突破固定维度限制，让神经微分方程可处理变维数据
PolyNODE: Variable-dimension Neural ODEs on M-polyfolds
- 将神经微分方程拓展至M-多流形，支持维度动态变化
- 在具有维度瓶颈的结构上实现数据重建与分类任务
- 首个可变维流模型，适合需要灵活表征空间的研究者
神经常微分方程（NODEs）是基于流形上向量场生成的动态系统的几何深度学习模型。尽管在流匹配等任务中表现优异，现有所有NODE模型均受限于流形固有维度，无法处理变维动态。本文将NODE扩展至M-多流形（可同时容纳不同维度并具备可微性），提出首个几何深度学习中的可变维流模型——PolyNODE。以具有维度瓶颈的显式M-多流形为例，构建基于参数化向量场的PolyNODE自编码器，使其穿越瓶颈。实验表明，该模型可有效训练完成重构任务，提取的潜在表示可用于下游分类。代码已公开于https://github.com/turbotage/PolyNODE。
原文摘要 · Abstract (English)
Neural ordinary differential equations (NODEs) are geometric deep learning models based on dynamical systems and flows generated by vector fields on manifolds. Despite numerous successful applications, particularly within the flow matching paradigm, all existing NODE models are fundamentally constrained to fixed-dimensional dynamics by the intrinsic nature of the manifold's dimension. In this paper, we extend NODEs to M-polyfolds (spaces that can simultaneously accommodate varying dimensions and a notion of differentiability) and introduce PolyNODEs, the first variable-dimensional flow-based model in geometric deep learning. As an example application, we construct explicit M-polyfolds featuring dimensional bottlenecks and PolyNODE autoencoders based on parametrised vector fields that traverse these bottlenecks. We demonstrate experimentally that our PolyNODE models can be trained to solve reconstruction tasks in these spaces, and that latent representations of the input can be extracted and used to solve downstream classification tasks. The code used in our experiments is publicly available at https://github.com/turbotage/PolyNODE .
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