# CHyLL: Learning Continuous Neural Representations of Hybrid Systems

arXiv: https://arxiv.org/abs/2512.10117 · HTML: https://mlsanity.com/paper/2512.10117/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 71 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 30 of 40)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [On the Continuity of Rotation Representations in Neural Networks](https://mlsanity.com/paper/1812.07035/) (2018)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [MuJoCo Playground](https://mlsanity.com/paper/2502.08844/) (2025)
- [Hybrid Internal Model: Learning Agile Legged Locomotion with Simulated Robot Response](https://mlsanity.com/paper/2312.11460/) (2023)
- [ContactNets: Learning Discontinuous Contact Dynamics with Smooth, Implicit Representations](https://mlsanity.com/paper/2009.11193/) (2020)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [Deep Learning of Koopman Representation for Control](https://mlsanity.com/paper/2010.07546/) (2020)
- [Neural Conditional Probability for Uncertainty Quantification](https://mlsanity.com/paper/2407.01171/) (2024)
- [Toward Safety-Aware Informative Motion Planning for Legged Robots](https://mlsanity.com/paper/2103.14252/) (2021)
- [Vysics: Object Reconstruction Under Occlusion by Fusing Vision and Contact-Rich Physics](https://mlsanity.com/paper/2504.18719/) (2025)
- [Simultaneous Learning of Contact and Continuous Dynamics](https://mlsanity.com/paper/2310.12054/) (2023)
- [GMKF: Generalized Moment Kalman Filter for Polynomial Systems with Arbitrary Noise](https://mlsanity.com/paper/2403.04712/) (2024)
- [Neural Hybrid Automata: Learning Dynamics with Multiple Modes and Stochastic Transitions](https://mlsanity.com/paper/2106.04165/) (2021)
- [Discrete-Time Hybrid Automata Learning: Legged Locomotion Meets Skateboarding](https://mlsanity.com/paper/2503.01842/) (2025)
- [Learning Neural Differential Algebraic Equations via Operator Splitting](https://mlsanity.com/paper/2403.12938/) (2024)
- [Ego-Vision World Model for Humanoid Contact Planning](https://mlsanity.com/paper/2510.11682/) (2025)
- [Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems](https://mlsanity.com/paper/2507.07222/) (2025)
- [Topologically-Informed Atlas Learning](https://mlsanity.com/paper/2110.00429/) (2021)
- [Max Entropy Moment Kalman Filter for Polynomial Systems with Arbitrary Noise](https://mlsanity.com/paper/2506.00838/) (2025)
- [Learning Linear Complementarity Systems](https://mlsanity.com/paper/2112.13284/) (2021)
- [Convex Geometric Motion Planning on Lie Groups via Moment Relaxation](https://mlsanity.com/paper/2305.13565/) (2023)
- [Legged Robot State Estimation within Non-inertial Environments](https://mlsanity.com/paper/2403.16252/) (2024)
- [A Generalized Metriplectic System via Free Energy and System~Identification via Bilevel Convex Optimization](https://mlsanity.com/paper/2410.06233/) (2024)

## Built on by (most cited first, 2 of 2)

- [Embedding Hybrid Systems into Continuous Latent Vector Fields](https://mlsanity.com/paper/2606.10596/) (2026)
- [Learning Continuous Neural Representation of Stochastic Hybrid Systems](https://mlsanity.com/paper/2609.38893/) (2026)

## Abstract (arXiv)

Learning the flows of hybrid systems that have both continuous and discrete time dynamics is challenging. The existing method learns the dynamics in each discrete mode, which suffers from the combination of mode switching and discontinuities in the flows. In this work, we propose CHyLL (Continuous Hybrid System Learning in Latent Space), which learns a continuous neural representation of a hybrid system without trajectory segmentation, event functions, or mode switching. The key insight of CHyLL is that the reset map glues the state space at the guard surface, reformulating the state space as a piecewise smooth quotient manifold where the flow becomes spatially continuous. Building upon these insights and the embedding theorems grounded in differential topology, CHyLL concurrently learns a singularity-free neural embedding in a higher-dimensional space and the continuous flow in it. We showcase that CHyLL can accurately predict the flow of hybrid systems with superior accuracy and identify the topological invariants of the hybrid systems. Finally, we apply CHyLL to the stochastic optimal control problem.