AI 中文总结
本文提出将随机混合系统近似为高维潜在空间中的连续SDE，通过辅助变量编码重置核实现拓扑粘合，并设计匹配分布损失的单一潜在SDE，无需模式标签或事件模拟即可恢复概率演化。
AI 中文摘要
随机混合系统（SHS）由描述连续动态的随机微分方程（SDE）和在保护面上触发的马尔可夫重置核共同支配。其概率演化可由混合福克-普朗克（HFP）方程描述，该方程包含对应于SDE的偏微分项和由重置核产生的积分项。本工作表明，此类SHS可以在更高维的潜在空间中被近似为一个SDE，其中样本路径是连续的。这一结果的关键在于使用辅助变量编码重置核的不同分支，将重置转换为确定性重置，从而实现拓扑粘合。通过嵌入定理，粘合后的流形可以嵌入到更高维的欧几里得空间中。我们证明，嵌入图像上的概率演化不再需要HFP方程中的显式重置项。基于该定理，我们设计了一个匹配演化状态分布的损失函数，使得单个潜在SDE能够恢复SHS的概率演化，而无需模式标签、轨迹分割或基于事件的模拟。
英文摘要
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.
