The workflow of the proposed probabilistic RUL prediction framework is illustrated in Fig. 2. The figure clearly depicts the end-to-end data flow, starting from the input of raw multi-sensor time-series data, followed by data preprocessing and sliding window segmentation, bidirectional feature encoding via the BiLSTM layer, calculation of key time-step weights by the attention mechanism layer, output of deterministic predictions through the fully connected layer, activation of Monte Carlo Dropout for multiple forward samplings to model uncertainty, and finally fusion of all sampling points using KDE to generate a probability density distribution. This distribution provides a basis for risk quantification in decision-making.

To address the challenge of predicting the RUL of equipment, a deep fusion network incorporating a BiLSTM architecture is proposed, with the aim of enhancing the extraction of temporal features and complex patterns. The core idea leverages the bidirectional temporal modeling capability of BiLSTM to fully capture contextual relationships in time-series data through the integration of both forward and backward directions. This mechanism overcomes the limitations of traditional unidirectional LSTM in modeling long-range dependencies and significantly improves the capture of dynamic correlations within time-series data. The workflow is illustrated in Fig. 3.

BiLSTM has demonstrated strong capabilities in capturing sequential dependencies. The computational process for the forward module of the model is given as follows [3]: (1)i→(t)=σ(ω→iix→t+ω→hih→(t−1)+bt→) (2)f→(t)=σ(ω→ifx→t+ω→hfh→(t−1)+b→f) (3)g→(t)=tanh⁡(ω→igx→t+ω→hgh→(t−1)+b→g) (4)o→(t)=σ(ω→iox→t+ω→hoh→(t−1)+b→o) (5)c→(t)=f→(t)⊙c→(t−1)+i→(t)⊙g→(t) (6)h→(t)=o→(t)⊙tanh⁡c→(t)

Similarly, formulation of the backward part of the BiLSTM albeit with a reversed temporal sequence: (7)i←(t)=σ(ω←iix←t+ω←hih←(t−1)+b←t) (8)f←(t)=σ(ω←ifx←t+ω←hfh←(t−1)+b←f) (9)g→(t)=tanh⁡(ω→igx→t+ω→hgh→(t−1)+b→g) (10)o←(t)=σ(ω←iox←t+ω←hoh←(t−1)+b←o) (11)c←(t)=f←(t)⊙c←(t−1)+i←(t)⊙g←(t) (12)h←(t)=o←(t)⊙tanh⁡c←(t)

In Eqs. (1)–(12), i(t), f(t), g(t), and o(t) represent the input gate, forget gate, candidate cell state, and output gate at time t, respectively; h(t) and c(t) denote the hidden state and cell state at time t, respectively; σ denotes the sigmoid function, ⊙ denotes the Hadamard product, w_ii, w_hi, w_if, w_hf, w_io, w_ho, w_ig, w_hg are learnable weights, b_i, b_f, b_g, b_o are bias terms, → indicates the forward direction, and ← indicates the backward direction. The above equations collectively constitute the core computational mechanism of the BiLSTM network. Through a series of differentiable control structures—including the input gate, forget gate, and output gate—the network adaptively manages the state of its internal memory cells. The forget gate f(t) determines how much information to retain from the historical memory cell c(t − 1) based on the current input and the previous hidden state; a value closer to 1 indicates more memory retention. The input gate i(t) works in conjunction with the candidate state g(t) to decide how much new information from the current input should be updated into the memory cell c(t). Finally, the output gate o(t) controls how much information from the current memory cell c(t) is output as the hidden state h(t) and passed to subsequent layers. By processing sequences along both forward and backward time axes, the BiLSTM can integrate contextual information at each time step.

The BiLSTM integrates two oppositely oriented LSTM layers to capture long-term temporal dependencies from bidirectional data. After processing through multiple LSTM layers, the model employs data smoothing to filter out noise interference from the raw input signals, thereby facilitating more accurate extraction of temporal features and improving the precision of RUL prediction. The architecture of the model is illustrated in Fig. 4, and its final output is expressed as follows:

(13)h(t)=h→(t)⊕h←(t)

The Additive Attention Layer, a classical implementation of the attention mechanism originally proposed by Bahdanau et al. in 2015 (also referred to as Bahdanau attention), computes the relevance between queries and keys through learnable parameters. Its core idea involves mapping queries and keys to the same dimension via separate linear transformations, followed by an additive operation and a nonlinear activation function to derive the attention weights [29]. For a given input sequence {x1,…,xt}, this encoder produces a forward hidden state sequence {h1→,…,ht→} and a backward hidden state sequence {h1←,…,ht←}. The encoded representation of any element x_j is formed by concatenating its corresponding forward and backward hidden states. The attention mechanism contains an alignment model that measures how well the annotation h_j from the input suits the hidden state s_i−1 from the output. The calculation formula is as follows: (14)eij=vaTtan⁡h(Wasi−1+Uahj)where Wa∈Rm×m and Ua∈Rm×2m are learnable weight matrix, va∈Rm is a weight vector used to compress to a scalar, and m is the dimension of the attention hidden layer.

