# PolyNODE: Variable-dimension Neural ODEs on M-polyfolds

arXiv: https://arxiv.org/abs/2602.15128 · HTML: https://mlsanity.com/paper/2602.15128/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 29 references that are in the map.
- First posted on arXiv 16 Feb 2026.

## Builds on (most cited first, 12 of 12)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [Neural Ordinary Differential Equations on Manifolds](https://mlsanity.com/paper/2006.06663/) (2020)
- [Equivariant Manifold Neural ODEs and Differential Invariants](https://mlsanity.com/paper/2401.14131/) (2024)

## Abstract (arXiv)

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 .