Turbofan Engine Remaining Useful Life Prediction with Reliable Prediction Intervals via LSTM-Based Quantile Regression and Conformal Calibration.

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Learn more: PMC Disclaimer | PMC Copyright Notice Sensors (Basel) . 2026 Apr 5;26(7):2249. doi: 10.3390/s26072249 Search in PMC Search in PubMed View in NLM Catalog Add to search Turbofan Engine Remaining Useful Life Prediction with Reliable Prediction Intervals via LSTM-Based Quantile Regression and Conformal Calibration Runsheng Diao Runsheng Diao 1 School of Automation, Shenyang Aerospace University, Shenyang 110136, China; [email protected] (R.D.); [email protected] (S.W.) Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization Find articles by Runsheng Diao 1 , Mingzhe Zhou Mingzhe Zhou 1 School of Automation, Shenyang Aerospace University, Shenyang 110136, China; [email protected] (R.D.); [email protected] (S.W.) Conceptualization, Methodology, Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition Find articles by Mingzhe Zhou 1, * , Guanglei Meng Guanglei Meng 1 School of Automation, Shenyang Aerospace University, Shenyang 110136, China; [email protected] (R.D.); [email protected] (S.W.) Validation, Resources, Writing – review & editing, Supervision, Funding acquisition Find articles by Guanglei Meng 1 , Shanze Wang Shanze Wang 1 School of Automation, Shenyang Aerospace University, Shenyang 110136, China; [email protected] (R.D.); [email protected] (S.W.) Software, Data curation Find articles by Shanze Wang 1 Editor: Fanglai Zhu 1 Author information Article notes Copyright and License information 1 School of Automation, Shenyang Aerospace University, Shenyang 110136, China; [email protected] (R.D.); [email protected] (S.W.) * Correspondence: [email protected] Roles Runsheng Diao : Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization Mingzhe Zhou : Conceptualization, Methodology, Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition Guanglei Meng : Validation, Resources, Writing – review & editing, Supervision, Funding acquisition Shanze Wang : Software, Data curation Fanglai Zhu : Academic Editor Received 2026 Mar 9; Revised 2026 Mar 28; Accepted 2026 Apr 3; Collection date 2026 Apr. © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license . PMC Copyright notice PMCID: PMC13075270  PMID: 41978034 Abstract To overcome the inability of point estimates to adequately characterize uncertainty and the unstable coverage of prediction intervals in turbofan engine remaining useful life (RUL) prediction, this study proposes an LSTM-based quantile regression framework (LSTM-QR). The framework generates a point prediction together with upper and lower predictive bounds in a single forward pass, thereby directly constructing a prediction interval with a nominal coverage of 80%. During training, a weighted pinball loss and an overestimation penalty are introduced to improve the robustness of quantile estimation. During inference, Conformalized Quantile Regression (CQR) is further applied for post hoc interval calibration. Experiments on the NASA C-MAPSS dataset show that the proposed method maintains stable point-prediction performance while substantially improving interval reliability after calibration. Under the same operating condition, PICP increases from 0.590 ± 0.035 to 0.800 ± 0.026 for FD001 → FD001 and from 0.722 ± 0.050 to 0.793 ± 0.032 for FD002 → FD002, corresponding to gains of 21.0 and 7.1 percentage points, respectively, with calibrated RMSE values of 16.235 ± 1.297 and 18.323 ± 0.411. Under cross-condition transfer, where the raw intervals exhibit clear under-coverage, CQR further raises PICP from 0.696 ± 0.046 to 0.806 ± 0.032 for FD001 → FD002 and from 0.593 ± 0.071 to 0.803 ± 0.021 for FD002 → FD001, corresponding to gains of 11.0 and 21.0 percentage points, respectively, while preserving RMSE at 21.758 ± 1.208 and 17.562 ± 0.062. These results indicate that the proposed method provides more reliable and interpretable prediction intervals under varying operating conditions, thereby offering effective support for predictive maintenance decision-making. Keywords: LSTM, quantile regression, uncertainty quantification, CQR, remaining useful life prediction 1. Introduction With the widespread use of turbofan engines in aviation, marine, and industrial systems, unplanned downtime caused by equipment failure can substantially increase maintenance costs and reduce system availability. Predictive maintenance (PdM) identifies signs of degradation before failure through continuous condition monitoring and trend analysis, thereby enabling maintenance to be scheduled proactively. Compared with corrective or fixed-interval maintenance, PdM can reduce the risk of unexpected failures while avoiding unnecessary maintenance costs. Within the PdM framework, remaining useful life (RUL) prediction is a key task that aims to estimate the time remaining before a system reaches a failure threshold [ 1 ]. In terms of modeling paradigms, RUL prediction methods can generally be divided into two categories: physics-based degradation models and statistical or machine-learning models based on historical monitoring data. For engine systems with complex structures and degradation mechanisms that are difficult to model, data-driven methods require less prior knowledge and can learn degradation patterns directly from multisource sensor data. As a result, they have become an important research direction in recent years [ 2 ]. Data-driven RUL prediction typically involves data acquisition, health indicator (HI) construction, and life prediction modeling. Traditional approaches often rely on handcrafted features or manually designed HIs, followed by regression using models such as support vector machines or random forests. In contrast, deep learning can automatically learn degradation representations from multivariate time series, thereby reducing the uncertainty and effort associated with feature engineering. In recent years, recurrent neural networks (RNNs) and their variants have been widely applied to RUL prediction. Among these models, long short-term memory (LSTM) networks introduce explicit memory cells through gating mechanisms, effectively alleviating the vanishing-gradient problem encountered by conventional RNNs when learning long-term dependencies. As a result, LSTM can capture latent states that reflect degradation evolution and has shown promising performance in various engine life prediction tasks [ 3 , 4 ]. The existing literature indicates that condition-monitoring-driven prognostics and health management (PHM) has developed into a relatively systematic methodological framework. Relevant review studies have summarized the major components and challenges of PdM and condition-based maintenance (CBM), while also categorizing engine prognostics, degradation modeling, and data-driven methods [ 5 , 6 , 7 ]. Regarding public benchmarks, NASA developed turbofan engine degradation datasets using the C-MAPSS simulation tool, and the PHM 2008 data challenge further promoted standardized and reproducible algorithm evaluation [ 8 , 9 , 10 ]. Early studies attempted to learn direct mappings from multisensor sequences to RUL using models such as RNNs [ 11 , 12 , 13 , 14 ]. With advances in end-to-end representation learning, numerous deep models have been developed for feature extraction and temporal modeling, including deep CNN-based degradation representation learning [ 15 , 16 ], LSTM-based life regression, and more recent architectures incorporating attention mechanisms and Transformer-based structures [ 17 ]. In addition, to address distribution shift caused by changing operating conditions, researchers have explored transfer learning and domain adaptation strategies to improve generalization in cross-condition scenarios [ 18 , 19 , 20 , 21 , 22 , 23 ]. Despite these advances, most existing RUL studies still emphasize point prediction accuracy, while paying relatively limited attention