# Isometric Transformation Invariant and Equivariant Graph Convolutional Networks

arXiv: https://arxiv.org/abs/2005.06316 · HTML: https://mlsanity.com/paper/2005.06316/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2020.

## Built on by (most cited first, 17 of 17)

- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions](https://mlsanity.com/paper/2205.11912/) (2022)
- [Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency](https://mlsanity.com/paper/2406.07598/) (2024)
- [Graph Neural PDE Solvers with Conservation and Similarity-Equivariance](https://mlsanity.com/paper/2405.16183/) (2024)
- [Physics-Regulated Deep Reinforcement Learning: Invariant Embeddings](https://mlsanity.com/paper/2305.16614/) (2023)
- [Autoequivariant Network Search via Group Decomposition](https://mlsanity.com/paper/2104.04848/) (2021)
- [Beyond permutation equivariance in graph networks](https://mlsanity.com/paper/2103.14066/) (2021)
- [Physics-constrained Unsupervised Learning of Partial Differential Equations using Meshes](https://mlsanity.com/paper/2203.16628/) (2022)
- [A Simple Strategy to Provable Invariance via Orbit Mapping](https://mlsanity.com/paper/2209.11916/) (2022)
- [Symmetry-driven graph neural networks](https://mlsanity.com/paper/2105.14058/) (2021)
- [Poisson-Dirac Neural Networks for Modeling Coupled Dynamical Systems across Domains](https://mlsanity.com/paper/2410.11480/) (2024)
- [Data efficiency in graph networks through equivariance](https://mlsanity.com/paper/2106.13786/) (2021)
- [Revisiting Transformation Invariant Geometric Deep Learning: An Initial Representation Perspective](https://mlsanity.com/paper/2112.12345/) (2021)
- [Phy-Taylor: Physics-Model-Based Deep Neural Networks](https://mlsanity.com/paper/2209.13511/) (2022)
- [CraftSPH: A high-accuracy and composable differentiable SPH solver implemented in PyTorch](https://mlsanity.com/paper/2609.38208/) (2026)

## Abstract (arXiv)

Graphs are one of the most important data structures for representing pairwise relations between objects. Specifically, a graph embedded in a Euclidean space is essential to solving real problems, such as physical simulations. A crucial requirement for applying graphs in Euclidean spaces to physical simulations is learning and inferring the isometric transformation invariant and equivariant features in a computationally efficient manner. In this paper, we propose a set of transformation invariant and equivariant models based on graph convolutional networks, called IsoGCNs. We demonstrate that the proposed model has a competitive performance compared to state-of-the-art methods on tasks related to geometrical and physical simulation data. Moreover, the proposed model can scale up to graphs with 1M vertices and conduct an inference faster than a conventional finite element analysis, which the existing equivariant models cannot achieve.