Verifier-Constrained Flow Expansion for Discovery Beyond the Data

2026-02-17

AI Summary

This work addresses the limitation of existing pre-trained flow models in generating out-of-distribution yet valid scientific designs—such as molecular structures—due to constrained training data. To overcome this, the authors propose the Flow Expander framework, which leverages strong and weak verifiers (e.g., chemical bonding rules) to perform density expansion on the flow model, thereby broadening the design space while preserving sample validity. The approach integrates mirror descent optimization, verifier-constrained flow expansion, and entropy maximization in a noisy state space, with theoretical guarantees of convergence and maximum entropy under constraints. Experimental results demonstrate that the method significantly enhances conformational diversity in molecular generation and visualization tasks without compromising validity.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Flow and diffusion models are typically pre-trained on limited available data (e.g., molecular samples), covering only a fraction of the valid design space (e.g., the full molecular space). As a consequence, they tend to generate samples from only a narrow portion of the feasible domain. This is a fundamental limitation for scientific discovery applications, where one typically aims to sample valid designs beyond the available data distribution. To this end, we address the challenge of leveraging access to a verifier (e.g., an atomic bonds checker), to adapt a pre-trained flow model so that its induced density expands beyond regions of high data availability, while preserving samples validity. We introduce formal notions of strong and weak verifiers and propose algorithmic frameworks for global and local flow expansion via probability-space optimization. Then, we present Flow Expander (FE), a scalable mirror descent scheme that provably tackles both problems by verifier-constrained entropy maximization over the flow process noised state space. Next, we provide a thorough theoretical analysis of the proposed method, and state convergence guarantees under both idealized and general assumptions. Ultimately, we empirically evaluate our method on both illustrative, yet visually interpretable settings, and on a molecular design task showcasing the ability of FE to expand a pre-trained flow model increasing conformer diversity while preserving validity.

Problem

• flow models
• scientific discovery
• distribution shift
• validity constraints
• design space expansion

Innovation

• flow expansion
• verifier-constrained optimization
• entropy maximization
• molecular design
• mirror descent