to interval reliability and deployment robustness under operating-condition shift. In particular, prediction intervals derived from quantile regression may suffer from coverage mismatch under finite-sample conditions and domain shift, which limits their practical value for maintenance-oriented decision-making. Therefore, a reliability-oriented interval prognostics framework is needed to provide both informative uncertainty estimates and improved robustness across operating conditions. Recent reliability-oriented PHM studies have increasingly emphasized that useful prognostics should not stop at point prediction, but should provide uncertainty-aware outputs that can be translated into maintenance decisions. For example, a recent study in Reliability Engineering & System Safety developed a multi-action predictive maintenance framework for lithium-ion batteries by coupling probabilistic RUL prediction with downstream maintenance optimization, thereby explicitly linking RUL uncertainty to maintenance actions for single- and multi-component systems [ 24 ]. In parallel, recent intelligent diagnosis studies have also moved toward reliability-aware learning under practical uncertainty and distribution shift. Representative examples include a multi-source sensor correlation adaptive fusion framework with uncertainty quantification for intelligent fault diagnosis, an evidential deep-learning-based test-time adaptation method for online machinery fault diagnosis, and a multi-source domain adaptation network based on transferable domain attributes and features for cross-domain fault diagnosis [ 25 , 26 , 27 ]. These studies collectively indicate that reliability, uncertainty awareness, and robustness to domain shift are becoming central concerns in PHM-related data-driven modeling. However, existing studies differ in task scope and application focus. The above battery-oriented work emphasizes downstream maintenance optimization based on probabilistic RUL outputs, whereas the present study focuses on the upstream prognostic module for turbofan engines. More specifically, this study is concerned with how to construct statistically meaningful and practically deployable RUL prediction intervals under operating-condition shift, so as to provide more trustworthy uncertainty-aware inputs for downstream maintenance decision-making. In this sense, the present work is positioned as a reliability-oriented interval prognostics approach for aero-engines rather than a complete prescriptive maintenance optimization framework. Motivated by this gap, this study proposes a reliability-oriented LSTM-based quantile regression framework with post hoc conformal calibration for turbofan engine RUL prediction. The framework jointly outputs a point estimate and the lower and upper bounds of a nominal prediction interval. To better reflect maintenance-oriented risk characteristics, low-RUL weighting and a mild overestimation penalty are incorporated during training. In addition, limited target-domain fine-tuning and inference-time conformalized quantile regression (CQR) calibration are introduced to improve interval reliability under operating-condition shift. In this way, the proposed method aims to provide more trustworthy uncertainty-aware prognostic outputs under both same-condition and cross-condition scenarios. The main contributions of this study are summarized as follows. (1) A reliability-oriented LSTM-based quantile regression framework is proposed for turbofan engine RUL prediction, which jointly outputs the point estimate and the lower and upper bounds of a nominal prediction interval in a single forward pass. Compared with conventional point-regression-based prognostics, the proposed framework provides more informative uncertainty cues for maintenance-oriented decision-making. (2) A risk-aware training objective is introduced by combining low-RUL weighting with a mild overestimation penalty. The former increases the optimization emphasis on maintenance-critical near-failure samples, while the latter explicitly suppresses overly optimistic life prediction from a safety perspective. (3) To improve deployment reliability under operating-condition shift, the framework combines limited target-domain fine-tuning with inference-time CQR calibration. This design enables the model to retain useful point prediction capability while bringing the empirical interval coverage closer to the nominal target under cross-condition scenarios. (4) Extensive experiments on the NASA C-MAPSS dataset, including same-condition evaluation, cross-condition transfer, ablation analysis, and multi-seed validation, demonstrate that the proposed framework achieves a favorable balance among point prediction accuracy, interval reliability, and practical interpretability. The remainder of this paper is organized as follows. Section 2 introduces the theoretical foundations of LSTM and quantile regression. Section 3 presents the structure, loss function, and training procedure of LSTM-QR. Section 4 describes the experimental settings and reports the results under both same-condition and cross-condition scenarios. Section 5 discusses the limitations of cross-condition prediction and possible directions for improvement. Section 6 concludes the paper and outlines future work. 2. Materials and Methods 2.1. LSTM Fundamentals RNNs are prone to gradient vanishing and explosion when trained on long sequences, which makes learning long-term dependencies difficult. LSTM introduces explicit memory cells through gating mechanisms to alleviate this problem and is therefore well suited for degradation modeling and RUL prediction tasks [ 28 , 29 ]. Given the input x t , the previous hidden state h t − 1 , and the previous cell state c t − 1 , a standard LSTM cell is updated as follows: f t = σ ( W f [ h t − 1 , x t ] + b f ) , (1) i t = σ ( W i ⋅ [ h t − 1 , x t ] + b i ) , (2) c ~ t = tanh ( W c [ h t − 1 , x t ] + b c ) , (3) c t = f t ⊙ c t − 1 + i t ⊙ c ~ t , (4) o t = σ ( W o [ h t − 1 , x t ] + b o ) , (5) h t = o t ⊙ tanh ( c t ) . (6) Here, σ ( ⋅ ) denotes the sigmoid function, tanh ( ⋅ ) denotes the hyperbolic tangent function, ⊙ represents the Hadamard product, and [ h t − 1 , x t ] denotes vector concatenation. Through the update mechanism defined in Equations (2)–(5), LSTM uses the cell state to preserve long-term memory and selectively regulates information flow through the forget, input, and output gates. By retaining and updating historical information in a gated manner, LSTM alleviates the vanishing-gradient problem commonly encountered by standard RNNs when modeling long sequences. This architecture can therefore learn discriminative long-term dependency features from degradation sequences and is well suited for RUL prediction. Figure 1 illustrates the architecture of an LSTM cell. The forget gate controls the retention of historical information, the input gate regulates the incorporation of current information, and the output gate determines how the cell state contributes to the hidden state. Through these gating mechanisms, LSTM can model long-term dependencies in degradation processes more effectively and stably [ 30 ]. Figure 1. Open in a new tab Schematic of a basic LSTM cell. 2.2. Quantile Regression and Prediction Intervals Quantile regression is a widely used uncertainty quantification method for regression tasks. Rather than estimating only the conditional mean, it estimates the conditional quantiles of the response variable given the input. Let Y denote the remaining useful life, and let X denote the input feature vector. The conditional quantile of Y given X = x at quantile level τ ∈ ( 0,1 ) is defined as follows [ 31 ]: Q τ ( Y ∣ X = x ) = inf { y : F Y ∣ X ( y ∣ x ) ≥ τ } (7) where F Y ∣ X ( y ∣ x ) denotes the conditional distribution function. According to Equation (7), when the input is x , q τ ( x ) satisfies P ( Y ≤ q τ ( x ) | X = x ) = τ , meaning that the probability mass to the left of this point equals τ . In practice, quantile