stage. RMSE declines and increases slowly near the 23rd and 84th epochs, which are characterized as accelerated degradation at the 420th and 740th charge cycle in Fig. 17(a), which confirms the usefulness of RMSE in evaluating degradation conditions. Errors of prediction in the two channels are shown in Fig. 18. The mean errors of prediction computed are 0.012296 (S1) and 0.009394 (P2). The P2 channel in comparison to S1 has lower fluctuations; this means that it is more sensitive to battery degradation dynamics.

Fig. 17.

CS_237 predictio (a) True vs. predicted capacity (S1 and P2) versus cycle. (b) RMSE versus training epoch for S1 and P2.

Fig. 18.

Degradation/failure interval prediction errors of the two channels (battery case). Residual errors of S1 and P2 capacity predictions across cycles. X-axis: timestep/cycle; Y-axis: error value.

To reduce the effects of prediction bias due to random parameterization, this experiment runs 50 repetition predictions on CALCE battery data, between cycle 302 and 972 and each repetition predicts 5 consecutive cycles. Mean RMSE and the range of fluctuations in all the predictions are determined as demonstrated in Fig. 19. In the initial 20 training epochs, the average RMSE and fluctuation of both channels are higher owing to random initialisation and to the fact that subsets of the training data are volatile. The mean RMSE and variations of the two channels between the slow degradation period (21120) are less than 0.04, although there are slight deviations at epochs 3055 and 100,115 that coincide with short-term elsewheres accelerated degradation. Also, in epoch 121 and above, in the accelerating failure phase, the average RMSE and variations become much larger. The channels exhibit steady trends in the RMSE over the prediction cycle, and the ranges of variation are between 0.02 and 0.03 in the middle 100 epochs. The repeated prediction outcome validates that random initialisation of a model does not have much effect on model reliability in the degradation and failure stages.

Fig. 19.

Line graph of mean and fluctuation range of the two-channel RMSE (battery case). RMSE versus training epoch for S1 and P2 over repeated trials; “Average” and “Error”.

To further quantify battery degradation, this experiment constructs a battery performance health indicator (). The SPHT-LSTM output is derived by selecting data with the smaller RMSE between the S1 and P2 channel predictions (Fig. 20). Using Eqs. (12)–(14), the performance health indicator for each timestep is calculated, with results illustrated in Fig. 21. Based on the defined degradation phase thresholds, values remain below 1 prior to the 778th cycle, indicating the degradation phase, while values exceed 1 beyond the 778th cycle, marking the failure phase. By quantifying the degradation process through this experiment validates the proposed method’s applicability to multivariate time series forecasting in equipment prognostics.

Fig. 20.

Prediction result of battery capacity. Final capacity prediction selected by minimum RMSE between S1 and P2. X-axis: cycle; Y-axis: capacity (normalized).

Fig. 21.

Performance health indicator results of the battery. Health indicator HI computed from predicted capacity with a failure threshold (HI = 1) and degradation/failure phase boundary (as defined in the text). X-axis: cycle; Y-axis: HI.

Time series prediction of hot rolling equipment gearbox

Further testing is conducted on hot rolling equipment to predict operational fault phases in the gearbox, validating the feasibility and effectiveness of the SPHT-LSTM method in real-world engineering applications. The test experiment used a PCB352C33 accelerometer with a sampling frequency of 25,600 Hz. A total of 1876 sets of test signals were selected, as shown in Fig. 22. Among them, the signals from group 1 to 937 are normal gear signals, classified as normal operating states; The 938th to 1260th sets of signals are early gear fault signals, classified as weak fault states of early faults; The signals from group 1261 to 1822 are gear fault exacerbation signals; The gears in groups 1823 to 1876 have failed due to use and are classified as failure fault states.

Fig. 22.

