Embedding Hybrid Systems into Continuous Latent Vector Fields

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.

Hybrid systems (hybrid automata) can model a wide range of dynamical systems that are governed by both discrete state transitions and continuous flows. Examples include humanoid locomotion, thermal stat control, task and motion planning, etc. Learning such a system is hard, as the discrete state transition will break the differentiability of ordinary differential equations (ODEs). This work proves that the hybrid systems can be described by continuous vector fields in a higher-dimensional latent space and proposes a latent ODE framework to learn them.