Integrating Survival-Based Aging Models with Data-Driven RUL Prognostics

This study presents a novel hybrid prognostic framework for uncertainty-aware Remaining Useful Life (RUL) estimation in turbofan engines using the NASA C-MAPSS dataset. The framework employs a state-aware strategy that bifurcates the engines operational lifespan into "healthy" and "degraded" regimes. An LSTM-based autoencoder, trained strictly on nominal data (RUL > 150 cycles), monitors reconstruction error to act as a robust state classifier. For the healthy regime, a Conditional Weibull Survival Analysis is used for Mean Residual Life estimation. For the degraded regime, a Probabilistic Neural Network with Monte Carlo Dropout captures both aleatoric and epistemic uncertainties. Rather than using rigid binary labels, a calibrated sigmoid function converts the autoencoders output into continuous state probabilities, dynamically weighting the final ensemble prediction. The primary strength of this framework is its generation of physically consistent uncertainty bands, yielding high-confidence predictions near end-of-life while accurately reflecting the inherent variance of early operation, providing a robust tool for risk-informed maintenance.

Data-driven remaining useful life (RUL) prediction requires complete degradation trajectories for training, yet such run-to-failure data are scarce and expensive. Practitioners currently lack principled guidance on how many failure examples suffice for a given model and accuracy target. This paper develops a sample complexity framework for RUL prediction comprising seven main results organised around three themes. First, we establish fundamental learning rates: a distribution-free generalization bound shows that the uniform deviation of the mean squared error decreases as $O(B^{2}\sqrt{p/n})$, where $p$ is the model complexity and $n$ the number of trajectories, and a minimax lower bound proves that the $Θ(p/n)$ rate is unimprovable.} \rev{Second, we quantify how domain knowledge accelerates learning: incorporating degradation physics reduces data requirements by up to two orders of magnitude for deep networks, a Bernstein-type analysis achieves the minimax-optimal $O(p/n)$ rate under high signal-to-noise conditions, and closed-form penalties reveal when an incorrectly assumed physics model hurts rather than helps. Third, we characterise the impact of data quality: fleet variability induces an irreducible bias$-$variance tradeoff, while right-censored observations suffer an efficiency loss that depends critically on the degradation class.} Closed-form expressions are provided for exponential, power-law, and stretched-exponential degradation. \rev{Cross-domain validation against published turbofan, battery, and bearing benchmarks confirms the theoretical predictions within a factor of 2$-$3 on average. The results yield practical guidelines for planning data collection, selecting model complexity, and evaluating physics model assumptions in prognostics applications.

Time-to-event modeling provides a systematic framework for estimating time-dependent failure risk, reliability, and remaining useful life (RUL) from longitudinal condition monitoring data. However, applying these models to distributed prognostics remains challenging because sensor trajectories and failure-time records are often stored across organizations or operational sites and cannot be centrally pooled due to privacy or proprietary constraints. Moreover, the classical Cox proportional hazards model relies on a nonseparable partial likelihood involving global risk sets, making direct optimization difficult under standard federated learning protocols. This paper presents a federated longitudinal-survival modeling framework for collaborative system failure prognostics. The proposed framework combines longitudinal sensor representation learning with a client-separable discrete-time hazard objective, enabling multiple clients to collaboratively train a prognostic model without sharing raw sensor measurements or individual failure records. Time-dependent representations extracted from multivariate sensor histories are used to estimate interval-specific failure hazards, reliability curves, and system RUL. Experiments on the four C-MAPSS turbofan engine degradation subsets under simulated decentralized settings demonstrate that the proposed framework consistently improves prognostic performance over isolated local training while maintaining performance comparable to centralized training across heterogeneous operating conditions and failure modes. These results demonstrate the potential of federated longitudinal-survival modeling for collaborative, data-aware condition monitoring and system failure prognostics.