# Equivariant Manifold Neural ODEs and Differential Invariants

arXiv: https://arxiv.org/abs/2401.14131 · HTML: https://mlsanity.com/paper/2401.14131/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 50 references that are in the map.
- First posted on arXiv 25 Jan 2024.

## Builds on (most cited first, 30 of 39)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)

## Built on by (most cited first, 2 of 2)

- [PolyNODE: Variable-dimension Neural ODEs on M-polyfolds](https://mlsanity.com/paper/2602.15128/) (2026)
- [Steerable Neural ODEs on Homogeneous Spaces](https://mlsanity.com/paper/2605.11133/) (2026)

## Abstract (arXiv)

In this paper, we develop a manifestly geometric framework for equivariant manifold neural ordinary differential equations (NODEs) and use it to analyse their modelling capabilities for symmetric data. First, we consider the action of a Lie group G on a smooth manifold M and establish the equivalence between equivariance of vector fields, symmetries of the corresponding Cauchy problems, and equivariance of the associated NODEs. We also propose a novel formulation, based on Lie theory for symmetries of differential equations, of the equivariant manifold NODEs in terms of the differential invariants of the action of G on M, which provides an efficient parameterisation of the space of equivariant vector fields in a way that is agnostic to both the manifold M and the symmetry group G. Second, we construct augmented manifold NODEs, through embeddings into flows on the tangent bundle TM, and show that they are universal approximators of diffeomorphisms on any connected M. Furthermore, we show that universality persists in the equivariant case and that the augmented equivariant manifold NODEs can be incorporated into the geometric framework using higher-order differential invariants. Finally, we consider the induced action of G on different fields on M and show how it can be used to generalise previous work, on, e.g., continuous normalizing flows, to equivariant models in any geometry.