# Learning Harmonic Molecular Representations on Riemannian Manifold

arXiv: https://arxiv.org/abs/2303.15520 · HTML: https://mlsanity.com/paper/2303.15520/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 75 references that are in the map.
- First posted on arXiv 27 Mar 2023.

## Builds on (most cited first, 28 of 28)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [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)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [PointContrast: Unsupervised Pre-training for 3D Point Cloud Understanding](https://mlsanity.com/paper/2007.10985/) (2020)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Deep Functional Maps: Structured Prediction for Dense Shape Correspondence](https://mlsanity.com/paper/1704.08686/) (2017)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence](https://mlsanity.com/paper/2003.14286/) (2020)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Docking](https://mlsanity.com/paper/2111.07786/) (2021)
- [Continuous and Orientation-preserving Correspondences via Functional Maps](https://mlsanity.com/paper/1806.04455/) (2018)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [A Rotation-Invariant Framework for Deep Point Cloud Analysis](https://mlsanity.com/paper/2003.07238/) (2020)
- [DPFM: Deep Partial Functional Maps](https://mlsanity.com/paper/2110.09994/) (2021)
- [Multi-Scale Representation Learning on Proteins](https://mlsanity.com/paper/2204.02337/) (2022)
- [Structure-based drug design with geometric deep learning](https://mlsanity.com/paper/2210.11250/) (2022)
- [End-to-End Learning on 3D Protein Structure for Interface Prediction](https://mlsanity.com/paper/1807.01297/) (2018)
- [Network Geometry](https://mlsanity.com/paper/2001.03241/) (2020)
- [DeepSurf: A surface-based deep learning approach for the prediction of ligand binding sites on proteins](https://mlsanity.com/paper/2002.05643/) (2020)

## Built on by (most cited first, 8 of 8)

- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [MolSpectra: Pre-training 3D Molecular Representation with Multi-modal Energy Spectra](https://mlsanity.com/paper/2502.16284/) (2025)
- [AtomSurf : Surface Representation for Learning on Protein Structures](https://mlsanity.com/paper/2309.16519/) (2023)
- [Effective Protein-Protein Interaction Exploration with PPIretrieval](https://mlsanity.com/paper/2402.03675/) (2024)
- [Deep Reinforcement Learning for Modelling Protein Complexes](https://mlsanity.com/paper/2405.02299/) (2024)
- [Manifold-Constrained Nucleus-Level Denoising Diffusion Model for Structure-Based Drug Design](https://mlsanity.com/paper/2409.10584/) (2024)
- [Recent advances in interpretable machine learning using structure-based protein representations](https://mlsanity.com/paper/2409.17726/) (2024)
- [From Surfaces to Volumes: Registered Geometry for Protein Representation Learning](https://mlsanity.com/paper/2609.36277/) (2026)

## Abstract (arXiv)

Molecular representation learning plays a crucial role in AI-assisted drug discovery research. Encoding 3D molecular structures through Euclidean neural networks has become the prevailing method in the geometric deep learning community. However, the equivariance constraints and message passing in Euclidean space may limit the network expressive power. In this work, we propose a Harmonic Molecular Representation learning (HMR) framework, which represents a molecule using the Laplace-Beltrami eigenfunctions of its molecular surface. HMR offers a multi-resolution representation of molecular geometric and chemical features on 2D Riemannian manifold. We also introduce a harmonic message passing method to realize efficient spectral message passing over the surface manifold for better molecular encoding. Our proposed method shows comparable predictive power to current models in small molecule property prediction, and outperforms the state-of-the-art deep learning models for ligand-binding protein pocket classification and the rigid protein docking challenge, demonstrating its versatility in molecular representation learning.