📄 中文摘要
神经常微分方程（NODEs）是基于动力系统和流形上向量场生成的流的几何深度学习模型。尽管在流匹配范式中取得了众多成功应用，但现有的NODE模型在本质上受到流形维度的限制，无法处理固定维度以外的动态。研究提出了将NODE扩展到M-多重流形（能够同时容纳可变维度和可微性概念的空间），并引入了PolyNODE，成为几何深度学习中首个可变维度的基于流的模型。作为应用示例，构建了具有维度瓶颈的显式M-多重流形和PolyNODE自编码器。
PolyNODE：基于 M-多重流形的可变维度神经常微分方程
出处: PolyNODE: Variable-dimension Neural ODEs on M-polyfolds
发布: 2026年2月18日
📄 English Summary
PolyNODE: Variable-dimension Neural ODEs on M-polyfolds
Neural ordinary differential equations (NODEs) are geometric deep learning models rooted in dynamical systems and flows generated by vector fields on manifolds. Despite numerous successful applications, particularly within the flow matching paradigm, existing NODE models are fundamentally constrained to fixed-dimensional dynamics due to the intrinsic nature of the manifold's dimension. This research extends NODEs to M-polyfolds, which can simultaneously accommodate varying dimensions and a notion of differentiability, introducing PolyNODEs as the first variable-dimensional flow-based model in geometric deep learning. An explicit construction of M-polyfolds featuring dimensional bottlenecks and a PolyNODE autoencoder is provided as an example application.