# Diffeomorphism-Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2602.06695 · HTML: https://mlsanity.com/paper/2602.06695/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 61 references that are in the map.
- First posted on arXiv 6 Feb 2026.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Revisiting Unreasonable Effectiveness of Data in Deep Learning Era](https://mlsanity.com/paper/1707.02968/) (2017)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Object Detection with Deep Learning: A Review](https://mlsanity.com/paper/1807.05511/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Deep Learning: A Critical Appraisal](https://mlsanity.com/paper/1801.00631/) (2018)
- [Unsupervised Learning of Probabilistic Diffeomorphic Registration for Images and Surfaces](https://mlsanity.com/paper/1903.03545/) (2019)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Relational Pooling for Graph Representations](https://mlsanity.com/paper/1903.02541/) (2019)
- [Deep Learning for Hyperspectral Image Classification: An Overview](https://mlsanity.com/paper/1910.12861/) (2019)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Recent advances and clinical applications of deep learning in medical image analysis](https://mlsanity.com/paper/2105.13381/) (2021)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Rotation-invariant convolutional neural networks for galaxy morphology prediction](https://mlsanity.com/paper/1503.07077/) (2015)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [Dreaming More Data: Class-dependent Distributions over Diffeomorphisms for Learned Data Augmentation](https://mlsanity.com/paper/1510.02795/) (2015)
- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [On Finite Difference Jacobian Computation in Deformable Image Registration](https://mlsanity.com/paper/2212.06060/) (2022)
- [Diffeomorphic Image Registration with Neural Velocity Field](https://mlsanity.com/paper/2202.12498/) (2022)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [On the Sample Complexity of Learning under Invariance and Geometric Stability](https://mlsanity.com/paper/2106.07148/) (2021)
- [Shape deformation analysis from the optimal control viewpoint](https://mlsanity.com/paper/1401.0661/) (2014)
- [On Reducing Negative Jacobian Determinant of the Deformation Predicted by Deep Registration Networks](https://mlsanity.com/paper/1907.00068/) (2019)
- [Medical Imaging Synthesis using Deep Learning and its Clinical Applications: A Review](https://mlsanity.com/paper/2004.10322/) (2020)

## Abstract (arXiv)

Incorporating group symmetries via equivariance into neural networks has emerged as a robust approach for overcoming the efficiency and data demands of modern deep learning. While most existing approaches, such as group convolutions and averaging-based methods, focus on compact, finite, or low-dimensional groups with linear actions, this work explores how equivariance can be extended to infinite-dimensional groups. We propose a strategy designed to induce diffeomorphism equivariance in pre-trained neural networks via energy-based canonicalisation. Formulating equivariance as an optimisation problem allows us to access the rich toolbox of already established differentiable image registration methods. Empirical results on segmentation and classification tasks confirm that our approach achieves approximate equivariance and generalises to unseen transformations without relying on extensive data augmentation or retraining.