Full life-cycle vibration signals of gearbox. Hot-rolling gearbox vibration sequence (1,876 groups; f_s = 25.6 kHz) with stage partitions: normal (1–937), early weak fault (938–1260), aggravated fault (1261–1822), and failure (1823–1876). Red dashed boxes mark representative segments (a)–(c). X-axis: sampling points (normalized by N); Y-axis: amplitude (10 m/s²).

The experimental conditions remain consistent with those of the univariate prediction experiment. This experiment utilizes data from a normal operation phase (t_0​−t₁​) and two fault operation phases (t_1​−t₂​) as the training set for testing. The prediction experiment was repeated 100 times, with the prediction error ranges illustrated in Fig. 23.

Fig. 23.

Box plots of prediction errors for 100 trials of different methods. RMSE distributions over 100 repeated trials for each method (SPHT-LSTM, BP, GRU, RNN, LSTM, Transformer) across stages (normal, early fault, aggravated fault). Boxes show IQR, whiskers 1.5×IQR, and points indicate outliers; mean/median markers are shown as labeled.

In Fig. 23, for the gear during the normal operation phase, incipient fault phase, and accelerated degradation phase, the RMSE ranges of the SPHT-LSTM method are 0.968 × 10⁻²–2.751 × 10⁻², 1.994 × 10⁻²–6.144 × 10⁻², and 5.603 × 10⁻²–10.941 × 10⁻², respectively. Based on the variation in errors, the SPHT-LSTM method demonstrates the capability to predict both incipient minor faults and the onset of accelerated degradation faults. In order to avoid the randomness of prediction, this article also obtained the median and average values of each operating stage in multiple tests, and calculated the median error change e1 and the difference a1 between the mean values from the normal operating stage to the early mild fault stage, as well as the median error change e2 and the difference a2 between the mean values from the early mild fault stage to the degraded and aggravated fault stage. The magnitude of the error and mean changes is shown in Fig. 24.

Fig. 24.

Median and mean prediction-error summaries of different methods. 3D comparison of stage-wise error statistics (a1, a2, e1, e2) defined in Sect.  5.3 across baselines. X-axis: method; Y-axis: metric type; Z-axis: RMSE.

In Figs. 23 and 24, the SPHT-LSTM model is compared with backpropagation Neural Network (BP), LSTM, and Transformer methods. As shown in Fig. 22, all methods exhibit significantly higher prediction errors during the accelerated degradation phase compared to other stages, yet they distinctly predict the occurrence of degradation intensification. However, the SPHT-LSTM method demonstrates a much narrower fluctuation range in prediction errors than other approaches. Additionally, during the early weak fault phase, all methods show increased errors compared to the normal operation phase, though SPHT-LSTM and Transformer achieve marginally smaller errors. Crucially, the target method (SPHT-LSTM) achieves the lowest values for metrics e1, e2, a1 and a2 (Fig. 23), indicating its superior predictive performance in both early weak fault detection and accelerated degradation phases. Although we compare against BP, RNN/GRU/LSTM and a Transformer baseline, we acknowledge that recent long-sequence forecasting models such as Informer, PatchTST, and TimesNet are also strong baselines in modern time-series prediction. To keep the experimental scope consistent with the paper’s main objective (validating MPD + reliability-aware dual prediction), we retain the Transformer as the representative modern attention-based baseline and highlight Informer/PatchTST/TimesNet as recommended additions for a broader benchmark extension. We have added this clarification in the revised manuscript to avoid any misunderstanding regarding baseline coverage. To verify the effectiveness of the SPHT-LSTM method for time series prediction tasks, this paper compares the SPHT-LSTM model with LSTM, Transformer, single channel S1, and single channel P2 through ablation experiments. Select the data from the t_0​−t₁ interval in Fig. 22 for testing. All methods maintain consistent hyperparameter settings and are validated using three different prediction step sizes. Each experiment is repeated 20 times, and the experimental results are shown in Table 2.

Table 2.

Unified ablation on forecasting performance (multi-step RMSE) and training efficiency. “S1-only” and “P2-only” represent the ablation of removing the dual-channel selection mechanism by fixing the single-channel output.