regression usually learns quantile estimates by minimizing the pinball loss. Given a sample ( x i , y i ) and model output q ^ τ ( x i ) , let the error be e i = y i − q ^ τ ( x i ) . The pinball loss is written as [ 31 ]: L τ ( y i , q ^ τ ( x i ) ) = max τ e i , ( τ − 1 ) e i = τ ( y i − q ^ τ ( x i ) ) , y i ≥ q ^ τ ( x i ) ( 1 − τ ) ( q ^ τ ( x i ) − y i ) , y i < q ^ τ ( x i ) (8) This loss imposes asymmetric penalties on underestimation and overestimation, such that its optimum statistically corresponds to the target quantile. It can therefore be used to learn prediction bounds under different risk preferences. To obtain both point estimates and interval information simultaneously, this study adopts joint learning of multiple quantiles. Let τ ∈ { 0.1 , 0.5 , 0.9 } . In one forward pass, the model outputs q ^ 0.1 ( x ) , q ^ 0.5 ( x ) and q ^ 0.9 ( x ) . Here, q ^ 0.5 ( x ) , i.e., the median, is commonly used as a robust point estimate, while [ q ^ 0.1 ( x ) , q ^ 0.9 ( x ) ] forms a central 80% prediction interval (PI). It should be emphasized that the statistical meaning of an 80% interval is that, under ideal conditions, it covers future samples with probability approximately 1 − α = 0.8 [ 32 , 33 ]. P Y ∈ [ q ^ α / 2 ( X ) , q ^ 1 − α / 2 ( X ) ] ≈ 1 − α , α = 0.2 (9) This interpretation provides a range for maintenance decision-making, spanning from a conservative lower bound to an optimistic upper bound: the former reflects the risk of earlier failure, whereas the latter reflects the possibility of later failure. Two issues merit attention. First, quantile crossing may occur because different quantiles are optimized independently under their respective loss terms. Under finite-sample conditions, occasional violations such as Q ^ 0.1 ( x ) > Q ^ 0.9 ( x ) can lead to invalid interval definitions. In practice, this issue can be addressed by post hoc monotonic reordering or by introducing monotonicity constraints during training. Second, coverage bias may still arise even when the quantiles are estimated correctly. Under finite-sample conditions or distribution shift, nominal coverage may deviate from empirical coverage, resulting in under-coverage or excessively conservative intervals. Therefore, a distribution-free post hoc calibration method is further employed in this study to align empirical coverage more closely with the target level and thereby improve interval reliability in cross-condition scenarios [ 34 ]. 2.3. Evaluation Metrics To quantitatively evaluate prediction interval quality, two commonly used metrics are interval coverage and interval width. The prediction interval coverage probability (PICP) denotes the proportion of ground-truth values that fall within the predicted intervals and reflects interval reliability. The mean prediction interval width (MPIW) denotes the average width of the predicted intervals and reflects interval sharpness. Suppose that there are N test samples. For the i -th sample, let the predicted lower and upper bounds be L i and U i , respectively, and let y i denote the ground-truth value [ 35 ]: Coverage metric: P I C P = 1 N ∑ i = 1 N I y i ∈ [ L ^ i , U ^ i ] (10) Mean interval width: M P I W = 1 N ∑ i = 1 N U ^ i − L ^ i (11) where L ^ i = q ^ 0.1 ( x i ) and U ^ i = q ^ 0.9 ( x i ) . For a given nominal confidence level (e.g., 80%), the ideal PICP should be close to 0.8. Provided that coverage is adequate, a smaller MPIW indicates a tighter prediction interval. Interval quality can also be evaluated using composite metrics that penalize insufficient coverage. In the following experiments, PICP and MPIW are reported to compare the uncertainty quantification performance of different models. When PICP is markedly lower than the nominal level, the interval exhibits under-coverage; when it is substantially higher, the interval becomes overly conservative [ 36 ]. 2.4. Point Prediction Metrics In addition to uncertainty metrics, point prediction accuracy remains a fundamental requirement for life prediction models. In this study, the root mean square error (RMSE) is used to quantify the average prediction error on the test set and is defined as follows: R M S E = 1 N ∑ i = 1 N ( y ^ i − y i ) 2 (12) where y ^ i denotes the predicted RUL of the i -th sample and y i denotes the true RUL. RMSE treats early and late prediction errors symmetrically. In real maintenance decision-making, however, the consequences of these two types of errors are asymmetric: late predictions are generally more hazardous because they may delay maintenance and increase the risk of failure. To reflect this asymmetry, the PHM 2008 challenge introduced an asymmetric scoring function, commonly referred to as the PHM Score, which imposes a heavier penalty on late predictions. The sample-wise PHM Score is defined as follows: d i = y ^ i − y i , s i = exp − d i 13 − 1 , d i < 0 exp d i 10 − 1 , d i ≥ 0 (13) Here, d i denotes the prediction error of the i -th sample. When d i ≥ 0 , the model yields a late prediction and incurs a larger exponential penalty; when d i < 0 , the penalty is relatively smaller. The overall PHM Score can be computed as either the sum or the mean of the sample-wise scores. In this study, the mean form is adopted to facilitate comparisons across test sets of different sizes. A lower PHM Score indicates better performance under a maintenance-oriented risk preference. In the following cross-condition experiments, both RMSE and PHM Score are reported to comprehensively evaluate point prediction performance and the control of late-prediction risk. 3. Proposed Framework 3.1. Overall Framework and Task Definition This study addresses the uncertainty quantification requirement in turbofan engine remaining useful life (RUL) prediction by constructing prediction intervals with empirically calibrated coverage while simultaneously providing point estimates. Unlike conventional regression methods that output only a single RUL estimate, the proposed framework jointly considers point prediction accuracy and interval reliability, where the former reflects life estimation error and the latter reflects interval coverage and sharpness. In this way, the model can provide more informative risk cues for maintenance-oriented decision-making. In terms of task settings, both same-condition and cross-condition scenarios are considered. The same-condition scenario is used to evaluate the baseline performance of the model when the data distribution is relatively consistent, whereas the cross-condition scenario is used to assess generalization ability and interval reliability under distribution shift. Under cross-condition transfer, differences in operating-condition combinations and sensor response distributions may induce point prediction bias and interval under-coverage. Therefore, limited target-domain supervision and post hoc interval calibration are introduced to improve prediction reliability. Let { x t } t = 1 T denote the multivariate time series of a unit, where x t ∈ R d is the observation vector at cycle t . Sliding windows are used to construct input samples, such that consecutive observations form an input sequence and the corresponding RUL at the end of the window is predicted. The model uses an LSTM encoder to extract temporal degradation features and a quantile regression head to jointly output three quantiles, namely q ^ 0.1 , q ^ 0.5 and q ^ 0.9 . The median quantile q ^ 0.5 is used as the point prediction, whereas [ q ^ 0.1 , q ^ 0.9 ] forms the raw prediction interval. Figure 2 illustrates the overall workflow of the proposed method. The pipeline consists of three stages. First, in the training phase, source-domain data are preprocessed and transformed into sliding-window sequences, which are then fed into the LSTM-QR model. The model is optimized using the weighted quantile loss together with an overestimation penalty to improve both quantile estimation quality and maintenance-oriented risk control. Second, under cross-condition settings, the trained model is further fine-tuned on a small labeled target-domain set to alleviate systematic errors caused by domain shift. Third, during inference, the fine-tuned model outputs raw quantile predictions, from which the raw interval is obtained. An independent calibration set is then used to compute nonconformity scores and estimate the CQR correction term δ , so that the raw interval [ q ^ 0.1 , q ^ 0.9 ] is expanded into the calibrated interval [ q ^ 0.1 − δ , q ^ 0.9 + δ ] . This three-stage design enables the framework to preserve point prediction capability while improving interval reliability under both same-condition and cross-condition scenarios. Figure 2. Open in a new tab Overall workflow of the proposed method. The pipeline includes source-domain training, target-domain fine-tuning, and inference-time CQR calibration. The model first outputs raw quantile predictions, where the median quantile is used as the point prediction and the lower and upper quantiles form the raw prediction interval. CQR then expands the raw interval based on the calibration set to improve empirical coverage consistency. 