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:2606. 10596v1 Announce Type: cross Abstract: 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 $m>2n$.
arXiv:2607. 27924v1 Announce Type: new Abstract: In the physical world we inhabit, space and time are fundamentally continuous.
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.
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.
arXiv:2606. 14284v1 Announce Type: cross Abstract: Time series prototype learning is fundamentally challenged by observational ambiguity.
arXiv:2608. 06595v1 Announce Type: cross Abstract: Neural networks applied to sequential decision-making tasks typically rely on latent representations of environment states.
arXiv:2410.23667v2 Announce Type: replace Abstract: Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known con...
arXiv:2609.37083v1 Announce Type: cross Abstract: We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing...
arXiv:2604. 16232v2 Announce Type: replace-cross Abstract: Understanding natural and engineered systems often relies on symbolic formulations, such as differential equations, which provide interpretability and transferability beyond black-box models.
arXiv:2607. 05280v1 Announce Type: new Abstract: Many real-world systems evolve continuously, yet most machine learning models interpret time series as discrete sequences.
arXiv:2602. 03670v2 Announce Type: replace-cross Abstract: Equilibrium Propagation (EP) is a physics-inspired learning algorithm that uses stationary states of a dynamical system both for inference and learning.
The paper presents a differentiable hybrid modelling framework that combines a JAX finite volume population balance solver with learnable neural network components. This approach learns constitutive laws and initial conditions directly from experimental data, improving the fidelity of transport models in chemical engineering. The framework’s differentiability also enables optimisation of experimental settings for desired process outcomes.