# Smooth, exact rotational symmetrization for deep learning on point clouds

arXiv: https://arxiv.org/abs/2305.19302 · HTML: https://mlsanity.com/paper/2305.19302/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 116 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 30 of 59)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Point Transformer](https://mlsanity.com/paper/2011.00931/) (2020)
- [KPConv: Flexible and Deformable Convolution for Point Clouds](https://mlsanity.com/paper/1904.08889/) (2019)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [PointCNN: Convolution On X-Transformed Points](https://mlsanity.com/paper/1801.07791/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [PCT: Point cloud transformer](https://mlsanity.com/paper/2012.09688/) (2020)
- [Multi-View 3D Object Detection Network for Autonomous Driving](https://mlsanity.com/paper/1611.07759/) (2016)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Multi-view Convolutional Neural Networks for 3D Shape Recognition](https://mlsanity.com/paper/1505.00880/) (2015)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Deep Learning for 3D Point Clouds: A Survey](https://mlsanity.com/paper/1912.12033/) (2019)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Are Transformers universal approximators of sequence-to-sequence functions?](https://mlsanity.com/paper/1912.10077/) (2019)
- [Rethinking Network Design and Local Geometry in Point Cloud: A Simple Residual MLP Framework](https://mlsanity.com/paper/2202.07123/) (2022)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [OctNet: Learning Deep 3D Representations at High Resolutions](https://mlsanity.com/paper/1611.05009/) (2016)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)

## Built on by (most cited first, 30 of 39)

- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [Orb: A Fast, Scalable Neural Network Potential](https://mlsanity.com/paper/2410.22570/) (2024)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [The Importance of Being Scalable: Improving the Speed and Accuracy of Neural Network Interatomic Potentials Across Chemical Domains](https://mlsanity.com/paper/2410.24169/) (2024)
- [Cartesian atomic cluster expansion for machine learning interatomic potentials](https://mlsanity.com/paper/2402.07472/) (2024)
- [The dark side of the forces: assessing non-conservative force models for atomistic machine learning](https://mlsanity.com/paper/2412.11569/) (2024)
- [PET-MAD, a lightweight universal interatomic potential for advanced materials modeling](https://mlsanity.com/paper/2503.14118/) (2025)
- [Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency](https://mlsanity.com/paper/2406.07598/) (2024)
- [Towards Fast, Specialized Machine Learning Force Fields: Distilling Foundation Models via Energy Hessians](https://mlsanity.com/paper/2501.09009/) (2025)
- [Probing the effects of broken symmetries in machine learning](https://mlsanity.com/paper/2406.17747/) (2024)
- [FlashMD: long-stride, universal prediction of molecular dynamics](https://mlsanity.com/paper/2505.19350/) (2025)
- [On the Completeness of Invariant Geometric Deep Learning Models](https://mlsanity.com/paper/2402.04836/) (2024)
- [Pushing the limits of unconstrained machine-learned interatomic potentials](https://mlsanity.com/paper/2601.16195/) (2026)
- [Weisfeiler Leman for Euclidean Equivariant Machine Learning](https://mlsanity.com/paper/2402.02484/) (2024)
- [How simple can you go? An off-the-shelf transformer approach to molecular dynamics](https://mlsanity.com/paper/2503.01431/) (2025)
- [A universal augmentation framework for long-range electrostatics in machine learning interatomic potentials](https://mlsanity.com/paper/2507.14302/) (2025)
- [Fourier Sliced-Wasserstein Embedding for Multisets and Measures](https://mlsanity.com/paper/2504.02544/) (2025)
- [MatRIS: Toward Reliable and Efficient Pretrained Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2603.02002/) (2026)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [Beyond Adam: SOAP and Muon for Faster, Label-Efficient Training of Machine Learning Interatomic Potentials](https://mlsanity.com/paper/2607.02499/) (2026)
- [Advancing Molecular Machine Learning Representations with Stereoelectronics-Infused Molecular Graphs](https://mlsanity.com/paper/2408.04520/) (2024)
- [Representing spherical tensors with scalar-based machine-learning models](https://mlsanity.com/paper/2505.05404/) (2025)
- [HydraScreen: A Generalizable Structure-Based Deep Learning Approach to Drug Discovery](https://mlsanity.com/paper/2311.12814/) (2023)
- [Lookup multivariate Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2509.07103/) (2025)
- [Training Dynamics of Learning 3D-Rotational Equivariance](https://mlsanity.com/paper/2512.02303/) (2025)
- [From Evaluation to Design: Using Potential Energy Surface Smoothness Metrics to Guide Machine Learning Interatomic Potential Architectures](https://mlsanity.com/paper/2602.04861/) (2026)
- [Quantitative Approximation Rates for Group Equivariant Learning](https://mlsanity.com/paper/2602.20370/) (2026)
- [How unconstrained machine-learning models learn physical symmetries](https://mlsanity.com/paper/2603.24638/) (2026)
- [Fourier Sliced-Wasserstein Embedding for Multisets and Measures](https://mlsanity.com/paper/2405.16519/) (2024)
- [Improving Molecular Modeling with Geometric GNNs: an Empirical Study](https://mlsanity.com/paper/2407.08313/) (2024)

## Abstract (arXiv)

Point clouds are versatile representations of 3D objects and have found widespread application in science and engineering. Many successful deep-learning models have been proposed that use them as input. The domain of chemical and materials modeling is especially challenging because exact compliance with physical constraints is highly desirable for a model to be usable in practice. These constraints include smoothness and invariance with respect to translations, rotations, and permutations of identical atoms. If these requirements are not rigorously fulfilled, atomistic simulations might lead to absurd outcomes even if the model has excellent accuracy. Consequently, dedicated architectures, which achieve invariance by restricting their design space, have been developed. General-purpose point-cloud models are more varied but often disregard rotational symmetry. We propose a general symmetrization method that adds rotational equivariance to any given model while preserving all the other requirements. Our approach simplifies the development of better atomic-scale machine-learning schemes by relaxing the constraints on the design space and making it possible to incorporate ideas that proved effective in other domains. We demonstrate this idea by introducing the Point Edge Transformer (PET) architecture, which is not intrinsically equivariant but achieves state-of-the-art performance on several benchmark datasets of molecules and solids. A-posteriori application of our general protocol makes PET exactly equivariant, with minimal changes to its accuracy.