3.2. LSTM-QR Model Structure Figure 3 illustrates the internal architecture of the proposed LSTM-QR model. The model takes multivariate time-series data in the form of sliding windows, encodes them using an LSTM, and jointly outputs multiple quantiles through a quantile regression head, thereby enabling both RUL point prediction and raw prediction interval construction. Let X t ∈ R L × d denote the input window, where L is the window length and d is the input feature dimension. Samples within the window are arranged chronologically to characterize the recent degradation dynamics of the equipment. The LSTM encoder models the temporal dependencies within the window, extracts sequential features of degradation evolution, and maps the window to a fixed-dimensional high-level representation. In this study, the hidden state at the final time step is used as the window representation for subsequent quantile regression. Figure 3. Open in a new tab Internal architecture of the proposed LSTM-QR model and illustration of raw prediction interval construction. The LSTM encoder extracts temporal degradation features from sliding-window inputs, and the quantile regression head jointly outputs the lower, median, and upper quantiles. At the output layer, the model jointly predicts three quantiles corresponding to the lower, median, and upper levels, which are set to 0.1, 0.5, and 0.9 in this study. Accordingly, q ^ 0.5 serves as the point prediction, whereas q ^ 0.1 and q ^ 0.9 define the lower and upper bounds of the nominal 80% prediction interval. This design enables the model to directly output interval estimates with explicit coverage semantics while preserving point prediction capability in a single forward pass. Because mild quantile crossing may occur under finite-sample training, the lower and upper bounds are consistently reordered during metric calculation and visualization; specifically, the smaller value is treated as the lower bound and the larger value is treated as the upper bound, thereby ensuring a valid and semantically consistent interval. It should be emphasized that this subsection describes only the raw quantile output structure of the model. The post hoc CQR procedure used to improve coverage consistency is introduced separately in Section 3.4 . Overall, the LSTM-QR model combines temporal feature encoding with joint quantile output, allowing it to provide RUL point estimates and raw prediction intervals simultaneously. This design forms the structural basis for the subsequent training objective and the inference-time CQR calibration. 3.3. Training Objective Design To jointly predict the lower, median, and upper quantiles, this study adopts multi-quantile learning based on the pinball loss. For a given quantile level τ ∈ { 0.1 , 0.5 , 0.9 } , let the prediction error be defined as e i τ = y i − q ^ τ ( x i ) (14) where y i is the ground-truth RUL of sample i , and q ^ τ ( x i ) denotes the predicted τ -th conditional quantile. The corresponding pinball loss is written as ρ τ ( e i τ ) = max ( τ e i τ , ( τ − 1 ) e i τ ) (15) By summing the loss terms over all target quantiles, the model can learn the lower, median, and upper bounds in a single forward pass. This formulation enables simultaneous point prediction and raw prediction interval construction within a unified regression framework. However, turbofan degradation samples are usually unevenly distributed over the life cycle. In particular, samples from the long-life stage often dominate the training set, whereas low-RUL samples that are more critical for maintenance decisions are relatively sparse. If all samples are treated equally, the model may focus excessively on the abundant early-stage data and underrepresent the degradation behavior near failure. To alleviate this imbalance, an RUL-dependent weight w i is introduced so that samples with smaller true RUL values receive larger weights. In this way, the contribution of near-failure samples is amplified during optimization, improving the learning emphasis on maintenance-critical operating regions. To preserve the overall loss scale and maintain training stability, the weights are mean-normalized so that their average equals one. The weighted multi-quantile loss is therefore defined as L w = 1 N ∑ i = 1 N w i ∑ τ ∈ { 0.1,0.5,0.9 } ρ τ ( e i , τ ) . (16) This weighting strategy does not change the statistical meaning of quantile regression itself; rather, it reallocates optimization emphasis toward low-RUL regions that are more relevant to practical prognostics. In addition to sample imbalance, RUL prediction also exhibits asymmetric engineering risk. In predictive maintenance, overestimating the remaining life may delay maintenance actions and thus cause more serious consequences than a similarly sized underestimation. To reflect this risk preference, a mild overestimation penalty is imposed on the median prediction q ^ 0.5 , which serves as the point estimate. Specifically, only the part where the predicted median exceeds the true RUL is penalized: L o v e r = 1 N ∑ i = 1 N max ( q ^ 0.5 ( x i ) − y i , 0 ) . (17) Unlike the weighted quantile loss, which mainly adjusts the learning emphasis across life stages, this term explicitly discourages hazardous late predictions from the perspective of maintenance risk control. Therefore, the two components play different roles: the former improves quantile learning under life-stage imbalance, whereas the latter introduces a safety-oriented bias against RUL overestimation. The final training objective is defined as L = L w + λ L o v e r (18) where λ > 0 is a hyperparameter controlling the trade-off between distributional quantile fitting and overestimation suppression. Overall, this risk-aware objective encourages the model to learn more informative interval bounds while reducing the chance of overly optimistic point predictions, thereby making the resulting outputs better aligned with maintenance-oriented decision requirements. 