Verifier-Constrained Flow Expansion for Discovery Beyond the Data

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.

Adapting generative foundation models, in particular diffusion and flow models, to optimize given reward functions (e.g., binding affinity) while satisfying constraints (e.g., molecular synthesizability) is fundamental for their adoption in real-world scientific discovery applications such as molecular design or protein engineering. While recent works have introduced scalable methods for reward-guided fine-tuning of such models via reinforcement learning and control schemes, it remains an open problem how to algorithmically trade-off reward maximization and constraint satisfaction in a reliable and predictable manner. Motivated by this challenge, we first present a rigorous framework for Constrained Generative Optimization, which brings an optimization viewpoint to the introduced adaptation problem and retrieves the relevant task of constrained generation as a sub-case. Then, we introduce Constrained Flow Optimization (CFO), an algorithm that automatically and provably balances reward maximization and constraint satisfaction by reducing the original problem to sequential fine-tuning via established, scalable methods. We provide convergence guarantees for constrained generative optimization and constrained generation via CFO. Ultimately, we present an experimental evaluation of CFO on both synthetic, yet illustrative, settings, and a molecular design task. Across these evaluations, CFO achieves consistent increases in reward while ensuring high constraint satisfaction, showcasing its practical utility for constrained generative optimization.

Molecular discovery is increasingly framed as an inverse design problem: identifying molecular structures that satisfy desired property profiles under feasibility constraints. While recent generative models provide continuous latent representations of chemical space, targeted optimization within these representations often leads to degraded validity, loss of structural fidelity, or unstable behavior. We introduce MoltenFlow, a modular framework that combines property-organized latent representations with flow-matching generative priors and gradient-based guidance. This formulation supports both conditioned generation and local optimization within a single latent-space framework. We show that guided latent flows enable efficient multi-objective molecular optimization under fixed oracle budgets with controllable trade-offs, while a learned flow prior improves unconditional generation quality.