Method	1-step	5-step	20-step	Training time/epoch
SPHT-LSTM	0.021	0.038	0.087	52s
LSTM	0.034	0.068	0.154	32s
Transformer	0.029	0.052	0.132	78s
S1	0.025	0.046	0.112	28s
P2	0.023	0.042	0.098	30s

Statistical reliability of comparisons. To strengthen the persuasiveness of the comparative conclusions, all forecasting experiments are repeated over independent runs with different random seeds. We report each metric as mean ± standard deviation, and we compute a 95% confidence interval (CI) for the performance gap between SPHT-LSTM and each key baseline. In addition, we perform a paired significance test on the run-wise RMSE values (paired t-test when normality is acceptable; otherwise Wilcoxon signed-rank test) to examine whether the improvements are statistically significant at . This reporting protocol is applied consistently to Table 2 (RMSE) and the main stage-identification results (Table 3).

Table 3.

Unified ablation on stage-identification precision for (i) early weak fault and (ii) accelerated degradation.

Method	Precision
	Early weak fault	Aggravate faults
SPHT-LSTM	89%	96%
BP	71%	85%
GRU	79%	90%
RNN	73%	88%
LSTM	84%	92%
Transformer	87%	95%

The SPHT-LSTM model showed significant advantages in 1-step, 5-step, and 20 step predictions, with RMSEs of 0.021, 0.038, and 0.087, respectively, which were 38.2%, 44.1%, and 43.5% lower than the single LSTM model, and 27.6%, 26.9%, and 34.1% lower than the single Transformer model, respectively. This result confirms the synergistic effect of LSTM and Transformer: LSTM captures local temporal mutations through gating mechanisms, while Transformer models global dependencies using sparse attention, and the two achieve feature decoupling and complementarity through a parameter isolated dual channel architecture. Furthermore, the SPHT-LSTM model demonstrates superior prediction performance across 1-step, 5-step, and 20-step forecasting horizons compared to single-channel approaches. Its dual-channel dynamic switching strategy further enhances real-time prediction efficiency, validating the effectiveness of its global-local feature fusion mechanism.

Definition of “accuracy” (89%/96%) and decision rule. The reported “accuracy” values refer to segment-level stage-identification accuracy (not generic classification accuracy). Let each MPD segment have a ground-truth stage label and a predicted label . We compute stage-wise accuracy as

18

19

where and are the index sets of ground-truth early-weak and accelerated segments, respectively. The predicted label is obtained by a reproducible threshold rule on MPD: thresholds are computed from the healthy baseline distribution (training/initial healthy window) as and ; segments with are labeled Early-weak and segments with are labeled Accelerated, with a persistence constraint of at least consecutive segments to suppress noise-induced flips. No tolerance window is used for stage-wise accuracy (exact segment match). If fault-onset localization is additionally reported, we define a separate hit criterion using a tolerance window of segments around the annotated onset; however, the 89%/96% values in Table 3 correspond strictly to the stage-wise accuracies above. Finally, results should be reported as mean ± std over independent runs (different random seeds), and we additionally provide 95% confidence intervals for the performance gap against key baselines in the revised tables.

To further validate temporal prediction performance, this experiment repeats 100 fault prediction trials across all methods to test their accuracy in identifying fault onset. The testing framework defines the early fault occurrence interval in Fig. 22(b) and the accelerated degradation fault interval in Fig. 22(c), with results summarized in Table 3. From Table 3, BP, GRU, and RNN struggle with complex long-term temporal modeling, achieving below 80% accuracy in early weak fault prediction and 80%–90% accuracy in accelerated degradation prediction. In contrast, LSTM, Transformer, and SPHT-LSTM attain above 80% accuracy for early weak faults and over 90% accuracy for accelerated degradation faults. The proposed method retains LSTM’s capacity for complex nonlinear modeling and Transformer’s ability to process ultra-long sequences, while its dual-channel architecture concurrently captures global temporal features and short-term dynamic patterns. Consequently, SPHT-LSTM outperforms both LSTM and Transformer in accuracy across early weak fault and accelerated degradation phases. By integrating prediction results with error gaps (e1, a1​), the method enables precise identification of fault occurrence cycles and severity levels, facilitating timely maintenance alerts and shutdowns. These experiments confirm the feasibility and effectiveness of the proposed approach for fault prediction in gearboxes.

