# Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation

arXiv: https://arxiv.org/abs/2311.16199 · HTML: https://mlsanity.com/paper/2311.16199/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 31 references that are in the map.
- First posted on arXiv 27 Nov 2023.

## Builds on (most cited first, 9 of 9)

- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Symmetry-adapted generation of 3d point sets for the targeted discovery of molecules](https://mlsanity.com/paper/1906.00957/) (2019)
- [Inverse design of 3d molecular structures with conditional generative neural networks](https://mlsanity.com/paper/2109.04824/) (2021)
- [Implicit-PDF: Non-Parametric Representation of Probability Distributions on the Rotation Manifold](https://mlsanity.com/paper/2106.05965/) (2021)
- [Symmetry-Aware Actor-Critic for 3D Molecular Design](https://mlsanity.com/paper/2011.12747/) (2020)
- [Reinforcement Learning for Molecular Design Guided by Quantum Mechanics](https://mlsanity.com/paper/2002.07717/) (2020)
- [Incompleteness of graph neural networks for points clouds in three dimensions](https://mlsanity.com/paper/2201.07136/) (2022)
- [Harmonic Exponential Families on Manifolds](https://mlsanity.com/paper/1505.04413/) (2015)

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

- [All-atom Diffusion Transformers: Unified generative modelling of molecules and materials](https://mlsanity.com/paper/2503.03965/) (2025)
- [Equivariant Neural Diffusion for Molecule Generation](https://mlsanity.com/paper/2506.10532/) (2025)
- [Equivariant Symmetry Breaking Sets](https://mlsanity.com/paper/2402.02681/) (2024)
- [A Recipe for Charge Density Prediction](https://mlsanity.com/paper/2405.19276/) (2024)
- [Straight-Line Diffusion Model for Efficient 3D Molecular Generation](https://mlsanity.com/paper/2503.02918/) (2025)
- [Scalable Autoregressive 3D Molecule Generation](https://mlsanity.com/paper/2505.13791/) (2025)
- [Zatom-1: Towards a Multimodal Foundation Model for 3D Molecules and Materials](https://mlsanity.com/paper/2602.22251/) (2026)
- [Equivariant Masked Position Prediction for Efficient Molecular Representation](https://mlsanity.com/paper/2502.08209/) (2025)
- [Guiding Diffusion Models with Reinforcement Learning for Stable Molecule Generation](https://mlsanity.com/paper/2508.16521/) (2025)
- [NEAT: Neighborhood-Guided, Efficient, Autoregressive Set Transformer for 3D Molecular Generation](https://mlsanity.com/paper/2512.05844/) (2025)
- [Equivariant Machine Learning on Graphs with Nonlinear Spectral Filters](https://mlsanity.com/paper/2406.01249/) (2024)
- [Generative Molecular Morphing for Flexible-Size Design via Unbalanced Optimal Transport](https://mlsanity.com/paper/2606.07239/) (2026)
- [AtomComposer: Discovering Chemical Space from First Principles with Reinforcement Learning](https://mlsanity.com/paper/2605.28287/) (2026)
- [Autoregressive latent diffusion for 3D molecule generation](https://mlsanity.com/paper/2607.09277/) (2026)
- [Equivariant Asynchronous Diffusion: An Adaptive Denoising Schedule for Accelerated Molecular Conformation Generation](https://mlsanity.com/paper/2603.10093/) (2026)
- [Generating Physically Consistent Molecules with Energy-Based Models](https://mlsanity.com/paper/2605.18381/) (2026)
- [GEqTrain: A Configuration-Driven Framework for Retargeting Equivariant Graph Neural Networks Across 3D Scientific Tasks](https://mlsanity.com/paper/2607.19083/) (2026)

## Abstract (arXiv)

We present Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree E(3)-equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.