# Hierarchical ODE: Learning Continuous-Time Physical Prototypes for Early Link Failure Detection

arXiv: https://arxiv.org/abs/2606.14284 · HTML: https://mlsanity.com/paper/2606.14284/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 12 Jun 2026.

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

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [This Looks Like That: Deep Learning for Interpretable Image Recognition](https://mlsanity.com/paper/1806.10574/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Time-Series Representation Learning via Temporal and Contextual Contrasting](https://mlsanity.com/paper/2106.14112/) (2021)
- [Environment Inference for Invariant Learning](https://mlsanity.com/paper/2010.07249/) (2020)
- [Unsupervised Representation Learning for Time Series with Temporal Neighborhood Coding](https://mlsanity.com/paper/2106.00750/) (2021)
- [Generalized Category Discovery](https://mlsanity.com/paper/2201.02609/) (2022)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [The Roadmap to 6G -- AI Empowered Wireless Networks](https://mlsanity.com/paper/1904.11686/) (2019)
- [CoST: Contrastive Learning of Disentangled Seasonal-Trend Representations for Time Series Forecasting](https://mlsanity.com/paper/2202.01575/) (2022)
- [AutoNovel: Automatically Discovering and Learning Novel Visual Categories](https://mlsanity.com/paper/2106.15252/) (2021)
- [On Disentangled Representations Learned From Correlated Data](https://mlsanity.com/paper/2006.07886/) (2020)
- [Coupled Oscillatory Recurrent Neural Network (coRNN): An accurate and (gradient) stable architecture for learning long time dependencies](https://mlsanity.com/paper/2010.00951/) (2020)
- [DeepDPM: Deep Clustering With an Unknown Number of Clusters](https://mlsanity.com/paper/2203.14309/) (2022)
- [Banyan: Improved Representation Learning with Explicit Structure](https://mlsanity.com/paper/2407.17771/) (2024)
- [FIC-TSC: Learning Time Series Classification with Fisher Information Constraint](https://mlsanity.com/paper/2505.06114/) (2025)
- [Self-Organizing Visual Prototypes for Non-Parametric Representation Learning](https://mlsanity.com/paper/2505.21533/) (2025)
- [IMTS is Worth Time × Channel Patches: Visual Masked Autoencoders for Irregular Multivariate Time Series Prediction](https://mlsanity.com/paper/2505.22815/) (2025)

## Abstract (arXiv)

Time series prototype learning is fundamentally challenged by observational ambiguity. Discrete architectures fail to resolve this, as they lack the capacity to decouple stochastic noise from continuous dynamics. Furthermore, rigid closed-set assumptions fail to capture unseen diversity. To address these limitations, we propose a hierarchical ordinary differential equation clustering network, which utilizes neural ordinary differential equation to model latent state evolution as a continuous integral curve. This formulation enforces temporal continuity to effectively disentangle smooth feature trends from stochastic noise, while our adaptive hierarchical mechanism autonomously determines the appropriate number of prototypes without rigid prior constraints. Validated on the early link failure detection task with irregularly sampled time series, the proposed method effectively extracts underlying physical prototypes, thereby enabling robust failure detection. Our code is available at https://github.com/NJ-LNN/Hierarchical-ODE.