# Learning Continuous Neural Representation of Stochastic Hybrid Systems

arXiv: https://arxiv.org/abs/2609.38893 · HTML: https://mlsanity.com/paper/2609.38893/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 48 references that are in the map.
- First posted on arXiv 30 Sep 2026.

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

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations](https://mlsanity.com/paper/2502.02472/) (2025)
- [Toward Safety-Aware Informative Motion Planning for Legged Robots](https://mlsanity.com/paper/2103.14252/) (2021)
- [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)
- [In-Context Learning of Stochastic Differential Equations with Foundation Inference Models](https://mlsanity.com/paper/2502.19049/) (2025)
- [Discrete-Time Hybrid Automata Learning: Legged Locomotion Meets Skateboarding](https://mlsanity.com/paper/2503.01842/) (2025)
- [Ego-Vision World Model for Humanoid Contact Planning](https://mlsanity.com/paper/2510.11682/) (2025)
- [Neural Stochastic Flows: Solver-Free Modelling and Inference for SDE Solutions](https://mlsanity.com/paper/2510.25769/) (2025)
- [Max Entropy Moment Kalman Filter for Polynomial Systems with Arbitrary Noise](https://mlsanity.com/paper/2506.00838/) (2025)
- [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)
- [Learning Stochastic Multiscale Models](https://mlsanity.com/paper/2506.22655/) (2025)
- [Learning Hybrid Dynamics via Convex Optimizations](https://mlsanity.com/paper/2509.24157/) (2025)
- [MePoly: Max Entropy Polynomial Policy Optimization](https://mlsanity.com/paper/2602.17832/) (2026)
- [Legged Robot State Estimation in Slippery Environments Using Invariant Extended Kalman Filter with Velocity Update](https://mlsanity.com/paper/2104.04238/) (2021)
- [Input Influence Matrix Design for MIMO Discrete-Time Ultra-Local Model](https://mlsanity.com/paper/2203.08729/) (2022)
- [Fully Proprioceptive Slip-Velocity-Aware State Estimation for Mobile Robots via Invariant Kalman Filtering and Disturbance Observer](https://mlsanity.com/paper/2209.15140/) (2022)
- [Convex Geometric Trajectory Tracking using Lie Algebraic MPC for Autonomous Marine Vehicles](https://mlsanity.com/paper/2305.09009/) (2023)
- [CHyLL: Learning Continuous Neural Representations of Hybrid Systems](https://mlsanity.com/paper/2512.10117/) (2025)
- [Stein Variational Ergodic Surface Coverage with SE(3) Constraints](https://mlsanity.com/paper/2603.09458/) (2026)
- [A Survey of Legged Robotics in Non-Inertial Environments: Past, Present, and Future](https://mlsanity.com/paper/2604.20990/) (2026)
- [Spectral Bayesian Estimation for General Stochastic Hybrid Systems](https://mlsanity.com/paper/2003.01201/) (2020)
- [Uncertainty Propagation for General Stochastic Hybrid Systems on Compact Lie Groups](https://mlsanity.com/paper/2203.02548/) (2022)
- [Embedding Hybrid Systems into Continuous Latent Vector Fields](https://mlsanity.com/paper/2606.10596/) (2026)

## Abstract (arXiv)

A stochastic hybrid system (SHS) is governed by a stochastic differential equation (SDE) describing the continuous dynamics and a Markov reset kernel triggered on the guard surface. Its probability evolution can be described by a hybrid Fokker-Planck (HFP) equation with a partial differential term corresponding to the SDE and an integral term arising from the reset kernel. This work shows that such an SHS can be approximated by an SDE in a higher-dimensional latent space where the sample paths are continuous. The key to this result is to encode different branches of the reset kernel using auxiliary variables, transforming the resets into deterministic ones that enable topological gluing. By the embedding theorem, the glued manifold can then be embedded into a higher-dimensional Euclidean space. We show that the probability evolution on the embedded image no longer requires explicit reset terms in the HFP equation. Building on this theorem, we design a loss that matches the evolving state distributions, enabling a single latent SDE to recover the probability evolution of the SHS without mode labeling, trajectory segmentation, or event-based simulations.