Researchers Develop Method to Embed Hybrid Systems into Continuous Vector Fields for Neural Network Learning
A new paper accepted to ICML 2026 proves that any n-dimensional hybrid dynamical system can be embedded into a continuous vector field in a higher-dimensional Euclidean space, enabling differentiable optimization of inherently discontinuous systems. The authors build on this theoretical result to train a latent Neural ODE with a consistency loss that accurately recovers hybrid system dynamics from time series data alone. This matters because hybrid systems—which mix continuous and discrete dynamics—are notoriously difficult to learn with gradient-based methods, and this work offers a principled, geometry-aware solution.
Hybrid dynamical systems, which combine continuous flows with discrete switching events, pose a fundamental challenge for machine learning because their discontinuities are incompatible with standard differentiable optimization. This paper establishes a formal embedding theorem showing that any n-dimensional hybrid system can be represented by a continuous vector field in an m-dimensional Euclidean space whenever m exceeds 2n, guaranteeing the existence of a well-posed continuous extrinsic representation. Leveraging this theoretical foundation, the authors propose a latent Neural ODE architecture augmented with a consistency loss applied in both the latent and original state spaces to faithfully recover hybrid system trajectories. Extensive experiments demonstrate that the method outperforms existing approaches across hybrid systems with varying geometric structures, using only observed time series as input. The work bridges dynamical systems theory and neural differential equations, with potential applications in robotics, control, and physics simulation where switching behaviors are common.
What's missing
It is unclear how the method handles noise or irregular sampling in real-world time series, or whether the learned embeddings remain interpretable. The generalization guarantees beyond the training distribution of hybrid system geometries are not addressed.
What different sources said
- arXiv cs.LGCenter
Embedding Hybrid Systems into Continuous Latent Vector Fields
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