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

arXiv: https://arxiv.org/abs/2509.00639 · HTML: https://mlsanity.com/paper/2509.00639/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 54 references that are in the map.
- First posted on arXiv 30 Aug 2025.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting](https://mlsanity.com/paper/2405.14616/) (2024)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Potential, Challenges and Future Directions for Deep Learning in Prognostics and Health Management Applications](https://mlsanity.com/paper/2005.02144/) (2020)
- [Neural Laplace: Learning diverse classes of differential equations in the Laplace domain](https://mlsanity.com/paper/2206.04843/) (2022)
- [Real-Time Model Calibration with Deep Reinforcement Learning](https://mlsanity.com/paper/2006.04001/) (2020)
- [A Comparison of Residual-based Methods on Fault Detection](https://mlsanity.com/paper/2309.02274/) (2023)
- [A Review of Machine Learning Methods Applied to Structural Dynamics and Vibroacoustic](https://mlsanity.com/paper/2204.06362/) (2022)
- [Graph Neural Networks for Virtual Sensing in Complex Systems: Addressing Heterogeneous Temporal Dynamics](https://mlsanity.com/paper/2407.18691/) (2024)
- [Physics-Informed Machine Learning for Battery Degradation Diagnostics: A Comparison of State-of-the-Art Methods](https://mlsanity.com/paper/2404.04429/) (2024)
- [Semi-Supervised Health Index Monitoring with Feature Generation and Fusion](https://mlsanity.com/paper/2312.02867/) (2023)
- [Deep Koopman Operator-based degradation modelling](https://mlsanity.com/paper/2308.01690/) (2023)
- [Health Index Estimation Through Integration of General Knowledge with Unsupervised Learning](https://mlsanity.com/paper/2405.04990/) (2024)
- [Efficient Simulation of Singularly Perturbed Systems Using a Stabilized Multirate Explicit Scheme](https://mlsanity.com/paper/2504.06371/) (2025)
- [Recent advances in data-driven methods for degradation modelling across applications](https://mlsanity.com/paper/2504.18164/) (2025)

## Built on by (most cited first, 3 of 3)

- [When, How Long and How Much? Interpretable Neural Networks for Time Series Regression by Learning to Mask and Aggregate](https://mlsanity.com/paper/2512.03578/) (2025)
- [Physics-Informed Condition Monitoring of SiC Power Modules](https://mlsanity.com/paper/2608.08363/) (2026)
- [Latent variable models for simultaneous EOV identification and removal in population-based SHM](https://mlsanity.com/paper/2608.11995/) (2026)

## Abstract (arXiv)

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.