# Graph Forward Distribution Matching for Molecular Inverse Design

arXiv: https://arxiv.org/abs/2609.32056 · HTML: https://mlsanity.com/paper/2609.32056/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 30 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 24 of 24)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [GuacaMol: Benchmarking Models for De Novo Molecular Design](https://mlsanity.com/paper/1811.09621/) (2018)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [DiffusionNFT: Online Diffusion Reinforcement with Forward Process](https://mlsanity.com/paper/2509.16117/) (2025)
- [Fr'echet ChemNet Distance: A metric for generative models for molecules in drug discovery](https://mlsanity.com/paper/1803.09518/) (2018)
- [Sample Efficiency Matters: A Benchmark for Practical Molecular Optimization](https://mlsanity.com/paper/2206.12411/) (2022)
- [MARS: Markov Molecular Sampling for Multi-objective Drug Discovery](https://mlsanity.com/paper/2103.10432/) (2021)
- [DeFoG: Discrete Flow Matching for Graph Generation](https://mlsanity.com/paper/2410.04263/) (2024)
- [Advantage Weighted Matching: Aligning RL with Pretraining in Diffusion Models](https://mlsanity.com/paper/2509.25050/) (2025)
- [Graph Diffusion Transformers for Multi-Conditional Molecular Generation](https://mlsanity.com/paper/2401.13858/) (2024)
- [Steering Masked Discrete Diffusion Models via Discrete Denoising Posterior Prediction](https://mlsanity.com/paper/2410.08134/) (2024)
- [Enhancing Reasoning for Diffusion LLMs via Distribution Matching Policy Optimization](https://mlsanity.com/paper/2510.08233/) (2025)
- [Classifier-free graph diffusion for molecular property targeting](https://mlsanity.com/paper/2312.17397/) (2023)
- [Iterative Distillation for Reward-Guided Fine-Tuning of Diffusion Models in Biomolecular Design](https://mlsanity.com/paper/2507.00445/) (2025)
- [Learning Flexible Forward Trajectories for Masked Molecular Diffusion](https://mlsanity.com/paper/2505.16790/) (2025)
- [Graph Diffusion Transformers are In-Context Molecular Designers](https://mlsanity.com/paper/2510.08744/) (2025)
- [Controllable Graph Generation with Diffusion Models via Inference-Time Tree Search Guidance](https://mlsanity.com/paper/2510.10402/) (2025)
- [Graph Diffusion that can Insert and Delete](https://mlsanity.com/paper/2506.15725/) (2025)
- [MolGuidance: Advanced Guidance Strategies for Conditional Molecular Generation with Flow Matching](https://mlsanity.com/paper/2512.12198/) (2025)
- [TMPO: Trajectory Matching Policy Optimization for Diverse and Efficient Diffusion Alignment](https://mlsanity.com/paper/2605.10983/) (2026)
- [Composable Score-based Graph Diffusion Model for Multi-Conditional Molecular Generation](https://mlsanity.com/paper/2509.09451/) (2025)
- [Controllable Molecular Generative Foundation Models](https://mlsanity.com/paper/2605.15354/) (2026)

## Abstract (arXiv)

Achieving precise control over multiple properties without sacrificing chemical validity remains a central challenge in molecular inverse design. Existing reinforcement learning (RL) methods fine-tune graph diffusion models by treating **reverse** sampling as a sequential policy, using a single terminal reward to optimize hundreds of coupled decisions. They often suffer from instability, validity collapse, and limited property gains. We introduce GraphFDM (Graph Forward Distribution Matching), a new online RL paradigm for graph diffusion that performs optimization through the **forward** process. GraphFDM uses valid generations to define a reward-tilted target distribution jointly optimized over graph size and molecular structure for each property condition, incorporating reinforcement signals into supervised learning without storing reverse trajectories. We derive the unique optimal target, prove a condition-wise improvement guarantee, and show that the fixed graph-size prior of standard graph diffusion leaves an irreducible matching gap. In multi-conditional polymer and small-molecule generation, GraphFDM achieves the lowest MAE on every target property, with reductions of up to 53.0% relative to the strongest baselines and chemical validity above 0.99. It further generalizes to out-of-distribution property combinations.