# PDE-based Group Equivariant Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2001.09046 · HTML: https://mlsanity.com/paper/2001.09046/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2020.

## Built on by (most cited first, 29 of 29)

- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [Dynamical systems' based neural networks](https://mlsanity.com/paper/2210.02373/) (2022)
- [Analysis of (sub-)Riemannian PDE-G-CNNs](https://mlsanity.com/paper/2210.00935/) (2022)
- [Non-linear manifold ROM with Convolutional Autoencoders and Reduced Over-Collocation method](https://mlsanity.com/paper/2203.00360/) (2022)
- [Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates](https://mlsanity.com/paper/2506.05513/) (2025)
- [Steerable Equivariant Representation Learning](https://mlsanity.com/paper/2302.11349/) (2023)
- [Translating Numerical Concepts for PDEs into Neural Architectures](https://mlsanity.com/paper/2103.15419/) (2021)
- [Optimal Transport on the Lie Group of Roto-translations](https://mlsanity.com/paper/2402.15322/) (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)
- [Capacity of Group-invariant Linear Readouts from Equivariant Representations: How Many Objects can be Linearly Classified Under All Possible Views?](https://mlsanity.com/paper/2110.07472/) (2021)
- [Predictions Based on Pixel Data: Insights from PDEs and Finite Differences](https://mlsanity.com/paper/2305.00723/) (2023)
- [PDE-CNNs: Axiomatic Derivations and Applications](https://mlsanity.com/paper/2403.15182/) (2024)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups](https://mlsanity.com/paper/2111.12139/) (2021)
- [Geometric Generative Models based on Morphological Equivariant PDEs and GANs](https://mlsanity.com/paper/2403.14897/) (2024)
- [Sigma Flows for Image and Data Labeling and Learning Structured Prediction](https://mlsanity.com/paper/2408.15946/) (2024)
- [On the VC dimension of deep group convolutional neural networks](https://mlsanity.com/paper/2410.15800/) (2024)
- [The Mechanics of CNN Filtering with Rectification](https://mlsanity.com/paper/2512.24338/) (2025)
- [Affine Invariance in Continuous-Domain Convolutional Neural Networks](https://mlsanity.com/paper/2311.09245/) (2023)
- [Semiring Activation in Neural Networks](https://mlsanity.com/paper/2405.18805/) (2024)
- [Generalization Bounds for Equivariant Networks on Markov Data](https://mlsanity.com/paper/2503.00292/) (2025)
- [Orientation Scores should be a Piece of Cake](https://mlsanity.com/paper/2504.00702/) (2025)
- [Roto-Translation Invariant Metrics on Position-Orientation Space](https://mlsanity.com/paper/2504.03309/) (2025)
- [Modelling and analysis of the 8 filters from the "master key filters hypothesis" for depthwise-separable deep networks in relation to idealized receptive fields based on scale-space theory](https://mlsanity.com/paper/2509.12746/) (2025)
- [Scale-covariant spiking wavelets](https://mlsanity.com/paper/2602.02020/) (2026)
- [DEGMC: Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutions](https://mlsanity.com/paper/2602.10221/) (2026)
- [Scale-invariant Gaussian derivative residual networks](https://mlsanity.com/paper/2603.02843/) (2026)
- [Varifold Moment Invariants for Sustainable and Explainable Contour Feature Extraction](https://mlsanity.com/paper/2606.07333/) (2026)
- [Mesh-Free Convolution: Learned Spectral Attenuation and Transport](https://mlsanity.com/paper/2610.15931/) (2026)

## Abstract (arXiv)

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation in addition to the standard translation equivariance of CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and non-linear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers, we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for linear convolution a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning based imaging applications with far fewer parameters than traditional CNNs.