Constraint-Guided Learning of Data-driven Health Indicator Models

This paper presents a constraint-guided deep learning (DL) framework to develop physically consistent health indicators (HIs) in bearing prognostics and health management. Conventional data-driven approaches often lack physical plausibility, while physics-based models are limited by incomplete knowledge of complex systems. To address this, we integrate domain knowledge into DL models via constraints, ensuring monotonicity, bounding output ranges between 1 and 0 (representing healthy to failed states, respectively), and maintaining consistency between signal energy trends and HI estimates. Using constraints eliminates the need for complex loss term balancing to incorporate domain knowledge. The constraint-guided gradient descent algorithm (CGGD) is used to train a DL model that satisfies specific constraints. Using time-frequency representations of accelerometer signals from the pronostia and XJTU-SY bearing datasets, the model learned using constraints generates more accurate and reliable representations of bearing health compared to conventional methods. It produces smoother degradation profiles that align with the expected physical behavior. Model performance is assessed using three metrics: trendability, robustness, and consistency. When compared to a conventional baseline model, the model learned using constraints shows a significant improvement in all three metrics. Another baseline incorporated the monotonicity behavior directly into the loss function using a soft-ranking approach. While this approach outperforms the model learned using constraints in trendability, due to its explicit monotonicity enforcement, the model learned with constraints performed better in robustness and consistency, providing stable and interpretable HI estimates over time. The ablation study confirms the importance of each constraint: the monotonicity constraint improves trendability, the boundary constraint ensures consistency, and the energy–HI consistency constraint enhances robustness. These findings demonstrate the effectiveness of CGGD in producing reliable and physically meaningful HIs for bearing prognostics and health management, offering a promising direction for future prognostic applications.