# MUDiff: Unified Diffusion for Complete Molecule Generation

arXiv: https://arxiv.org/abs/2304.14621 · HTML: https://mlsanity.com/paper/2304.14621/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 48 references that are in the map.
- First posted on arXiv 28 Apr 2023.

## Builds on (most cited first, 30 of 33)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [A Generalization of Transformer Networks to Graphs](https://mlsanity.com/paper/2012.09699/) (2020)
- [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)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Can Graph Neural Networks Count Substructures?](https://mlsanity.com/paper/2002.04025/) (2020)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Symmetry-adapted generation of 3d point sets for the targeted discovery of molecules](https://mlsanity.com/paper/1906.00957/) (2019)
- [Revisiting Heterophily For Graph Neural Networks](https://mlsanity.com/paper/2210.07606/) (2022)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Directional Graph Networks](https://mlsanity.com/paper/2010.02863/) (2020)
- [Is Heterophily A Real Nightmare For Graph Neural Networks To Do Node Classification?](https://mlsanity.com/paper/2109.05641/) (2021)
- [One Transformer Can Understand Both 2D & 3D Molecular Data](https://mlsanity.com/paper/2210.01765/) (2022)
- [Break the Ceiling: Stronger Multi-scale Deep Graph Convolutional Networks](https://mlsanity.com/paper/1906.02174/) (2019)
- [MDM: Molecular Diffusion Model for 3D Molecule Generation](https://mlsanity.com/paper/2209.05710/) (2022)

## Built on by (most cited first, 30 of 42)

- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation](https://mlsanity.com/paper/2404.19739/) (2024)
- [GenMol: A Drug Discovery Generalist with Discrete Diffusion](https://mlsanity.com/paper/2501.06158/) (2025)
- [Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions](https://mlsanity.com/paper/2402.15602/) (2024)
- [SemlaFlow -- Efficient 3D Molecular Generation with Latent Attention and Equivariant Flow Matching](https://mlsanity.com/paper/2406.07266/) (2024)
- [Cometh: A continuous-time discrete-state graph diffusion model](https://mlsanity.com/paper/2406.06449/) (2024)
- [FlowMol3: Flow Matching for 3D De Novo Small-Molecule Generation](https://mlsanity.com/paper/2508.12629/) (2025)
- [NExT-Mol: 3D Diffusion Meets 1D Language Modeling for 3D Molecule Generation](https://mlsanity.com/paper/2502.12638/) (2025)
- [A Survey of Graph Transformers: Architectures, Theories and Applications](https://mlsanity.com/paper/2502.16533/) (2025)
- [ReactZyme: A Benchmark for Enzyme-Reaction Prediction](https://mlsanity.com/paper/2408.13659/) (2024)
- [Probing Equivariance and Symmetry Breaking in Convolutional Networks](https://mlsanity.com/paper/2501.01999/) (2025)
- [TABASCO: A Fast, Simplified Model for Molecular Generation with Improved Physical Quality](https://mlsanity.com/paper/2507.00899/) (2025)
- [A Directional Diffusion Graph Transformer for Recommendation](https://mlsanity.com/paper/2404.03326/) (2024)
- [What Is Missing In Homophily? Disentangling Graph Homophily For Graph Neural Networks](https://mlsanity.com/paper/2406.18854/) (2024)
- [3D Vessel Graph Generation Using Denoising Diffusion](https://mlsanity.com/paper/2407.05842/) (2024)
- [LayerDAG: A Layerwise Autoregressive Diffusion Model for Directed Acyclic Graph Generation](https://mlsanity.com/paper/2411.02322/) (2024)
- [Straight-Line Diffusion Model for Efficient 3D Molecular Generation](https://mlsanity.com/paper/2503.02918/) (2025)
- [Learning Flexible Forward Trajectories for Masked Molecular Diffusion](https://mlsanity.com/paper/2505.16790/) (2025)
- [Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation](https://mlsanity.com/paper/2602.15022/) (2026)
- [Zero Shot Molecular Generation via Similarity Kernels](https://mlsanity.com/paper/2402.08708/) (2024)
- [E(3)-equivariant models cannot learn chirality: Field-based molecular generation](https://mlsanity.com/paper/2402.15864/) (2024)
- [Representation Learning on Heterophilic Graph with Directional Neighborhood Attention](https://mlsanity.com/paper/2403.01475/) (2024)
- [Effective Protein-Protein Interaction Exploration with PPIretrieval](https://mlsanity.com/paper/2402.03675/) (2024)
- [EnzymeFlow: Generating Reaction-specific Enzyme Catalytic Pockets through Flow Matching and Co-Evolutionary Dynamics](https://mlsanity.com/paper/2410.00327/) (2024)
- [PowerGrow: Feasible Co-Growth of Structures and Dynamics for Power Grid Synthesis](https://mlsanity.com/paper/2509.12212/) (2025)
- [Revealing the Pitfalls and Re-Evaluating the Advancement of Heterophilic Graph Learning](https://mlsanity.com/paper/2409.05755/) (2024)
- [Reaction-conditioned De Novo Enzyme Design with GENzyme](https://mlsanity.com/paper/2411.16694/) (2024)
- [Clifford Group Equivariant Diffusion Models for 3D Molecular Generation](https://mlsanity.com/paper/2504.15773/) (2025)
- [Unraveling the Potential of Diffusion Models in Small Molecule Generation](https://mlsanity.com/paper/2507.08005/) (2025)
- [NEAT: Neighborhood-Guided, Efficient, Autoregressive Set Transformer for 3D Molecular Generation](https://mlsanity.com/paper/2512.05844/) (2025)

## Abstract (arXiv)

Molecule generation is a very important practical problem, with uses in drug discovery and material design, and AI methods promise to provide useful solutions. However, existing methods for molecule generation focus either on 2D graph structure or on 3D geometric structure, which is not sufficient to represent a complete molecule as 2D graph captures mainly topology while 3D geometry captures mainly spatial atom arrangements. Combining these representations is essential to better represent a molecule. In this paper, we present a new model for generating a comprehensive representation of molecules, including atom features, 2D discrete molecule structures, and 3D continuous molecule coordinates, by combining discrete and continuous diffusion processes. The use of diffusion processes allows for capturing the probabilistic nature of molecular processes and exploring the effect of different factors on molecular structures. Additionally, we propose a novel graph transformer architecture to denoise the diffusion process. The transformer adheres to 3D roto-translation equivariance constraints, allowing it to learn invariant atom and edge representations while preserving the equivariance of atom coordinates. This transformer can be used to learn molecular representations robust to geometric transformations. We evaluate the performance of our model through experiments and comparisons with existing methods, showing its ability to generate more stable and valid molecules. Our model is a promising approach for designing stable and diverse molecules and can be applied to a wide range of tasks in molecular modeling.