# PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions

arXiv: https://arxiv.org/abs/2007.10408 · HTML: https://mlsanity.com/paper/2007.10408/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2020.

## Built on by (most cited first, 30 of 34)

- [Is Attention Better Than Matrix Decomposition?](https://mlsanity.com/paper/2109.04553/) (2021)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutions](https://mlsanity.com/paper/2107.14519/) (2021)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNs](https://mlsanity.com/paper/2104.03584/) (2021)
- [Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration](https://mlsanity.com/paper/2312.15701/) (2023)
- [Symmetry Discovery for Different Data Types](https://mlsanity.com/paper/2410.09841/) (2024)
- [Moving Frame Net: SE(3)-Equivariant Network for Volumes](https://mlsanity.com/paper/2211.03420/) (2022)
- [MEPNet: A Model-Driven Equivariant Proximal Network for Joint Sparse-View Reconstruction and Metal Artifact Reduction in CT Images](https://mlsanity.com/paper/2306.14274/) (2023)
- [Rotation-Equivariant Self-Supervised Method in Image Denoising](https://mlsanity.com/paper/2505.19618/) (2025)
- [RSF-Conv: Rotation-and-Scale Equivariant Fourier Parameterized Convolution for Retinal Vessel Segmentation](https://mlsanity.com/paper/2309.15638/) (2023)
- [A Regularization-Guided Equivariant Approach for Image Restoration](https://mlsanity.com/paper/2505.19799/) (2025)
- [Rotation Equivariant Arbitrary-scale Image Super-Resolution](https://mlsanity.com/paper/2508.05160/) (2025)
- [PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNs](https://mlsanity.com/paper/2208.03720/) (2022)
- [Empowering Networks With Scale and Rotation Equivariance Using A Similarity Convolution](https://mlsanity.com/paper/2303.00326/) (2023)
- [Incorporating Arbitrary Matrix Group Equivariance into KANs](https://mlsanity.com/paper/2410.00435/) (2024)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [SRE-CNN: A Spatiotemporal Rotation-Equivariant CNN for Cardiac Cine MR Imaging](https://mlsanity.com/paper/2409.08537/) (2024)
- [Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variations](https://mlsanity.com/paper/2409.11140/) (2024)
- [Vanilla Group Equivariant Vision Transformer: Simple and Effective](https://mlsanity.com/paper/2602.08047/) (2026)
- [DASS: Differentiable Architecture Search for Sparse neural networks](https://mlsanity.com/paper/2207.06968/) (2022)
- [H-NeXt: The next step towards roto-translation invariant networks](https://mlsanity.com/paper/2311.01111/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [FILTRA: Rethinking Steerable CNN by Filter Transform](https://mlsanity.com/paper/2105.11636/) (2021)
- [Rotation Equivariant Mamba for Vision Tasks](https://mlsanity.com/paper/2603.09138/) (2026)
- [Image-to-Image Translation Framework Embedded with Rotation Symmetry Priors](https://mlsanity.com/paper/2604.12805/) (2026)
- [Warped Convolutional Networks: Bridge Homography to sl(3) algebra by Group Convolution](https://mlsanity.com/paper/2206.11657/) (2022)
- [Learning from Pattern Completion: Self-supervised Controllable Generation](https://mlsanity.com/paper/2409.18694/) (2024)
- [DUN-SRE: Deep Unrolling Network with Spatiotemporal Rotation Equivariance for Dynamic MRI Reconstruction](https://mlsanity.com/paper/2506.10309/) (2025)
- [Progressive Alignment Degradation Learning for Pansharpening](https://mlsanity.com/paper/2506.20179/) (2025)

## Abstract (arXiv)

Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more general continuous group, the n-dimension Euclidean group. In implementation, we discretize the system using the numerical schemes of PDOs, deriving approximately equivariant convolutions (PDO-eConvs). Theoretically, the approximation error of PDO-eConvs is of the quadratic order. It is the first time that the error analysis is provided when the equivariance is approximate. Extensive experiments on rotated MNIST and natural image classification show that PDO-eConvs perform competitively yet use parameters much more efficiently. Particularly, compared with Wide ResNets, our methods result in better results using only 12.6% parameters.