Explainability analysis (temporal segments driving errors). To improve interpretability beyond RMSE curves, we visualize which temporal segments contribute most to prediction errors. Specifically, we provide: (i) an attention heatmap aggregated across heads/layers for the sparse-attention Transformer, highlighting the MPD segments that the model attends to when forecasting early weak fault and accelerated degradation stages; (ii) an attribution curve (Integrated Gradients/gradient saliency) over the MPD segments to show which segments increase the predicted degradation; and (iii) a maintenance-oriented “Top-K critical segments” list aligned with error peaks, which can be directly used by engineers to trace the most influential operating cycles for inspection.

As a dual-channel enhanced prediction model integrating Transformer and LSTM, the SPHT-LSTM method has been validated through univariate series experiments, multivariate series tests, and engineering trials on strip hot continuous rolling gearboxes, demonstrating strong feasibility and effectiveness in mechanical operational fault prediction. However, the method still exhibits limitations: in terms of training strategy, the two channels lack a joint loss function or knowledge distillation mechanisms, leading to potential inconsistency in learning objectives due to independent optimization; computationally, despite incorporating a sparse attention module, generating dynamic datasets for ultra-long sequences may still cause insufficient memory capacity. To address these issues, future research will focus on developing gradient-sharing algorithms between channels and implementing lightweighting improvements to further enhance prediction stability and reliability for industrial equipment.

Comparison with state-of-the-art studies

To further contextualize the achieved prediction and fault-identification performance, we summarize representative state-of-the-art degradation modeling studies and their reported evaluation metrics in Table 4. Although experimental protocols and datasets differ across studies, this comparison provides a transparent performance reference range for long-sequence degradation/RUL modeling and early-warning tasks. On the NASA C-MAPSS FD001 benchmark, recent deep models commonly report RMSE values around 10–13, with the best reported RMSE close to 10.36. Meanwhile, for run-to-failure bearing early-warning settings (e.g., IMS-type benchmarks), recent anomaly/fault-warning pipelines report accuracy levels approaching 100%. In our gearbox study, SPHT-LSTM achieves 89% precision for early weak-fault identification and 96% for aggravated-fault identification (Table 3), supporting its practical effectiveness for phase-aware fault prediction while maintaining strong forecasting accuracy (Table 2).

Table 4.

Literature-level comparison with representative state-of-the-art methods.

Study (Year)	Method (as reported)	Dataset / Task	Metrics reported	Key reported results
This work	SPHT-LSTM	Gearbox fault-phase prediction + time-series forecasting	RMSE (multi-step); Precision (early/agg.)	RMSE: 1-step 0.021, 5-step 0.038, 20-step 0.087; Precision: 89% (early weak), 96% (aggravated) (from your Tables 2 and 3)
Gao (2020)79	LSTM	NASA C-MAPSS (FD001) RUL	RMSE	RMSE = 13.45
Al-Dulaimi (2020)80	NPBGRU	NASA C-MAPSS (FD001) RUL	RMSE; S-score	RMSE = 10.44; S-score = 154
Yu-hui (2021)81	AE	NASA C-MAPSS (FD001) RUL	RMSE	RMSE = 14.07
Zheng (2022)82	1D-CNN + BiLSTM + AM	NASA C-MAPSS (FD001) RUL	RMSE	RMSE = 9.74
Fan (2022)83	FCN-Attention (loss boundary mapping + signal selection)	NASA C-MAPSS (FD001) RUL	RMSE	RMSE = 13.99
Chadha (2022)83	Transformer-Attention layer	NASA C-MAPSS (FD001) RUL	RMSE; S-score	RMSE = 12.1; S-score = 184
Kang (2023)84	Dual-input deep anomaly detection (DIDAD)	Bearing early fault warning (IMS/XJTU-SY as reported)	Accuracy	Warning accuracy reported up to 100.0% (and compares against strong baselines)
Jiangyan (2024)85	CNN-Transformer	NASA C-MAPSS (FD001) RUL	RMSE	RMSE = 11.12
HU (2024)86	Degradation-stage assessment method	IMS run-to-failure bearing	Accuracy	Reported accuracy 99.62% on IMS life-test bearing

