# NeurEPDiff: Neural Operators to Predict Geodesics in Deformation Spaces

arXiv: https://arxiv.org/abs/2303.07115 · HTML: https://mlsanity.com/paper/2303.07115/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 13 Mar 2023.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [DeepFLASH: An Efficient Network for Learning-based Medical Image Registration](https://mlsanity.com/paper/2004.02097/) (2020)
- [Complex-valued Neural Networks with Non-parametric Activation Functions](https://mlsanity.com/paper/1802.08026/) (2018)
- [Geo-SIC: Learning Deformable Geometric Shapes in Deep Image Classifiers](https://mlsanity.com/paper/2210.13704/) (2022)
- [Learning a smooth kernel regularizer for convolutional neural networks](https://mlsanity.com/paper/1903.01882/) (2019)

## Built on by (most cited first, 8 of 8)

- [textttNePhi: Neural Deformation Fields for Approximately Diffeomorphic Medical Image Registration](https://mlsanity.com/paper/2309.07322/) (2023)
- [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023)
- [TLRN: Temporal Latent Residual Networks For Large Deformation Image Registration](https://mlsanity.com/paper/2407.11219/) (2024)
- [Learning Geodesics of Geometric Shape Deformations From Images](https://mlsanity.com/paper/2410.18797/) (2024)
- [CoRLD: Contrastive Representation Learning Of Deformable Shapes In Images](https://mlsanity.com/paper/2503.17162/) (2025)
- [4DLoG: Generative Modeling of Neurodegenerative Brain Anatomy with 4D Longitudinal Diffusion Model](https://mlsanity.com/paper/2604.22700/) (2026)
- [FineMorphs: Affine-diffeomorphic sequences for regression](https://mlsanity.com/paper/2305.17255/) (2023)
- [Geodesic-informed Generative Diffusion Model For Topology-preserved Image Video Generation](https://mlsanity.com/paper/2609.08153/) (2026)

## Abstract (arXiv)

This paper presents NeurEPDiff, a novel network to fast predict the geodesics in deformation spaces generated by a well known Euler-Poincar'e differential equation (EPDiff). To achieve this, we develop a neural operator that for the first time learns the evolving trajectory of geodesic deformations parameterized in the tangent space of diffeomorphisms(a.k.a velocity fields). In contrast to previous methods that purely fit the training images, our proposed NeurEPDiff learns a nonlinear mapping function between the time-dependent velocity fields. A composition of integral operators and smooth activation functions is formulated in each layer of NeurEPDiff to effectively approximate such mappings. The fact that NeurEPDiff is able to rapidly provide the numerical solution of EPDiff (given any initial condition) results in a significantly reduced computational cost of geodesic shooting of diffeomorphisms in a high-dimensional image space. Additionally, the properties of discretiztion/resolution-invariant of NeurEPDiff make its performance generalizable to multiple image resolutions after being trained offline. We demonstrate the effectiveness of NeurEPDiff in registering two image datasets: 2D synthetic data and 3D brain resonance imaging (MRI). The registration accuracy and computational efficiency are compared with the state-of-the-art diffeomophic registration algorithms with geodesic shooting.