Disentangling Slow and Fast Temporal Dynamics in Degradation Inference with Hierarchical Differential Models

Abstract: Reliable inference of system degradation from sensor data is fundamental to condition monitoring and prognostics in engineered systems. Since degradation is rarely observable and measurable, it must be inferred to enable accurate health assessment and decision-making. This is particularly challenging because operational variations dominate system behavior, while degradation introduces only subtle, long-term changes. Consequently, sensor data mainly reflect short-term operational variability, making it difficult to disentangle the underlying degradation process. Residual-based methods are widely employed, but the residuals remain entangled with operational history, often resulting in noisy and unreliable degradation estimation, particularly in systems with dynamic responses. Neural Ordinary Equations (NODEs) offer a promising framework for inferring latent dynamics, but the time-scale separation in slow-fast systems introduces numerical stiffness and complicates training, while degradation disentanglement remains difficult. To address these limitations, we propose a novel Hierarchical Controlled Differential Equation (H-CDE) framework that incorporates a slow (degradation) and a fast (operation) CDE component in a unified architecture. It introduces three key innovations: a multi-scale time integration scheme to mitigate numerical stiffness; a learnable path transformation that extracts latent degradation drivers to control degradation evolution; and a novel activation function that enforces monotonicity on inferred degradation as a regularizer for disentanglement. Through comprehensive evaluations on both dynamic response (e.g., bridges) and steady state (e.g., aero-engine) systems, we demonstrate that H-CDE effectively disentangles degradation from operational dynamics and outperforms residual-based baselines, yielding more accurate, robust, and interpretable inference.