Continuous models simplify probability tracking in hybrid stochastic systems
Learning Continuous Neural Representation of Stochastic Hybrid Systems
Machine LearningArtificial Intelligence
Summary
Stochastic hybrid systems combine predictable continuous changes with sudden, random resets. The authors found a way to represent these complex systems as smoother processes in a space with more dimensions. This approach removes the need to separately handle the sudden resets and makes the system's probability behavior easier to model and predict. Their method uses extra variables to encode the resets, turning them into continuous transitions that fit together neatly. This lets one model capture the whole system's probability changes without breaking the behavior into parts or tracking reset events explicitly.
What this means in practice
- •For control system engineers: Model complex systems with mixed continuous and discrete behaviors without needing event-driven simulations.
- •For robotics developers: Simulate hybrid robotic systems’ behavior more efficiently by using smooth continuous models in higher dimensions.
Authors
Sangli Teng, Hang Liu, Koushil Sreenath
Abstract
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.