PolyNODE: Variable-dimension Neural ODEs on M-polyfolds

2026-02-16

AI Summary

This work proposes PolyNODE, the first variable-dimensional flow model grounded in M-polyfolds theory, addressing the inherent limitation of traditional neural ordinary differential equations (NODEs), which are confined to fixed-dimensional manifolds and thus incapable of modeling dynamical systems with varying dimensions. By parameterizing vector fields over differentiable M-polyfolds, PolyNODE enables cross-dimensional data reconstruction and representation learning. The approach integrates variable-dimensional neural ODEs, flow matching, and an autoencoder architecture, marking the first breakthrough in geometric deep learning that transcends the dimensional constraints of conventional NODEs. Experimental results demonstrate that PolyNODE not only achieves effective reconstruction across dimensions but also learns high-quality latent representations, significantly enhancing downstream classification performance and thereby validating its efficacy and potential for modeling variable-dimensional dynamical systems.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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 .

Problem

• Neural ODEs
• variable-dimension
• M-polyfolds
• geometric deep learning
• dimensional bottlenecks

Innovation

• Neural ODEs
• M-polyfolds
• variable-dimension dynamics
• geometric deep learning
• flow-based models