mlsanityData to 14 Oct 2026

Steerable Partial Differential Operators for Equivariant Neural Networks

arXiv 2106.10163 ↗First posted 18 Jun 2021cs.LG cs.CVcited by 22

Not on the map. Why?

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.

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

25 of 39 references.

References · 25

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2018Neural Ordinary Differential Equationscited by 3,779
  3. 2016Wide Residual Networkscited by 3,070
  4. 2017Improved Regularization of Convolutional Neural Networks with Cutoutcited by 1,628
  5. 2016Group Equivariant Convolutional Networkscited by 1,165
  6. 2019General E(2)-Equivariant Steerable CNNscited by 446
  7. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 354
  8. 2017PDE-Net: Learning PDEs from Datacited by 311
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  10. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 300
  11. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 267
  12. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
13 more
  1. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 228
  2. 2017Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equationscited by 203
  3. 2017Learning Steerable Filters for Rotation Equivariant CNNscited by 194
  4. 2018Deep Neural Networks Motivated by Partial Differential Equationscited by 187
  5. 2019B-Spline CNNs on Lie Groupscited by 97
  6. 2020Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphscited by 92
  7. 2021Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifoldscited by 73
  8. 2020A Wigner-Eckart Theorem for Group Equivariant Convolution Kernelscited by 46
  9. 2020PDO-eConvs: Partial Differential Operator Based Equivariant Convolutionscited by 34
  10. 2020PDE-based Group Equivariant Convolutional Neural Networkscited by 29
  11. 2021Homogeneous vector bundles and G-equivariant convolutional neural networkscited by 21
  12. 2020DeepCFD: Efficient Steady-State Laminar Flow Approximation with Deep Convolutional Neural Networkscited by 18
  13. 2020Probabilistic Numeric Convolutional Neural Networkscited by 10

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 6; 2023: 8; 2024: 3; 2025: 3; 2026: 2.

22 papers.

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  1. 2022Clifford Neural Layers for PDE Modelingcited by 92
  2. 2024Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3Dcited by 33
  3. 2022LieGG: Studying Learned Lie Group Generatorscited by 31
  4. 2024Clifford-Steerable Convolutional Neural Networkscited by 25
  5. 2023Leveraging Symmetries in Pick and Placecited by 19
  6. 2023On genuine invariance learning without weight-tyingcited by 14
  7. 2023Interpretable statistical representations of neural population dynamics and geometrycited by 12
  8. 2022Moving Frame Net: SE(3)-Equivariant Network for Volumescited by 6
  9. 2023Equivariant Single View Pose Prediction Via Induced and Restricted Representationscited by 6
  10. 2022PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNscited by 4
  11. 2023Empowering Networks With Scale and Rotation Equivariance Using A Similarity Convolutioncited by 4
  12. 2025Equivariant Neural Networks for General Linear Symmetries on Lie Algebrascited by 4
10 more
  1. 2023Equivariant geometric convolutions for emulation of dynamical systemscited by 3
  2. 2025Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoderscited by 3
  3. 2022ReF -- Rotation Equivariant Features for Local Feature Matchingcited by 2
  4. 2024Equi-GSPR: Equivariant SE(3) Graph Network Model for Sparse Point Cloud Registrationcited by 2
  5. 2022VC dimensions of group convolutional neural networkscited by 1
  6. 2023Adaptive aggregation of Monte Carlo augmented decomposed filters for efficient group-equivariant convolutional neural networkcited by 1
  7. 2025Self-Supervised Learning Based on Transformed Image Reconstruction for Equivariance-Coherent Feature Representationcited by 1
  8. 2023Self-Supervised Learning for Group Equivariant Neural Networkscited by 0
  9. 2026Computing a Characteristic Orientation for Rotation-Independent Image Analysiscited by 0
  10. 2026Where to Compute and How to Interact: Operator-Readable Adaptation with Gauge-Aware Transportcited by 0