# Neural Ordinary Differential Equations on Manifolds

arXiv: https://arxiv.org/abs/2006.06663 · HTML: https://mlsanity.com/paper/2006.06663/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2020.

## Built on by (most cited first, 30 of 31)

- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [Moser Flow: Divergence-based Generative Modeling on Manifolds](https://mlsanity.com/paper/2108.08052/) (2021)
- [Matching Normalizing Flows and Probability Paths on Manifolds](https://mlsanity.com/paper/2207.04711/) (2022)
- [Introduction to Normalizing Flows for Lattice Field Theory](https://mlsanity.com/paper/2101.08176/) (2021)
- [Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes](https://mlsanity.com/paper/2310.07216/) (2023)
- [Riemannian Residual Neural Networks](https://mlsanity.com/paper/2310.10013/) (2023)
- [Equivariant Manifold Flows](https://mlsanity.com/paper/2107.08596/) (2021)
- [Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups](https://mlsanity.com/paper/2405.16381/) (2024)
- [Scaling Riemannian Diffusion Models](https://mlsanity.com/paper/2310.20030/) (2023)
- [Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modeling](https://mlsanity.com/paper/2304.03937/) (2023)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifolds](https://mlsanity.com/paper/2510.21608/) (2025)
- [Density estimation on smooth manifolds with normalizing flows](https://mlsanity.com/paper/2106.03500/) (2021)
- [Deep Efficient Continuous Manifold Learning for Time Series Modeling](https://mlsanity.com/paper/2112.03379/) (2021)
- [Learning Distributions on Manifolds with Free-Form Flows](https://mlsanity.com/paper/2312.09852/) (2023)
- [Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs](https://mlsanity.com/paper/2305.08849/) (2023)
- [Non-Perturbative Trivializing Flows for Lattice Gauge Theories](https://mlsanity.com/paper/2410.13161/) (2024)
- [Learning ODEs via Diffeomorphisms for Fast and Robust Integration](https://mlsanity.com/paper/2107.01650/) (2021)
- [Equivariant Manifold Neural ODEs and Differential Invariants](https://mlsanity.com/paper/2401.14131/) (2024)
- [Are Euler angles a useful rotation parameterisation for pose estimation with Normalizing Flows?](https://mlsanity.com/paper/2511.02277/) (2025)
- [Manifold Aware Denoising Score Matching (MAD)](https://mlsanity.com/paper/2603.02452/) (2026)
- [Optimal Potential Shaping on SE(3) via Neural ODEs on Lie Groups](https://mlsanity.com/paper/2401.15107/) (2024)
- [Geometry-Preserving Neural Architectures on Manifolds with Boundary](https://mlsanity.com/paper/2602.03082/) (2026)
- [PolyNODE: Variable-dimension Neural ODEs on M-polyfolds](https://mlsanity.com/paper/2602.15128/) (2026)
- [An Invariant Compiler for Neural ODEs in AI-Accelerated Scientific Simulation](https://mlsanity.com/paper/2603.23861/) (2026)
- [Steerable Neural ODEs on Homogeneous Spaces](https://mlsanity.com/paper/2605.11133/) (2026)
- [Learning Unbiased Permutations via Flow Matching](https://mlsanity.com/paper/2605.16755/) (2026)
- [Riemannian Neural Hamiltonian Flows: Geodesic Symplectic Transport and Interpretability](https://mlsanity.com/paper/2609.21647/) (2026)

## Abstract (arXiv)

Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately current approaches fall short when the underlying space has a non trivial topology, and are only available for the most basic geometries. Recently normalizing flows in Euclidean space based on Neural ODEs show great promise, yet suffer the same limitations. Using ideas from differential geometry and geometric control theory, we describe how neural ODEs can be extended to smooth manifolds. We show how vector fields provide a general framework for parameterizing a flexible class of invertible mapping on these spaces and we illustrate how gradient based learning can be performed. As a result we define a general methodology for building normalizing flows on manifolds.