Problem
Research questions and friction points this paper is trying to address.
Stochastic Hybrid Systems
Hybrid Fokker-Planck Equation
Continuous Neural Representation
Latent Space Embedding
Probability Evolution
Learning Continuous Neural Representation of Stochastic Hybrid Systems
🤖 AI Summary
This study addresses the challenge of modeling probability evolution under discrete resets in stochastic hybrid systems by proposing a latent-space continuousization method. By introducing auxiliary variables to encode reset branches, the original system is embedded into a high-dimensional manifold and reformulated as a continuous stochastic differential equation (SDE), where topological gluing eliminates explicit reset terms. End-to-end optimization is achieved through a hybrid Fokker-Planck equation, neural representations, and distribution-matching losses. Without requiring mode labeling or event simulation, this approach accurately reconstructs the continuous probabilistic dynamics using a single latent SDE, providing a unified and efficient modeling framework for complex stochastic hybrid systems.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Stochastic Hybrid Systems
Continuous Neural Representation
Latent SDE
Topological Gluing
Hybrid Fokker-Planck Equation
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