# Steerable Partial Differential Operators for Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2106.10163 · HTML: https://mlsanity.com/paper/2106.10163/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 39 references that are in the map.
- First posted on arXiv 18 Jun 2021.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [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)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://mlsanity.com/paper/2010.10952/) (2020)
- [PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions](https://mlsanity.com/paper/2007.10408/) (2020)
- [PDE-based Group Equivariant Convolutional Neural Networks](https://mlsanity.com/paper/2001.09046/) (2020)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [DeepCFD: Efficient Steady-State Laminar Flow Approximation with Deep Convolutional Neural Networks](https://mlsanity.com/paper/2004.08826/) (2020)
- [Probabilistic Numeric Convolutional Neural Networks](https://mlsanity.com/paper/2010.10876/) (2020)

## Built on by (most cited first, 22 of 22)

- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3D](https://mlsanity.com/paper/2401.12046/) (2024)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Leveraging Symmetries in Pick and Place](https://mlsanity.com/paper/2308.07948/) (2023)
- [On genuine invariance learning without weight-tying](https://mlsanity.com/paper/2308.03904/) (2023)
- [Interpretable statistical representations of neural population dynamics and geometry](https://mlsanity.com/paper/2304.03376/) (2023)
- [Moving Frame Net: SE(3)-Equivariant Network for Volumes](https://mlsanity.com/paper/2211.03420/) (2022)
- [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https://mlsanity.com/paper/2307.03704/) (2023)
- [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)
- [Equivariant Neural Networks for General Linear Symmetries on Lie Algebras](https://mlsanity.com/paper/2510.22984/) (2025)
- [Equivariant geometric convolutions for emulation of dynamical systems](https://mlsanity.com/paper/2305.12585/) (2023)
- [Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoders](https://mlsanity.com/paper/2505.13569/) (2025)
- [ReF -- Rotation Equivariant Features for Local Feature Matching](https://mlsanity.com/paper/2203.05206/) (2022)
- [Equi-GSPR: Equivariant SE(3) Graph Network Model for Sparse Point Cloud Registration](https://mlsanity.com/paper/2410.05729/) (2024)
- [VC dimensions of group convolutional neural networks](https://mlsanity.com/paper/2212.09507/) (2022)
- [Adaptive aggregation of Monte Carlo augmented decomposed filters for efficient group-equivariant convolutional neural network](https://mlsanity.com/paper/2305.10110/) (2023)
- [Self-Supervised Learning Based on Transformed Image Reconstruction for Equivariance-Coherent Feature Representation](https://mlsanity.com/paper/2503.18753/) (2025)
- [Self-Supervised Learning for Group Equivariant Neural Networks](https://mlsanity.com/paper/2303.04427/) (2023)
- [Computing a Characteristic Orientation for Rotation-Independent Image Analysis](https://mlsanity.com/paper/2602.20930/) (2026)
- [Where to Compute and How to Interact: Operator-Readable Adaptation with Gauge-Aware Transport](https://mlsanity.com/paper/2609.15620/) (2026)

## Abstract (arXiv)

Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are usually defined by convolutions with a kernel, whereas they are partial differential operators (PDOs) in physics. Developing the theory of equivariant PDOs in the context of deep learning could bring these subjects even closer together and lead to a stronger flow of ideas. In this work, we derive a G-steerability constraint that completely characterizes when a PDO between feature vector fields is equivariant, for arbitrary symmetry groups G. We then fully solve this constraint for several important groups. We use our solutions as equivariant drop-in replacements for convolutional layers and benchmark them in that role. Finally, we develop a framework for equivariant maps based on Schwartz distributions that unifies classical convolutions and differential operators and gives insight about the relation between the two.