# Continuous normalizing flows on manifolds

arXiv: https://arxiv.org/abs/2104.14959 · HTML: https://mlsanity.com/paper/2104.14959/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 56 references that are in the map.
- First posted on arXiv 14 Mar 2021.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Implicit Reparameterization Gradients](https://mlsanity.com/paper/1805.08498/) (2018)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [The Power Spherical distribution](https://mlsanity.com/paper/2006.04437/) (2020)
- [Reparameterization Gradients through Acceptance-Rejection Sampling Algorithms](https://mlsanity.com/paper/1610.05683/) (2016)
- [The embedding dimension of Laplacian eigenfunction maps](https://mlsanity.com/paper/1605.01643/) (2016)
- [Increasing Expressivity of a Hyperspherical VAE](https://mlsanity.com/paper/1910.02912/) (2019)

## Built on by (most cited first, 10 of 10)

- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [Learning Distributions on Manifolds with Free-Form Flows](https://mlsanity.com/paper/2312.09852/) (2023)
- [Non-Perturbative Trivializing Flows for Lattice Gauge Theories](https://mlsanity.com/paper/2410.13161/) (2024)
- [Discovering Symmetry Groups with Flow Matching](https://mlsanity.com/paper/2512.20043/) (2025)
- [Cartan flow matching](https://mlsanity.com/paper/2605.03588/) (2026)
- [Intrinsic Flow Matching on Quantum Pure-State Manifolds with Phase-Aligned Transport](https://mlsanity.com/paper/2606.21256/) (2026)

## Abstract (arXiv)

Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions. Unfortunately, current approaches are only available for the most basic geometries and fall short when the underlying space has a nontrivial topology, limiting their applicability for most real-world data. Using fundamental ideas from differential geometry and geometric control theory, we describe how the recently introduced Neural ODEs and continuous normalizing flows can be extended to arbitrary smooth manifolds. We propose a general methodology for parameterizing vector fields on these spaces and demonstrate how gradient-based learning can be performed. Additionally, we provide a scalable unbiased estimator for the divergence in this generalized setting. Experiments on a diverse selection of spaces empirically showcase the defined framework's ability to obtain reparameterizable samples from complex distributions.