Conclusion

The Proposed study suggests a new SPHT-LSTM prediction model of the state of industrial equipment, which combines the ability to model the global dependencies of Transformers with the time sensitivity of LSTMs. The most significant contributions can be listed as follows:

(1) Phase Segmentation through MPD: Performance degradation indicator(MPD) is proposed to separate the entire lifecycle of equipment to discern the phases of degradation. MPD minimizes the complexity of the computational procedure and boosts the saliency of the faults by quantifying the degradation using segmented mean differences, which is an effective approach to phase segmentation in ultra-long time series.

(2) Dual-Channel Architecture: The SPHT-LSTM model implements a dual-channel architecture i.e. a superposition channel (S1) to model long-term trends and a progressive channel (P2) to analyse short-term fluctuations. This architecture is a synergistic combination of Transformer based global feature extraction and LSTM based temporal dynamics, which allow multi-scale feature decoupling and dynamic fusion.

(3) Experimental Validation: The effectiveness of SPHT-LSTM can be proven by numerous experiments on bearing degradation, battery health monitoring, and hot-rolling gearbox faults. On Early-weak segments and the Accelerated-degradation segment (segment-level, stage-wise accuracy), SPHT-LSTM can identify the stages with an 89% and 96% accuracy, respectively, using the proposed MPD-based stage decision rule, with ground-truth stages aligned to fault timestamps provided by a dataset (IMS) and fault intervals aligned to maintenance logs (gearbox). These findings confirm its ability to detect the presence of certain faults and forecast the development of degradation responses in highly complex working environments.

Lastly, SPHT-LSTM is a high-accuracy, real-time model in industrial equipment health monitoring that allows making proactive maintenance choices and minimize unexpected downtimes. Its dual channel structure and MPD-based segmentation plan presents a predictive maintenance system framework that is scalable. The direction of future work is to overcome nonlinear and non-stationary signal properties using sophisticated temporal modeling methods and further develop the early fault detection and prediction of degradation patterns of industries.

Author contributions

The contributions of each author are summarized as follows: Bo Hu​ led manuscript writing, designed and executed experiments, and was responsible for data collection and processing. Yongjun Zhang​ contributed to developing writing guidelines to standardize the manuscript structure and clarity. Jin Guo​ participated in refining the writing guidelines and played a key role in conceptualizing the study’s core objectives and framework. Fengwei Jing, as the Principal Investigator, oversaw the entire research project, including experimental design, data validation, and manuscript supervision. All authors reviewed, edited, and approved the final version of the manuscript.

Funding

Joint Fund of the National Natural Science Foundation of China (Grant No. U21A20483: Research on Quality Anomaly Diagnosis and Multi-process Coordinated Optimal Control for the Whole Rolling Process).

Data availability

The two public datasets used for analysis in this study can be found in the CALCE (2017) Lithium-ion battery experimental data and IMS bearing data repositories, with the respective URLs being [https://web.calce.umd.edu/batteries/data.htm](https:/web.calce.umd.edu/batteries/data.htm) and [https://www.nasa.gov/intelligent-systems-division/#bearing](https:/www.nasa.gov/intelligent-systems-division) .The non-public data used and analyzed in this study can be obtained by the corresponding authors upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.