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 mechanical and infrastructural systems. Since degradation is rarely directly observable and measurable, it must be inferred to enable accurate health assessment and decision-making. This is particularly challenging because operational and environmental variations dominate system behavior, while degradation introduces only subtle, long-term changes. Consequently, sensor data primarily reflect short-term operational variability, making it difficult to disentangle the underlying degradation process. Most unsupervised degradation inference methods learn nominal system behavior and use residuals as degradation proxies. However, residuals remain strongly entangled with operational history, yielding noisy and unreliable degradation estimates, particularly in infrastructural systems with dominant transient dynamics. Neural Ordinary Differential Equations (NODEs) offer a flexible framework for modeling latent dynamics, but in degraded systems, they suffer from numerical stiffness and degradation disentanglement remains difficult. To address these challenges, we propose a Hierarchical Controlled Differential Equation (H-CDE) framework that jointly models slow degradation dynamics and fast operational dynamics. H-CDE improves numerical efficiency through separate time integration of slow and fast components. Through a learnable path transformation mapping raw inputs to a latent degradation-relevant control path and a monotonicity-enforcing activation function that regularizes the inferred degradation dynamics, H-CDE enables effective disentangled degradation inference. Evaluations on mechanical and infrastructural systems demonstrate that H-CDE outperforms residual-based baselines, yielding more accurate, robust, and interpretable degradation inference in an unsupervised setting.