Embedding Hybrid Systems into Continuous Latent Vector Fields

2026-06-09

AI Summary

This study addresses the challenge of representing hybrid systems—characterized by intrinsic discontinuities—in a continuous form amenable to differential optimization. The authors theoretically establish that any hybrid system can be embedded into a Euclidean space with a continuous vector field, provided the embedding dimension exceeds twice the dimensionality of the system’s state space. Building on this result, they propose a latent-variable neural ODE model that learns hybrid system dynamics from time-series data alone by jointly optimizing a consistency loss between the state space and the latent space. This work provides the first theoretical foundation for the existence of continuous extrinsic representations of hybrid systems and demonstrates high-fidelity flow reconstruction across diverse geometric configurations, significantly outperforming existing methods.

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 $m>2n$. This result suggests that an intrinsically discontinuous hybrid system generically admits a continuous extrinsic representation that is well-posed for differentiable optimization. Building on this existence theorem, we show that a latent Neural ODE with consistency loss in both the latent and state space can accurately recover the flow of hybrid systems. Extensive experiments suggest the proposed method outperforms the existing method in learning hybrid systems with varying geometries from only time series data.

Problem

• hybrid systems
• continuous representation
• embedding
• differentiable optimization
• latent vector fields

Innovation

• hybrid systems
• continuous latent vector fields
• Neural ODE
• embedding theorem
• differentiable optimization