The weights are calculated by applying the softmax operation to the alignment scores, and they represent the probability that the input x_j is aligned with the output y_i. The weight distribution can be obtained by normalizing the scores with softmax. (15)αij=softmax(eijm)

As output of the attention mechanism, the context vector is calculated as the annotations h_j: (16)Coni=∑j=1Nαijhj

The model in Eq. (14) learns to evaluate the relevance between the hidden state h_j (at the j-th time step of the input sequence) and the prediction target for the current prediction task. Subsequently, these scores are normalized into a probability distribution αij via the Softmax function in Eq. (15). Finally, as shown in Eq. (16), the hidden states of all time steps are weighted and summed according to their attention weights, generating a context vector c_ij that condenses the key information of the sequence.

The innovation of the proposed prediction framework lies in the synergistic mechanism between residual learning and stochastic regularization. The residual pathway enables cross-layer propagation of raw signals through parallel linear transformations, whose mathematical essence establishes a high-speed channel for gradient flow. Experimental results demonstrate that this design effectively mitigates the gradient vanishing problem when handling long-term temporal dependencies in sensor signals. Meanwhile, LayerNorm standardizes the activation values of neurons, maintaining the data distribution of each layer’s output within a stable range. Since each forward propagation effectively samples from the posterior distribution of network weights, the predictive distribution obtained through multiple sampling iterations naturally incorporates model epistemic uncertainty. This elegantly introduces Bayesian inference into conventional neural networks [30] and enables probabilistic outputs without altering the base architecture. The selection of the KDE method is primarily based on two considerations. First, as a non-parametric method, KDE does not require prior assumptions about the distribution shape of predictions, enabling more flexible fitting of complex distribution patterns (e.g., skewed or multimodal distributions) that may occur in real-world data. Second, it offers high computational efficiency, making it particularly suitable for combination with Monte Carlo Dropout sampling and thus more feasible and practical for engineering applications.

Theoretical analysis indicates that the generalization capability of this architecture stems from the synergy of multiple regularization mechanisms: Dropout provides a model averaging effect in deep networks, LayerNorm maintains feature distribution stability, and residual connections ensure optimization feasibility. Each forward pass utilizes a randomly generated Dropout mask, effectively sampling a distinct sub-network from the learned weight posterior. This process yields a discrete set of T prediction points, capturing the epistemic uncertainty arising from model parameters. Residual connections are incorporated after the second BiLSTM layer, with their outputs summed to those of the attention layer to mitigate gradient vanishing in deep networks. Stochastic regularization is implemented via Dropout layers applied after each BiLSTM layer and the first fully-connected layer, with a dropout rate of 0.2 determined through grid search.

For practical decision-making, key quantiles can be easily derived from the probability density distribution to define a predictive confidence interval. By evaluating this failure risk, strategies like condition-based maintenance can be optimized to balance the costs of unexpected downtime against those of premature component replacement. The main workflow is illustrated in Fig. 5.

Following the approach in reference [31], in Bayesian learning, for a given input x*, obtaining its predictive distribution requires marginalizing over all model parameters: (17)f(y∗|x∗,X,Y)=∫f(y∗|x∗,ω)f(ω|X,Y)dωwhere ω={N^k}k=1L denotes the set of random variables for an L-layer model.

To approximate the predictive distribution, T sets of vectors are sampled from a Bernoulli distribution with probability {p1,…,pL}, enabling empirical estimation of the first two moments: (18)Ef(y∗|x∗)(y∗)≈1T∑t=1Ty^∗(x∗,N^kt,…,N^Lt)

Monte Carlo estimator is referred to as MC-Dropout. In practice, it is equivalent to performing T forward passes through the deep network and averaging the outputs. We provide a novel derivation for this result, which allows us to derive well-founded uncertainty estimates. Empirically, the integration can be approximated by averaging the network weights (i.e., scaling each W_i by p_i at test time). Similarly, for regression tasks, the predictive mean and variance can be computed as: (19)Ef(y∗|x∗)((y∗)T(y∗))≈ς−1ID+1T∑t=1Ty^∗(x∗,N^1t,…,N^Lt)Ty^∗(x∗,N^1t,…,N^Lt) (20)Varf(y∗|x∗)(y∗)≈ς−1ID+1T∑t=1Ty^∗(x∗,N^1t,…,N^Lt)Ty^∗(x∗,N^1t,…,N^Lt)−Ef(y∗|x∗)(y∗)TEf(y∗|x∗)(y∗)where the predictive uncertainty is estimated via the sample variance over T forward passes plus the inverse model precision, denoted by ς. Since the model precision is often related to the ratio between learning rate r₁ and weight decay r₂, we define ς=r1/r2.

