A Unified Density Operator View of Flow Control and Merging

• URL: http://arxiv.org/abs/2602.08012v1
• Date: Sun, 08 Feb 2026 15:27:28 GMT
• Title: A Unified Density Operator View of Flow Control and Merging
• Authors: Riccardo De Santi, Malte Franke, Ya-Ping Hsieh, Andreas Krause,
• Abstract summary: We introduce a unifying probability-space framework that subsumes both as limit cases, and enables reward-guided flow merging.<n>We also introduce Reward-Guided Flow Merging (RFM), a mirror-descent scheme that reduces reward-guided flow merging to a sequence of standard fine-tuning problems.<n>We provide first-of-their-kind theoretical guarantees for reward-guided and pure flow merging via RFM.
• Score: 37.902481322917396
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent progress in large-scale flow and diffusion models raised two fundamental algorithmic challenges: (i) control-based reward adaptation of pre-trained flows, and (ii) integration of multiple models, i.e., flow merging. While current approaches address them separately, we introduce a unifying probability-space framework that subsumes both as limit cases, and enables reward-guided flow merging, allowing principled, task-aware combination of multiple pre-trained flows (e.g., merging priors while maximizing drug-discovery utilities). Our formulation renders possible to express a rich family of operators over generative models densities, including intersection (e.g., to enforce safety), union (e.g., to compose diverse models), interpolation (e.g., for discovery), their reward-guided counterparts, as well as complex logical expressions via generative circuits. Next, we introduce Reward-Guided Flow Merging (RFM), a mirror-descent scheme that reduces reward-guided flow merging to a sequence of standard fine-tuning problems. Then, we provide first-of-their-kind theoretical guarantees for reward-guided and pure flow merging via RFM. Ultimately, we showcase the capabilities of the proposed method on illustrative settings providing visually interpretable insights, and apply our method to high-dimensional de-novo molecular design and low-energy conformer generation.

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