# Equivariant geometric convolutions for emulation of dynamical systems

arXiv: https://arxiv.org/abs/2305.12585 · HTML: https://mlsanity.com/paper/2305.12585/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 54 references that are in the map.
- First posted on arXiv 21 May 2023.

## Builds on (most cited first, 30 of 30)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [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)
- [A guide to convolution arithmetic for deep learning](https://mlsanity.com/paper/1603.07285/) (2016)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces](https://mlsanity.com/paper/2206.08362/) (2022)
- [Data Augmentation vs. Equivariant Networks: A Theory of Generalization on Dynamics Forecasting](https://mlsanity.com/paper/2206.09450/) (2022)
- [ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds](https://mlsanity.com/paper/2111.15341/) (2021)
- [Universal Approximation Theorem for Equivariant Maps by Group CNNs](https://mlsanity.com/paper/2012.13882/) (2020)
- [Towards fully covariant machine learning](https://mlsanity.com/paper/2301.13724/) (2023)

## Built on by (most cited first, 3 of 3)

- [Generalization vs. Memorization in Autoregressive Deep Learning: Or, Examining Temporal Decay of Gradient Coherence](https://mlsanity.com/paper/2509.00024/) (2025)
- [Autoregressive deep learning for real-time simulation of soft tissue dynamics during virtual neurosurgery](https://mlsanity.com/paper/2601.13676/) (2026)
- [Loss Landscape Geometry of Partial Differential Equation Emulators: Or, Symmetry Learning via Gradient Alignment](https://mlsanity.com/paper/2601.20172/) (2026)

## Abstract (arXiv)

Machine learning methods are increasingly being employed as surrogate models in place of computationally expensive and slow numerical integrators for a bevy of applications in the natural sciences. However, while the laws of physics are relationships between scalars, vectors, and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet, and a UNet. In numerical experiments emulating 2D compressible Navier-Stokes, we see better accuracy and improved stability compared to baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any CNN-based method applied to an appropriate class of problems