Neural Ordinary Differential Equations on Manifolds

Luca Falorsi; Patrick Forr\'e

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Neural Ordinary Differential Equations on Manifolds

Luca Falorsi, Patrick Forré

Machine Learning · Submitted 11 Jun 2020

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim As a result we define a general methodology for building normalizing flows on manifolds.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,834 input tokens.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.14 · human participants 0.02 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.84 · mentions code or data availability 0.02.

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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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Normalizing flows are a powerful technique for obtaining reparameterizable samples from complex multimodal distributions.	background 1.00	0.06	0.03	asserted 1.00
s1	Unfortunately current approaches fall short when the underlying space has a non trivial topology, and are only available for the most basic geometries.	gap 0.89	0.06	0.03	asserted 1.00
s2	Recently normalizing flows in Euclidean space based on Neural ODEs show great promise, yet suffer the same limitations.	background 0.68	0.07	0.03	asserted 0.96
s3	Using ideas from differential geometry and geometric control theory, we describe how neural ODEs can be extended to smooth manifolds.	contribution 0.89	0.85	0.16	asserted 0.97
s4	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.	method 0.66	0.92	0.20	asserted 1.00
s5	As a result we define a general methodology for building normalizing flows on manifolds.	contribution 0.94	0.91	0.78	asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from proof or derivation 0.57.

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C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

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.

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References

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Riemannian Score-Based Generative Modelling
cs.LG 2022-02 conditional novelty 7.0 of 10

Score-based generative models can be defined directly on Riemannian manifolds by running the noising diffusion on the manifold and simulating its time-reversal with geodesic random walks, which scales better than Mose...
HyperFlow: Class-Incremental Semantic Change Detection via Hyperbolic Flow
cs.CV 2026-11 conditional novelty 6.0 of 10

A model that treats land-cover change as deviations from a learned hyperbolic flow extends semantic change detection to class-incremental settings, alongside a new hierarchical remote sensing dataset.
Riemannian Flow Models with Reinforcement Learning for Molecular Crystal Structure Prediction
cs.LG 2026-09 conditional novelty 6.0 of 10

CG-OMatG, a coarse-grained Riemannian flow model with manifold policy-gradient RL, lifts the OMC25-MCF solved rate from 3.9% to 12.7%, though RL adds no solved blind-test target and the reward shares its energy model ...
Introduction to Normalizing Flows for Lattice Field Theory
hep-lat 2021-01 conditional novelty 2.0 of 10

A hands-on Jupyter notebook tutorial that implements normalizing-flow samplers for 2D scalar and U(1) gauge lattice field theories, including gauge-equivariant coupling layers and an independence-Metropolis correction step.

How the literature uses it

25 arXiv papers cite this in 35 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 16 sentences without a confident label

Contests it 1 paper

Mathieu et al. [ref] , Lou et al. [ref] and Falorsi et al. [ref] propose continuous normalizing flows for general Riemannian manifold, without considering any property of $$ space, which leads to unsatisfactory performance for probabilistic rotation modeling.

Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modeling · Introduction · contest 0.83

Compares with it 1 paper

We note in passing that some authors [ref] have reported competitive or even superior performance in a similar setting by replacing the projection method with continuous normalizing flows for Riemannian manifolds.

A Review of Change of Variable Formulas for Generative Modeling · Data on Embedded Manifolds · 0.85

Cites it as support 1 paper

This exponential amplification carries over to learned vector fields, where projection or retraction steps are needed to keep iterates close to the constraint set [ref] , so feasibility degrades with both network expressivity (which controls $L$) and integration horizon.

Learning Unbiased Permutations via Flow Matching · Related Work · 0.98

Cites it as background 14 papers

A manifold CNF, $Φ:$, is an orientation preserving diffeomorphism from the manifold to itself [ref] .

Moser Flow: Divergence-based Generative Modeling on Manifolds · Preliminaries · 0.91
In a parallel line of works, [ref] developed CNFs over Riemannian manifolds.

Matching Normalizing Flows and Probability Paths on Manifolds · Introduction · 0.98
Works that provide an answer to this question can be roughly divided between those that leverage known local geometric information of the manifold [ref] and those that learn its topological structure [ref] .

Density estimation on smooth manifolds with normalizing flows · Introduction · 0.94

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