将不连续的混合系统嵌入连续向量场，实现可微优化求解。
Embedding Hybrid Systems into Continuous Latent Vector Fields
- 通过构造连续向量场，将n维混合系统嵌入2n维以上空间。
- 用带一致性损失的隐变量神经ODE，准确恢复混合系统的流形轨迹。
- 适用于仅从时间序列学习复杂几何混合系统的研究者。
本工作证明，当维度m>2n时，任意n维混合系统均可嵌入m维欧几里得空间，并在其像上定义连续向量场。该结果表明，本质上不连续的混合系统通常存在一个对可微优化良好的连续外部表示。基于此存在性定理，我们提出一种在隐空间与状态空间均施加一致性损失的隐变量神经微分方程（Neural ODE），能精确恢复混合系统的流。大量实验表明，该方法在仅从时间序列数据学习具有不同几何结构的混合系统时，优于现有方法。
原文摘要 · Abstract (English)
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.
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