This prediction framework utilizes multiple forward passes with Dropout enabled, performing N predictions for each model in the ensemble to approximate Bayesian inference. The final prediction is obtained by aggregating all sampled outputs through averaging, while the resulting probability density distribution reflects the model’s predictive uncertainty. Theoretical studies have demonstrated that residual connections can enhance the optimization behavior of the network. Dropout can be interpreted as a Bayesian approximation of Gaussian processes, with its key advantage lying in the provision of predictive distributions rather than point estimates—this is crucial for risk assessment in RUL prediction applications.

In engineering practice, when applying KDE, an ensemble strategy is adopted: each of the 10 base models performs 100 independent predictions, resulting in a total of 1000 predictive samples. These predictions are processed via KDE to generate probability distribution curves of the RUL. Research indicates that under large-sample conditions, the choice of kernel function has limited influence on the shape of the probability density function. The mathematical formulation can be expressed as follows: (21)f^(x)=1n∑i=1n1hκ(x−xih)=1n∑i=1n1h2πe−12(x−xih)2

In the Eq. (21), κ represents the Gaussian kernel function, x−xih denotes the normalized distance variable, and h is the bandwidth parameter, which controls the smoothness of the kernel. We adopted the commonly used Scott’s rule to adaptively determine the optimal bandwidth, which provides excellent smoothing for data approximating a normal distribution. The specific formula is h=n−1d+4, where n denotes the number of samples and d represents the dimension.

Unlike deterministic RUL prediction models, the proposed framework employs KDE to model the probability density distribution of the RUL prediction, thereby enabling uncertainty quantification—an essential feature for supporting maintenance decision-making by management.

To validate the effectiveness of the proposed RUL prediction method, time-series data from 21 sensors in the C-MAPSS dataset [32] were utilized, with the aircraft engine model illustrated in Fig. 6. These sensors capture measurements such as temperature, pressure, and rotational speed from various locations within the engine. By simulating turbofan engine degradation under diverse conditions and faults, the dataset captures complex performance dynamics. The significant variation in complexity and fault types among its subsets makes them an ideal testbed for a thorough evaluation of model adaptability and generalizability. Detailed information on the four subsets of C-MAPSS is provided in Table 2. As shown in the table, FD002 and FD004 are more complex than the other subsets, containing more training and testing trajectories and involving six different operating conditions. Following the setup in [4], sensor outputs that remain constant throughout the lifecycle—providing no useful information for RUL prediction. Thus, 14 sensor signals were selected for both training and testing, specifically from sensors numbered 2, 3, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17, 20, and 21.

In accordance with [4], a sliding window of size 30 was applied to segment the sensor data. For sequences shorter than 30 time steps, the segmented data were padded using the first available value. To direct the model’s focus toward the critical degradation phase near failure, the RUL labels in early stages were truncated to a constant value. A sliding step of 1 was adopted to maximize training data utilization by generating a large number of overlapping windows, thereby enriching the training samples. Data normalization, which has been shown to be crucial in RUL prediction, was performed using min-max scaling on all sensor readings, with the maximum RUL value set to 125.

The hyperparameter configuration of the proposed deep fusion network reflects a carefully balanced design tailored for time-series regression tasks. The input window length was set to 30 time steps, sufficiently capturing engine degradation trends without introducing excessive noise. The corresponding 14 input features represent the most predictive sensor indicators retained after preprocessing. The hidden architecture comprises two LSTM layers, each with 50 units, providing adequate memory capacity to learn temporal dependencies while controlling model complexity to avoid overfitting. Subsequent fully connected layers consist of 96 and 128 neurons, respectively. This progressively expanding structure facilitates hierarchical feature abstraction, enabling the extraction of high-level patterns from sequential encodings for accurate prediction.

The Adam optimizer was selected for its ability to adaptively adjust the learning rate without manual scheduling. The MSE loss function was used to heavily penalize large prediction errors. An early stopping mechanism with a patience of 10 epochs was employed to terminate training promptly when validation loss ceased to improve and restore the optimal weights, effectively preventing overfitting. A learning rate decay mechanism was also adopted, reducing the learning rate to one-tenth of its original value if no improvement was observed after 5 epochs. For model training, the batch size was set to 32 and the number of training epochs to 100, with all experiments conducted under the TensorFlow framework. Training was performed on a workstation equipped with an NVIDIA RTX 3080 GPU, using an initial learning rate of 0.001 and a learning rate scheduling strategy. The complete training process took approximately 1–1.5 h, with slight variations across different dataset subsets.