3.4. CQR Calibration Method Although quantile regression can directly produce a nominal prediction interval, the empirical coverage of the raw interval may deviate from the target level under finite-sample conditions and, more importantly, under cross-condition distribution shift. Recent studies have also attempted to improve conditional coverage in quantile-based uncertainty estimation through more structured quantile-learning strategies [ 37 ]. Therefore, this study employs Conformalized Quantile Regression (CQR) as a post hoc calibration step at inference time to improve interval reliability. This procedure does not retrain or modify the model parameters. Instead, it adjusts only the interval boundaries in a data-driven manner based on an independent calibration set. For a test sample x i , let the raw quantile outputs of the trained model be q ^ 0.1 ( x i ) , q ^ 0.5 ( x i ) and q ^ 0.9 ( x i ) corresponding to the lower bound, median, and upper bound, respectively. The raw prediction interval is defined as [ q ^ 0.1 ( x i ) , q ^ 0.9 ( x i ) ] . Because mild quantile crossing may occasionally occur under finite-sample training, the lower and upper bounds are first reordered when necessary so that the smaller value is treated as the lower bound and the larger value is treated as the upper bound. This step ensures a valid raw interval before calibration. Given an independent calibration set { x i , y i } i = 1 n , the nonconformity score for each sample is defined as the extent to which the ground-truth value falls outside the raw interval [ 38 , 39 , 40 ]: s i = max ( q ^ 0.1 ( x i ) − y i , y i − q ^ 0.9 ( x i ) , 0 ) (19) A larger score indicates that the raw interval is less compatible with the observed target. Based on the empirical distribution of { s i } i = 1 n , a calibration value δ is estimated according to the desired coverage level. For a target coverage level of 1 − α , the calibration value is defined as the k-th order statistic of the nonconformity scores, where k = [ ( n + 1 ) ( 1 − α ) ] - th . This quantity represents the amount of interval expansion needed to bring the empirical coverage closer to the desired level. The calibrated interval for a new sample x is then constructed as q ^ 0.1 ( x ) − δ , q ^ 0.9 ( x ) + δ It should be emphasized that this post hoc calibration changes only the lower and upper interval boundaries and leaves the median prediction q ^ 0.5 ( x ) unchanged. Consequently, point-prediction metrics such as RMSE and PHM Score remain unchanged before and after calibration, whereas interval-related metrics such as PICP and MPIW vary with the interval adjustment. This distinction is important because the role of CQR in this study is not to improve point prediction accuracy, but to correct coverage mismatch and enhance the reliability of predictive intervals. In the same-condition setting, the raw quantile interval may already achieve empirical coverage close to the nominal level, so the calibration effect is typically limited. In contrast, under cross-condition transfer, distribution shift often causes under-coverage of the raw interval, making calibration more necessary. In this study, a small labeled target-domain set is used not only for fine-tuning but also for building the calibration set, so that the calibration samples are more consistent with the deployment scenario. As a result, CQR provides a simple and distribution-free mechanism to improve interval reliability under changing operating conditions, although this gain is usually accompanied by some interval widening. 4. Experiments and Results 4.1. Dataset and Implementation Details The proposed method is validated using the NASA C-MAPSS dataset generated by the C-MAPSS simulation tool developed at NASA Ames Research Center, Moffett Field, CA, USA. All experiments were implemented in Python (version 3.10.19), and the model was developed using PyTorch (version 2.11.0+cu128). Data processing and visualization were performed using NumPy (version 2.2.6), pandas (version 2.3.3), and Matplotlib (version 3.10.8). FD001 and FD002 are selected for both same-condition and cross-condition experiments. Each cycle includes three operating-condition variables and twenty-one sensor variables, resulting in a total of 24 input features. The data split follows the official file structure: train_FD00x is used for training, test_FD00x for testing, and the true terminal RUL values of the test units are provided in RUL_FD00x. Training labels are generated by backtracking from the failure cycle, whereas test labels for the full trajectories are generated by backtracking from the provided terminal RUL values. To stabilize optimization and mitigate the long-tail effect in the early-life stage, the RUL target is uniformly capped at 125 cycles. For clarity and engineering interpretability, Table 1 summarizes the common physical abbreviations, descriptions, and units of the 21 sensor variables. Table 1. Description of sensor variables. Sensor ID Abbreviation Description Unit s1 T2 Fan inlet total temperature °R s2 T24 LPC outlet total temperature °R s3 T30 HPC outlet total temperature °R s4 T50 LPT outlet total temperature °R s5 P2 Fan inlet pressure psia s6 P15 Bypass-duct total pressure psia s7 P30 HPC outlet total pressure psia s8 Nf Fan physical speed rpm s9 Nc Core physical speed rpm s10 epr Engine pressure ratio — s11 Ps30 HPC outlet static pressure psia s12 phi Fuel-to-P30 ratio pps/psi s13 NRf Corrected fan speed rpm s14 NRc Corrected core speed rpm s15 BPR Bypass ratio — s16 farB Burner fuel–air ratio — s17 htBleed Bleed enthalpy — s18 Nf_dmd Demanded fan speed rpm s19 PCNfR_dmd Demanded core speed rpm s20 W31 HPT coolant bleed flow lbm/s s21 W32 LPT coolant bleed flow lbm/s Open in a new tab Preprocessing includes z-score normalization and the removal of near-constant features; an additional 1 × 10 − 8 is introduced during normalization to avoid division by zero. For same-condition experiments, scaler_mode = source is adopted, meaning that the target domain reuses the source-domain normalization statistics. For cross-condition experiments, scaler_mode = per_domain is used, meaning that the target domain applies its own normalization statistics, while the source-domain feature mask is retained to ensure dimensional consistency and mitigate scale shift. The training procedure begins with source-domain pretraining using a unit-level validation split (val_ratio = 0.1), and the best checkpoint is selected accordingly. When finetune_ratio > 0, supervised fine-tuning is performed on a subset of target-domain training units. Adam is used as the optimizer (weight decay = 1 × 10 − 5 ), together with gradient clipping (max-norm = 1.0). The model jointly outputs three quantiles and is evaluated using RMSE, the PHM 2008 Score, PICP, and MPIW. When CQR is enabled, a calibration set is constructed from randomly truncated windows sampled from target-domain training trajectories to estimate δ , after which the lower and upper interval bounds are symmetrically expanded. Therefore, CQR does not alter the median quantile prediction; accordingly, RMSE and the PHM 2008 Score remain unchanged, whereas PICP and MPIW vary with interval adjustment. The default hyperparameters are seq_len = 30 and batch_size = 256. A two-stage learning-rate schedule is adopted, with 1 × 10 − 3 for pretraining and 1 × 10 − 4 for fine-tuning. The best model is selected by monitoring the RMSE of the median quantile on the source-domain validation set. The numbers of training epochs are set to 25/30 (pretraining/fine-tuning) for same-condition experiments, 60/50 for FD001 → FD002, and 70/70 for FD002 → FD001 to ensure adequate convergence. 4.2. Same-Condition Results and Analysis Table 2 summarizes the results under the same operating condition. To reduce the influence of stochastic factors such as random initialization and data shuffling, all values are reported as the mean ± standard deviation over three random seeds. Overall, the proposed method achieves stable performance in both same-condition settings, and FD001 remains easier than FD002 in terms of both point prediction and interval estimation. Specifically, FD001 → FD001 achieves RMSE = 16.235 ± 1.297, PHM Score = 394.9 ± 5.0, PICP = 0.800 ± 0.026, and MPIW = 33.766 ± 0.811, whereas FD002 → FD002 achieves RMSE = 18.323 ± 0.411, PHM Score = 1658.1 ± 367.2, PICP = 0.793 ± 0.032, and MPIW = 37.203 ± 3.583. These results indicate that the model can learn a more stable degradation mapping under a single operating condition, while the more complex multi-condition setting of FD002 leads to higher prediction error and greater interval uncertainty. Table 2. Overall results under the same-condition setting (mean ± standard deviation over three random seeds). Setting RMSE PHM PICP MPIW FD001 → FD001 (Uncalibrated) 16.235 ± 1.297 394.9 ± 5.0 0.590 ± 0.035 22.822 ± 2.025 FD001 → FD001 (Calibrated) 16.235 ± 