# Active Flow Expansion for Out-of-Distribution Discovery: from Theory to Molecules

arXiv: https://arxiv.org/abs/2606.08802 · HTML: https://mlsanity.com/paper/2606.08802/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 61 references that are in the map.
- First posted on arXiv 7 Jun 2026.

## Builds on (most cited first, 30 of 41)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution](https://mlsanity.com/paper/2310.16834/) (2023)
- [RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment](https://mlsanity.com/paper/2304.06767/) (2023)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Flow-GRPO: Training Flow Matching Models via Online RL](https://mlsanity.com/paper/2505.05470/) (2025)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Reinforced Self-Training (ReST) for Language Modeling](https://mlsanity.com/paper/2308.08998/) (2023)
- [Your Absorbing Discrete Diffusion Secretly Models the Conditional Distributions of Clean Data](https://mlsanity.com/paper/2406.03736/) (2024)
- [Safe Model-based Reinforcement Learning with Stability Guarantees](https://mlsanity.com/paper/1705.08551/) (2017)
- [Synthetic Data from Diffusion Models Improves ImageNet Classification](https://mlsanity.com/paper/2304.08466/) (2023)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Imitating Human Behaviour with Diffusion Models](https://mlsanity.com/paper/2301.10677/) (2023)
- [Self-Consuming Generative Models Go MAD](https://mlsanity.com/paper/2307.01850/) (2023)
- [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020)
- [Masked Diffusion Models are Secretly Time-Agnostic Masked Models and Exploit Inaccurate Categorical Sampling](https://mlsanity.com/paper/2409.02908/) (2024)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Review](https://mlsanity.com/paper/2407.13734/) (2024)
- [Inference-Time Alignment in Diffusion Models with Reward-Guided Generation: Tutorial and Review](https://mlsanity.com/paper/2501.09685/) (2025)
- [Self-Play Fine-Tuning of Diffusion Models for Text-to-Image Generation](https://mlsanity.com/paper/2402.10210/) (2024)
- [Stagewise Safe Bayesian Optimization with Gaussian Processes](https://mlsanity.com/paper/1806.07555/) (2018)
- [Safe Exploration in Finite Markov Decision Processes with Gaussian Processes](https://mlsanity.com/paper/1606.04753/) (2016)
- [DIME:Diffusion-Based Maximum Entropy Reinforcement Learning](https://mlsanity.com/paper/2502.02316/) (2025)

## Built on by (most cited first, 3 of 3)

- [BOReFT: Manifold Steering of Language Models for Black-box Optimization](https://mlsanity.com/paper/2609.33722/) (2026)
- [Triage-PiEvo: Evidence Admission for Principle-Evolving Scientific Agents](https://mlsanity.com/paper/2610.14683/) (2026)
- [LLM-to-Flow Distillation: Teaching Natural Language Goals to Scientific Generators](https://mlsanity.com/paper/2610.17246/) (2026)

## Abstract (arXiv)

Standard flow and diffusion pre-training matches the distribution of available data (e.g., molecules), which often covers only a small fraction of the valid design space. In generative discovery, however, one aims to sample valid new-to-nature designs, assigned negligible probability under, and thus inaccessible to, standard models fitted to the observed data. To overcome this limitation, we depart from data distribution matching and view a generative model through its generable set: the region it covers with non-negligible probability. This allows to introduce a new learning principle for out-of-distribution flow modeling: enlarging a model's generable set to increase coverage of the valid design space. We propose Active Flow Expansion (ActFlow), a continued pre-training method that employs verifier feedback to expand a pre-trained model over new valid regions by iteratively adapting to synthetic data generated through active exploration in the learned flow representation. Theoretically, we establish to our knowledge first-of-their-kind statistical learning guarantees for out-of-distribution flow modeling, analyzing generable set expansion as a local-to-global reachability process over a learned representation. Empirically, we assess ActFlow with suitable out-of-distribution generative modeling metrics across small organic molecules, mid-sized drug-like molecules, therapeutic peptides, and protein sequence design tasks. Results show that ActFlow expands valid coverage far beyond the region modeled by the initial pre-trained model, significantly outperforming widely adopted synthetic flow pre-training methods.