Analytical Health Indices: Towards Reliability-Informed Deep Learning for PHM

Deep learning has demonstrated significant potential for prognostics in complex systems (Fink et al., 2020). Recent advances in physics-informed machine learning have integrated physics-of-failure principles within data-driven models (AriasChao, Kulkarni, Goebel, & Fink, 2022). Beyond physical laws, fleet-level time-to-failure (TTF) distributions provide valuable prior knowledge for individual asset life predictions.In this paper we derive a probabilistic analytical health index(HI) model based on power-law degradation, enabling a probabilistic description that reconciles individual variability with fleet-wide trends. We show that, under Weibull, Gamma, and Pareto-distributed TTFs, the HI evolution follows an analytical form, allowing explicit characterization of time to reach intermediate degradation levels. Therefore, this work provides a theoretical foundation for integrating reliability principles with deep learning, advancing towards Reliability-Informed Deep Learning (RIDL). The approach is validated on synthetic turbofan engine data and real-world battery degradation datasets. This work establishes a rigorous basis for embedding reliability engineering principles into deep learning, improving predictive maintenance and remaining useful life (RUL) estimation.