1.297 394.9 ± 5.0 0.800 ± 0.026 33.766 ± 0.811 FD002 → FD002 (Uncalibrated) 18.323 ± 0.411 1658.1 ± 367.2 0.722 ± 0.050 33.339 ± 4.928 FD002 → FD002 (Calibrated) 18.323 ± 0.411 1658.1 ± 367.2 0.793 ± 0.032 37.203 ± 3.583 Open in a new tab From the perspective of statistical stability, the standard deviations under the same-condition setting are generally moderate, suggesting that the proposed framework is not overly sensitive to random seeds when the training and test distributions are matched. In particular, the PICP values of both FD001 and FD002 remain close to the nominal 80% level across repeated runs, indicating that the calibrated intervals maintain reliable coverage with limited fluctuation. Although FD002 exhibits a larger variation in PHM Score and MPIW than FD001, this behavior is consistent with its higher complexity and stronger operating-condition heterogeneity. Therefore, the multi-seed results further support the conclusion that the proposed method provides stable point prediction performance and reasonably reliable prediction intervals under same-condition scenarios. Figure 4 , Figure 5 , Figure 6 , Figure 7 and Figure 8 further visualize the prediction behavior under the same-condition setting. Specifically, Figure 4 presents the sorted ribbon plots, Figure 5 shows representative engine degradation trajectories, Figure 6 reports the interval calibration diagnostics, and Figure 7 and Figure 8 summarize the stage-wise prediction error and interval behavior for FD001 and FD002, respectively. Figure 4. Open in a new tab Sorted ribbon plots of test results under the same operating condition. Panels ( a , b ) correspond to the results for FD001 and FD002, respectively. The blue dashed line indicates the capped maximum RUL value of 125 cycles used in the data preprocessing. Figure 5. Open in a new tab RUL degradation trajectories of representative engines under the same operating condition. Panels ( a , b ) show the representative engine trajectories for FD001 and FD002, respectively. The blue dashed line indicates the capped maximum RUL value of 125 cycles used in the data preprocessing. Figure 6. Open in a new tab Interval calibration diagnostics at the nominal 80% level under the same-condition setting. Error bars denote 95% confidence intervals of the empirical coverage. The dashed horizontal line indicates the nominal coverage level of 0.80. Panels ( a , b ) correspond to FD001 → FD001 and FD002 → FD002, respectively. Figure 7. Open in a new tab Prediction error and interval behavior across different life stages on FD001. Panels ( a – c ) show the binned mean absolute error (MAE), mean prediction interval width (MPIW), and prediction interval coverage probability (PICP), respectively, as functions of the true RUL. Figure 8. Open in a new tab Prediction error and interval behavior across different life stages on FD002. Panels ( a – c ) show the binned mean absolute error (MAE), mean prediction interval width (MPIW), and prediction interval coverage probability (PICP), respectively, as functions of the true RUL. 4.3. Cross-Condition Results and Analysis To evaluate generalization under changing operating-condition distributions, cross-condition experiments are conducted in both directions, FD001 → FD002 and FD002 → FD001, and the results are summarized in Table 3 . Similarly to Table 2 , all values are reported as the mean ± standard deviation over three random seeds. Under distribution shift, the proposed framework achieves RMSE = 21.758 ± 1.208, PHM Score = 2943.7 ± 782.6, PICP = 0.806 ± 0.032, and MPIW = 50.952 ± 0.462 for FD001 → FD002, while for FD002 → FD001 it achieves RMSE = 17.562 ± 0.062, PHM Score = 633.3 ± 12.7, PICP = 0.803 ± 0.021, and MPIW = 40.888 ± 2.336. Compared with the same-condition setting, the cross-condition results show larger performance variability, which is expected because operating-condition shift affects both point prediction bias and interval calibration difficulty. Table 3. Overall results under the cross-condition setting (mean ± standard deviation over three random seeds). Setting RMSE PHM PICP MPIW FD001 → FD002 (Uncalibrated) 21.758 ± 1.208 2943.7 ± 782.6 0.696 ± 0.046 43.108 ± 3.900 FD001 → FD002 (Calibrated) 21.758 ± 1.208 2943.7 ± 782.6 0.806 ± 0.032 50.952 ± 0.462 FD002 → FD001 (Uncalibrated) 17.562 ± 0.062 633.3 ± 12.7 0.593 ± 0.071 25.218 ± 3.209 FD002 → FD001 (Calibrated) 17.562 ± 0.062 633.3 ± 12.7 0.803 ± 0.021 40.888 ± 2.336 Open in a new tab Nevertheless, an important observation is that the empirical coverage remains consistently close to the nominal 80% level in both transfer directions. This suggests that the combination of limited target-domain fine-tuning and inference-time CQR calibration is effective in improving interval reliability even when the source and target distributions are mismatched. At the same time, the two transfer directions still exhibit noticeable asymmetry. FD001 → FD002 shows a larger PHM Score and a wider interval, whereas FD002 → FD001 achieves a lower RMSE and a narrower interval on average. This indicates that cross-condition transfer difficulty is direction-dependent and that good point prediction performance alone does not necessarily imply well-calibrated uncertainty estimation. Overall, the multi-seed results confirm that the proposed framework maintains not only acceptable predictive accuracy but also relatively robust calibrated coverage under cross-condition scenarios. Figure 9 , Figure 10 and Figure 11 further visualize the calibration effect and prediction behavior under cross-condition transfer. Figure 9 presents the interval calibration diagnostics, Figure 10 shows the sorted ribbon plots for the two transfer directions, and Figure 11 illustrates representative engine degradation trajectories. Figure 9. Open in a new tab Interval calibration diagnostics at the nominal 80% level under the cross-condition setting. Figure 10. Open in a new tab Sorted ribbon plots of test results under the cross-condition setting. Panels ( a , b ) correspond to the results for FD001 → FD002 and FD002 → FD001, respectively. The blue dashed line indicates the capped maximum RUL value of 125 cycles used in the data preprocessing. Figure 11. Open in a new tab RUL degradation trajectories of representative engines in the cross-condition setting. Panel ( a ) shows the prediction trajectory of a representative engine under FD002 → FD001, and panel ( b ) shows the prediction trajectory of a representative engine under FD001 → FD002. The blue dashed line indicates the capped maximum RUL value of 125 cycles used in the data preprocessing. 4.4. Comparison with Baseline Methods To further verify the relative advantages of the proposed method in both point prediction accuracy and uncertainty quantification, this subsection compares four representative deep-learning baselines for predictive uncertainty estimation: MC Dropout-LSTM, Gaussian NLL-LSTM, TCN-QR and Deep Ensemble Gaussian-LSTM [ 41 , 42 , 43 , 44 ]. All methods are evaluated under the same preprocessing scheme, window construction strategy, and evaluation protocol. RMSE, the PHM 2008 Score, PICP and MPIW for the nominal 80% prediction interval are reported to jointly assess point prediction accuracy, coverage reliability, and interval sharpness. To improve the statistical reliability of the comparison, all competing baselines were also repeated over three random seeds, and the results are reported as the mean ± standard deviation. The same-condition results are presented in Table 4 and Table 5 . On FD001, MC Dropout-LSTM and TCN-QR achieve slightly lower RMSE values than the proposed method, but both suffer from evident under-coverage, with PICP = 0.640 ± 0.020 and 0.700 ± 0.115, respectively. By contrast, Deep Ensemble Gaussian-LSTM achieves higher coverage (PICP = 0.880 ± 0.070), but at the cost of a substantially wider interval (MPIW = 48.612 ± 9.386). Gaussian NLL-LSTM achieves PICP close to the nominal level, but its interval width and variance are both