Hierarchical Controlled Differential Equations (H-CDE)

Official implementation of "Disentangling Slow and Fast Temporal Dynamics in Degradation Inference with Hierarchical Differential Models", accepted at Reliability Engineering & System Safety (RESS).

Mengjie Zhao, Olga Fink

[arXiv]

Overview

Degradation introduces only subtle, long-term changes in sensor observations, while fast operational variations dominate the signal. H-CDE is a hierarchical controlled differential equation framework that explicitly disentangles slow degradation dynamics from fast operational dynamics through learnable path transformations with monotonicity-enforcing activation functions. The method achieves state-of-the-art performance over residual-based approaches on both mechanical (N-CMAPSS aero-engine) and infrastructural (bridge) systems.

Environment Setup

Python 3.11 and PyTorch 2.2.0 are required. We recommend conda:

conda create -n hcde python=3.11
conda activate hcde
pip install -r requirements.txt

Note: numpy must stay below 2.0 for compatibility with PyTorch 2.2.0. The requirements.txt pins numpy==1.26.4.

Then set the project root on your PYTHONPATH:

export PYTHONPATH=/path/to/hcde

Datasets

Dataset	Domain	Config dir
N-CMAPSS	Aero-engine RUL	run/configs/ncmapss/
Bridge (synthetic)	Structural degradation	run/configs/bridge/

Bridge dataset

The bridge dataset is a synthetic dataset generated via a physics-based finite element simulation implemented in src/simulation/beam.py. The model represents a simply-supported 2D Euler–Bernoulli beam subject to:

• Distributed traffic load and ambient temperature time series as inputs
• Rayleigh damping and a Newmark-beta dynamic solver for structural response
• Continuum damage mechanics (CDM) for progressive stiffness degradation: Young's modulus E and second moment of area I decay with accumulated damage D
• Thermal loading (axial expansion + bending moment from through-depth temperature gradient via an exponential filter)

The simulation outputs per-record displacement, velocity, and acceleration at selected span locations, which form the sensor observations used for training and evaluation.

The N-CMAPSS dataset must be downloaded separately. Place all dataset files under the datasets/ directory matching the dataset.dir field in the corresponding config YAML.

Running Experiments

The entry point is run/main.py, controlled by a YAML config file:

python run/main.py --cfg run/configs/<dataset>/<model>.yaml

N-CMAPSS (aero-engine)

# Full H-CDE model
python run/main.py --cfg run/configs/ncmapss/hcde.yaml

# Ablation: without path transformation
python run/main.py --cfg run/configs/ncmapss/hcde-p=f.yaml

# Ablation: without monotonic activation
python run/main.py --cfg run/configs/ncmapss/hcde-pt=f.yaml

# FNN baseline
python run/main.py --cfg run/configs/ncmapss/fnn.yaml

Bridge

# Full H-CDE model
python run/main.py --cfg run/configs/bridge/hcde.yaml

# Ablation: without path transformation
python run/main.py --cfg run/configs/bridge/hcde-p=f.yaml

# Ablation: without monotonic activation
python run/main.py --cfg run/configs/bridge/hcde-pt=f.yaml

# FNN baseline
python run/main.py --cfg run/configs/bridge/fnn.yaml

Results and checkpoints are written to the directory specified by out_dir in the config. TensorBoard logs are saved alongside:

tensorboard --logdir results/

Citation

If you find this work useful, please cite:

@article{zhao2026hcde,
  title     = {Disentangling Slow and Fast Temporal Dynamics in Degradation Inference with Hierarchical Differential Models},
  author    = {Zhao, Mengjie and Fink, Olga},
  journal   = {Reliability Engineering \& System Safety},
  year      = {2026},
  note      = {arXiv:2509.00639},
  doi       = {10.48550/arXiv.2509.00639}
}