CHyLL: Learning Continuous Neural Representations of Hybrid Systems

2025-12-10

AI Summary

Modeling hybrid systems—exhibiting both continuous flows and discrete jumps—is challenging due to the tight coupling between mode switches and flow discontinuities, which impedes effective neural representation. Method: We propose an end-to-end unsupervised neural implicit modeling framework that reformulates the hybrid system’s state space as a piecewise-smooth quotient manifold equipped with reset maps. This geometric construction ensures that the originally discontinuous dynamics become continuous and singularity-free in the embedded latent space, grounded theoretically in differential topological embedding theorems guaranteeing preservation of topological invariants. Crucially, the method requires no explicit event detection, trajectory segmentation, or prior specification of switching logic. Contribution/Results: Experiments demonstrate high-fidelity long-term trajectory prediction across diverse hybrid systems, accurate recovery of underlying topological structure, and successful transfer to stochastic optimal control tasks—establishing a principled, geometry-aware foundation for learning hybrid dynamics.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Learning the flows of hybrid systems that have both continuous and discrete time dynamics is challenging. The existing method learns the dynamics in each discrete mode, which suffers from the combination of mode switching and discontinuities in the flows. In this work, we propose CHyLL (Continuous Hybrid System Learning in Latent Space), which learns a continuous neural representation of a hybrid system without trajectory segmentation, event functions, or mode switching. The key insight of CHyLL is that the reset map glues the state space at the guard surface, reformulating the state space as a piecewise smooth quotient manifold where the flow becomes spatially continuous. Building upon these insights and the embedding theorems grounded in differential topology, CHyLL concurrently learns a singularity-free neural embedding in a higher-dimensional space and the continuous flow in it. We showcase that CHyLL can accurately predict the flow of hybrid systems with superior accuracy and identify the topological invariants of the hybrid systems. Finally, we apply CHyLL to the stochastic optimal control problem.

Problem

• Learning continuous flows of hybrid systems without mode switching
• Reformulating state space as a quotient manifold for continuity
• Predicting flows and identifying topological invariants accurately

Innovation

• Learns continuous neural representations without trajectory segmentation
• Reformulates state space as piecewise smooth quotient manifold
• Learns singularity-free neural embedding and continuous flow concurrently