This work proves that an $n$-dimensional hybrid system can be embedded into an $m$-dimensional Euclidean space equipped with a continuous vector field on its embedded image whenever $m>2n$. This result suggests that an intrinsically discontinuous hybrid system generically admits a continuous extrinsic representation that is well-posed for differentiable optimization.
arXiv:2607. 27924v1 Announce Type: new Abstract: In the physical world we inhabit, space and time are fundamentally continuous.
By Dongxiu Liu, Haoyi Niu, Peng Cheng, Yuan Gao, Xirui Kang, Sangli Teng, Koushil Sreenath, Xianyuan Zhan
In the physical world we inhabit, space and time are fundamentally continuous. However, existing machine learning paradigms for world modeling are largely confined to discrete-time prediction, thereby exhibiting significant inefficiency in capturing the dynamics of physical world.
arXiv:2608. 06595v1 Announce Type: cross Abstract: Neural networks applied to sequential decision-making tasks typically rely on latent representations of environment states.
By Mohamed Ghanem, Bernd Finkbeiner
The paper demonstrates that a stochastic hybrid system (SHS), which combines continuous dynamics governed by a stochastic differential equation (SDE) with discrete resets triggered by a Markov kernel, can be approximated by a single SDE in a higher‑dimensional latent space. By encoding the reset branches with auxiliary variables, the resets become deterministic, allowing the system’s manifold to be glued and embedded into Euclidean space. This embedding eliminates the need for explicit reset terms in the hybrid Fokker‑Planck equation, and the authors propose a loss function that matches evolving state distributions, enabling the latent SDE to recover the SHS’s probability evolution without mode labeling, trajectory segmentation, or event‑based simulations.
By Sangli Teng, Hang Liu, Koushil Sreenath
arXiv:2606. 14284v1 Announce Type: cross Abstract: Time series prototype learning is fundamentally challenged by observational ambiguity.
By Jiaen Lv, Leran Qi, Shaowei Wang