relatively large. Compared with these methods, the proposed LSTM-QR achieves PICP = 0.800 ± 0.026 and MPIW = 33.766 ± 0.811, yielding the most balanced trade-off between coverage reliability and interval sharpness while maintaining competitive point prediction performance. Table 4. Baseline comparison under the same-condition setting (FD001; mean ± standard deviation over three random seeds). FD001 RMSE PHM PICP MPIW MC Dropout-LSTM 15.456 ± 0.669 414.2 ± 3.4 0.640 ± 0.020 20.906 ± 0.913 Gaussian NLL-LSTM 19.143 ± 6.811 556.7 ± 43.2 0.807 ± 0.055 51.029 ± 19.296 TCN-QR 15.509 ± 0.568 443.3 ± 13.3 0.700 ± 0.115 28.120 ± 6.165 Deep Ensemble Gaussian-LSTM 16.007 ± 0.666 296.7 ± 6.6 0.880 ± 0.070 48.612 ± 9.386 LSTM-QR 16.235 ± 1.297 394.9 ± 5.0 0.800 ± 0.026 33.766 ± 0.811 Open in a new tab Table 5. Baseline comparison under the same-condition setting (FD002; mean ± standard deviation over three random seeds). FD002 RMSE PHM PICP MPIW MC Dropout-LSTM 17.468 ± 0.562 1740.2 ± 251.2 0.781 ± 0.046 34.238 ± 2.350 Gaussian NLL-LSTM 15.777 ± 0.673 1490.2 ± 72.1 0.747 ± 0.045 29.747 ± 2.847 TCN-QR 17.219 ± 0.378 1786.7 ± 227.2 0.708 ± 0.028 31.655 ± 2.990 Deep Ensemble Gaussian-LSTM 15.556 ± 0.064 1466.7 ± 25.2 0.762 ± 0.030 30.836 ± 2.016 LSTM-QR 18.323 ± 0.411 1658.1 ± 367.2 0.793 ± 0.032 37.203 ± 3.583 Open in a new tab In the more complex FD002 scenario ( Table 5 ), several baseline methods achieve lower RMSE values than the proposed method, but their interval reliability is either insufficient or less balanced. For example, Gaussian NLL-LSTM and Deep Ensemble Gaussian-LSTM achieve lower RMSE values, but their PICP values remain below the nominal 80% level. In contrast, the proposed LSTM-QR achieves PICP = 0.793 ± 0.032, which is the closest to the target coverage among the compared methods, although its RMSE is not the lowest. These results indicate that under more complex multi-condition settings, the proposed method remains advantageous in terms of interval reliability and overall calibration quality rather than pure point prediction accuracy alone. The cross-condition results are presented in Table 6 and Table 7 . For FD001 → FD002, the proposed LSTM-QR achieves the lowest RMSE among all compared methods (21.758 ± 1.208) while maintaining calibrated coverage close to the nominal level (PICP = 0.806 ± 0.032). Its interval width (MPIW = 50.952 ± 0.462) is also comparable to that of TCN-QR and smaller than those of MC Dropout-LSTM, Gaussian NLL-LSTM, and Deep Ensemble Gaussian-LSTM. These results indicate that, in this transfer direction, the proposed method achieves the best balance between point prediction accuracy and interval reliability, although its PHM Score still leaves room for improvement. Table 6. Baseline comparison under the cross-condition setting (FD001 → FD002; mean ± standard deviation over three random seeds). FD001 → FD002 RMSE PHM PICP MPIW MC Dropout-LSTM 23.932 ± 0.513 5356.7 ± 354.18 0.855 ± 0.027 54.823 ± 2.943 Gaussian NLL-LSTM 25.540 ± 1.474 1436.7 ± 202.1 0.787 ± 0.036 57.736 ± 4.985 TCN-QR 22.176 ± 0.904 2273.3 ± 511.9 0.846 ± 0.049 50.552 ± 2.525 Deep Ensemble Gaussian-LSTM 23.610 ± 0.397 1266.7 ± 225.5 0.821 ± 0.015 56.544 ± 1.124 LSTM-QR 21.758 ± 1.208 2943.7 ± 782.6 0.806 ± 0.032 50.952 ± 0.462 Open in a new tab Table 7. Baseline comparison under the cross-condition setting (FD002 → FD001; mean ± standard deviation over three random seeds). FD002 → FD001 RMSE PHM PICP MPIW MC Dropout-LSTM 39.969 ± 3.656 605.4 ± 180.4 0.823 ± 0.015 60.915 ± 5.081 Gaussian NLL-LSTM 45.240 ± 1.407 472.5 ± 35.2 0.809 ± 0.021 59.596 ± 2.017 TCN-QR 45.457 ± 2.708 585.7 ± 115.3 0.820 ± 0.017 66.487 ± 7.686 Deep Ensemble Gaussian-LSTM 44.588 ± 1.570 690.5 ± 66.4 0.807 ± 0.038 59.406 ± 2.040 LSTM-QR 17.562 ± 0.062 633.3 ± 12.7 0.803 ± 0.021 40.888 ± 2.336 Open in a new tab For FD002 → FD001, the advantage of the proposed method becomes even more pronounced. The proposed LSTM-QR achieves RMSE = 17.562 ± 0.062, which is substantially lower than those of all baseline methods, while also maintaining calibrated coverage close to the nominal level (PICP = 0.803 ± 0.021). Moreover, its MPIW = 40.888 ± 2.336 is markedly smaller than those of the competing methods, all of which require much wider intervals to achieve comparable coverage. Although the PHM Score is not the lowest, the proposed method clearly provides the best overall compromise between point prediction accuracy, interval reliability, and interval compactness in this transfer direction. Overall, the comparative experiments indicate that some baseline methods tend to produce excessively wide intervals in pursuit of higher coverage, whereas others suffer from under-coverage when maintaining relatively narrow intervals. In contrast, the proposed LSTM-QR consistently maintains coverage close to the nominal level and demonstrates stronger overall competitiveness under cross-condition settings, making it more suitable for interval-based RUL prediction in maintenance-oriented applications. 4.5. Ablation Study To clarify the individual roles of low-RUL weighting, the overestimation penalty, fine-tuning, and CQR calibration, ablation experiments were conducted under the two cross-condition settings, namely FD001 → FD002 and FD002 → FD001. The compared variants include Plain QR, Weighting only, Over-penalty only, Full risk-aware, and their corresponding combinations with fine-tuning and CQR. This design allows the separate and joint effects of the four components to be examined under distribution shift. Three main observations can be drawn from Table 8 and Table 9 . First, fine-tuning is the dominant factor for effective cross-condition transfer. Without fine-tuning, all variants perform poorly in both directions. For example, in FD001 → FD002, Plain QR yields RMSE = 51.985 and PICP = 0.261, whereas Plain QR + FT reduces RMSE to 20.725 and improves PICP to 0.806. Similarly, in FD002 → FD001, Plain QR improves from RMSE = 48.645 and PICP = 0.260 to RMSE = 15.297 and PICP = 0.600 after fine-tuning. These results indicate that source-domain quantile learning alone is insufficient under operating-condition shift, and that limited target-domain adaptation is essential for recovering cross-condition predictive capability. Table 8. Ablation results of the proposed framework under the FD001 → FD002 setting. “√” means used; “–” means not used. Setting Low-RUL Weighting Overestimation Penalty Fine-Tuning CQR RMSE PHM Score PICP MPIW Plain QR – – – – 51.985 1,098,574.0 0.261 36.968 Weighting only √ – – – 55.307 1,938,596.8 0.194 21.315 Over-penalty only – √ – – 53.001 1,246,954.0 0.281 37.248 Full risk-aware √ √ – – 49.309 945,916.4 0.281 36.862 Plain QR + FT – – √ – 20.725 3411.8 0.806 48.055 Weighting only + FT √ – √ – 24.374 6890.2 0.593 35.372 Over-penalty only + FT – √ √ – 20.841 2982.9 0.810 48.236 Full risk-aware + FT √ √ √ – 21.050 2272.3 0.747 46.830 Plain QR + FT + CQR – – √ √ 20.725 3411.8 0.822 49.831 Full risk-aware + FT + CQR √ √ √ √ 21.050 2272.3 0.806 51.438 Open in a new tab Table 9. Ablation results of the proposed framework under the FD002 → FD001 setting. “√” means used; “–” means not used. Setting Low-RUL Weighting Overestimation Penalty Fine-Tuning CQR RMSE PHM Score PICP MPIW Plain QR – – – – 48.645 61,369.6 0.260 23.683 Weighting only √ – – – 48.376 62,367.8 0.280 25.494 Over-penalty only – √ – – 48.481 61,657.2 0.280 27.689 Full risk-aware √ √ – – 47.968 57,022.9 0.250 17.297 Plain QR + FT – – √ – 15.297 360.3 0.600 24.994 Weighting only + FT √ – √ – 14.876 338.6 0.610 25.373 Over-penalty only + FT – √ √ – 15.984 422.8 0.630 25.171 Full risk-aware + FT √ √ √ – 17.491 644.2 0.530 21.969 Plain QR + FT + CQR – – √ √ 15.297 360.3 0.910 50.090 Full risk-aware + FT + CQR √ √ √ √ 17.491 644.2 0.810 43.554 Open in a new tab Second, the risk-aware objective mainly reshapes the trade-off between point accuracy and maintenance-oriented risk, rather than uniformly improving all metrics. In FD001 → FD002, Full risk-aware + FT gives a slightly