To validate the effectiveness and reliability of the proposed model, the following evaluation metrics were employed: Root Mean Square Error (RMSE) and Score function. 1.

RMSE: This metric measures the deviation between predicted values and actual values. It is defined as follows: (22)RMSE=1N∑k=1N(Δk)2

In Eqs. (22) and (23), Δk=RUL^k−RULk, N represents the number of samples, while RUL^k and RULk denote the predicted and true values of RUL, respectively.

This section presents a comparative analysis between the proposed BiLSTM-based fusion network and several state-of-the-art methods referenced in Table 3, including LSTM-BS [17], MCLSTM [18], GAT [19], RNN-AE [20], BiGRU-TSAM [21], and BLCNN [22]. The BiLSTM-based fusion network demonstrates superior performance over most comparative methods across multiple subsets. Notably, it achieves significant improvements in both RMSE and Score metrics on the more challenging and complex subsets, FD003 and FD004.

To verify the statistical significance and robustness of the proposed method, we conducted systematic replicate experiments. For the four subsets of the C-MAPSS dataset, the model was independently run under identical experimental settings using 10 different random seeds (42, 123, 456, 789, 999, 111, 222, 333, 444, 555). Each experiment included a complete workflow of data preprocessing, model training, and evaluation to ensure strict consistency of experimental conditions. The significance test analysis of the proposed method is presented in Table 4, which includes the mean, coefficient of variation, and 95% confidence interval of the RMSE and Score metrics, providing a comprehensive evaluation of the model’s robustness across different datasets.

To further analyze the RUL prediction accuracy on the C-MAPSS dataset, visualizations of the prediction results for all four subsets are provided in Figs. 7 and 8. Fig. 7 illustrates the overall prediction performance across all test engines within each subset, while Fig. 8 depicts the results for individual engines. As shown in the subplots of Fig. 7, a noticeable deviation between the predicted and actual RUL values is observed when the true RUL is high. However, as the true RUL approaches zero, the predicted and actual curves converge, demonstrating improved alignment and smoother fitting. The overall trends captured in Fig. 8 reveal that the predicted curves consistently follow the true RUL trajectories, particularly during the critical degradation phase of the engines. This indicates the model’s capability to accurately reflect the gradual decline in RUL and capture the underlying degradation trends. These results visually underscore the robustness of the model in scenarios involving rapid health deterioration, highlighting its practical suitability for providing timely and accurate early warnings in industrial fault prediction systems.

In addition, the probability density prediction distribution for individual engine lifetimes within four subsets was visualized, as shown in Figs. 9–12. The results demonstrate that the proposed method achieves accurate RUL predictions under various operating conditions and scenarios, with prediction errors consistently lower than those of the traditional LSTM model. The probability density curves exhibit a Gaussian distribution, and the predicted RUL means show minimal deviation from the actual values. From a maintenance perspective, the shape and spread of these distributions are highly informative. For instance, in Fig. 10a,b, the narrow and sharp distribution indicates high prediction confidence, suggesting that maintenance can be scheduled with high certainty around the mean RUL. The proposed framework not only maintains a low mean squared error but also generates probability distribution curves, which are unavailable through traditional method. This advancement shifts the prediction paradigm from “point estimation” to “risk quantification,” highlighting the value of probabilistic prediction in maintenance decision-making.

The model exhibits two main types of RUL prediction bias: (1) slight overestimation during early-to-mid operation due to weak degradation signals being masked by noise. Although the attention mechanism aims to focus on key information, the model struggles to extract strongly predictive early degradation patterns from complex multi-sensor signals when degradation features are not yet prominent. (2) occasional significant underestimation near failure end for units with abrupt performance drops, likely due to insufficient training data on such rapid degradation patterns. This is likely because training data contains relatively few examples of rapid failure, preventing the model from adequately learning the dynamics of fast degradation.

To quantitatively evaluate the individual contribution of each key component in the proposed framework—namely the Attention mechanism, Residual Connections, and the KDE-based probabilistic framework, a comprehensive ablation study was conducted. The objective is to isolate the impact of these components on the model’s overall prediction performance and its uncertainty quantification capability.

The ablation experiments were performed on four subsets under identical experimental settings (e.g., hyperparameters, training-testing split) as the full model. We designed three model variants for comparison: Variant A (w/o Attention), Variant B (w/o Residual), and Variant C (w/o KDE). The results of the ablation study are summarized in Table 5.

In conclusion, the attention mechanism enables temporal feature selection, residual connections ensure stable training of deep networks, and the KDE framework provides essential uncertainty quantification for risk-conscious decision-making. Their synergistic integration in the full model is justified by its superior and robust performance across all evaluation scenarios.