higher RMSE than Plain QR + FT (21.050 vs. 20.725), but achieves a clearly lower PHM Score (2272.3 vs. 3411.8), indicating better control of maintenance-related late-prediction risk. In the same direction, Over-penalty only + FT achieves RMSE = 20.841, PHM Score = 2982.9, and PICP = 0.810, suggesting that the overestimation penalty helps suppress overly optimistic predictions while maintaining competitive point accuracy. In FD002 → FD001, however, weighting only + FT achieves the best RMSE and PHM Score among the non-CQR fine-tuned variants, whereas Over-penalty only + FT attains the highest PICP. These results suggest that low-RUL weighting and overestimation control influence different aspects of prediction behavior, and that their effects are direction-dependent under cross-condition transfer. Third, CQR mainly serves to correct coverage mismatch, typically at the cost of interval widening. In FD001 → FD002, Plain QR + FT + CQR increases PICP from 0.806 to 0.822, while MPIW increases from 48.055 to 49.831; Full risk-aware + FT + CQR raises PICP from 0.747 to 0.806, with MPIW increasing from 46.830 to 51.438. In FD002 → FD001, the effect is more pronounced: Plain QR + FT + CQR improves PICP from 0.600 to 0.910, but also expands MPIW from 24.994 to 50.090. In comparison, Full risk-aware + FT + CQR achieves PICP = 0.810 with a narrower MPIW of 43.554, suggesting a more balanced compromise between target coverage and interval sharpness. Therefore, CQR should be understood as a post hoc reliability correction step rather than a mechanism for improving point prediction accuracy. Overall, the ablation study shows that fine-tuning is the key contributor to cross-condition transferability, the risk-aware objective mainly affects the accuracy–risk trade-off, and CQR is primarily responsible for restoring empirical coverage toward the nominal target. Taken together, these components support the proposed framework as a reliability-oriented interval prognostics pipeline rather than a design that uniformly optimizes every metric. 5. Discussion The results suggest that the proposed framework is useful not only for improving predictive accuracy, but also for supporting maintenance-oriented decision-making through interval-aware prognostics. Unlike a single point estimate, the predicted interval provides an explicit risk range for the remaining life. In practice, the lower bound can be interpreted as a conservative estimate associated with earlier failure risk, whereas the upper bound reflects a more optimistic estimate associated with later failure. Therefore, when the lower bound approaches a maintenance threshold, a more cautious inspection or replacement strategy may be preferred; when the interval is wide, the system operator may choose to increase monitoring frequency or combine the prediction with additional diagnostic evidence before making a maintenance decision. In this sense, the proposed method provides more informative support for maintenance planning than point prediction alone. Another important implication is that reliability calibration is especially necessary under operating-condition shift. The cross-condition results show that raw quantile intervals may suffer from under-coverage, indicating that quantile regression alone cannot guarantee the target confidence level when the deployment distribution differs from the training distribution. In this context, CQR plays a practical role as a lightweight post hoc correction mechanism: it does not change the point prediction itself, but adjusts the interval boundaries so that empirical coverage is brought closer to the nominal level. This property is valuable for deployment because it improves interval reliability without requiring a complete redesign of the prediction model. However, the benefit is typically accompanied by wider intervals, which reflects the inherent trade-off between coverage and sharpness. It should also be noted that the proposed framework is intended as a reliability-oriented prognostic module rather than a complete maintenance optimization system. In practical industrial settings, interval RUL prediction would need to be further integrated with maintenance thresholds, resource constraints, inspection cost, and mission scheduling. Therefore, the main contribution of this study lies in providing a more reliable uncertainty-aware input for downstream maintenance decision processes, rather than directly solving the full prescriptive maintenance problem. 6. Conclusions To address the limitation of point estimates in turbofan engine remaining useful life prediction, this study proposed an LSTM-based interval prediction framework that jointly outputs point estimates and the bounds of a nominal 80% prediction interval. By combining weighted quantile learning, a mild overestimation penalty, limited target-domain fine-tuning, and post hoc CQR calibration, the proposed method improves interval reliability while preserving point prediction capability. Experiments on the NASA C-MAPSS dataset showed that the proposed framework achieves a favorable balance between point prediction accuracy and interval quality under both same-condition and cross-condition settings. In particular, the results indicate that fine-tuning is crucial for cross-condition transfer, whereas CQR is effective in correcting coverage mismatch and improving interval reliability under distribution shift. From a practical perspective, the proposed method can provide more informative support for maintenance-oriented prognostics than point prediction alone by offering a calibrated uncertainty range together with the RUL estimate. Nevertheless, this study is still limited by the use of the C-MAPSS dataset, the absence of explicit monotonicity constraints, and the possible interval widening introduced by calibration. Future work will focus on domain adaptation, conditional calibration, monotonicity-aware learning, and validation on datasets and decision-oriented evaluation protocols that are closer to real engineering practice. Abbreviations The following abbreviations are used in this manuscript: RUL Remaining Useful Life LSTM Long Short-Term Memory QR Quantile Regression LSTM-QR LSTM-based Quantile Regression CQR Conformalized Quantile Regression PdM Predictive Maintenance Open in a new tab Author Contributions Conceptualization, R.D. and M.Z.; methodology, R.D. and M.Z.; software, R.D. and S.W.; validation, R.D. and G.M.; formal analysis, R.D.; investigation, R.D.; resources, M.Z. and G.M.; data curation, R.D. and S.W.; writing—original draft preparation, R.D.; writing—review and editing, M.Z. and G.M.; visualization, R.D.; supervision, M.Z. and G.M.; project administration, M.Z.; funding acquisition, M.Z. and G.M. All authors have read and agreed to the published version of the manuscript. Institutional Review Board Statement Not applicable. Informed Consent Statement Not applicable. Data Availability Statement The data used in this study are publicly available from the NASA C-MAPSS dataset [ https://www.kaggle.com/datasets/behrad3d/nasa-cmaps (accessed on 3 January 2026)]. The processed data and code used to support the findings of this study are available from the corresponding author upon reasonable request. Conflicts of Interest The authors declare no conflicts of interest. Funding Statement This research was funded by the National Natural Science Foundation of China, grant number 62003337, and the Aviation Science Fund, grant number 20240048054001. Footnotes Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. References 1. 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Data Availability Statement The data used in this study are publicly available from the NASA C-MAPSS dataset [ https://www.kaggle.com/datasets/behrad3d/nasa-cmaps (accessed on 3 January 2026)]. The processed data and code used to support the findings of this study are available from the corresponding author upon reasonable request. 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