A Unified Density Operator View of Flow Control and Merging
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915784821833728 |
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| author | De Santi, Riccardo Franke, Malte Hsieh, Ya-Ping Krause, Andreas |
| author_facet | De Santi, Riccardo Franke, Malte Hsieh, Ya-Ping Krause, Andreas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08012 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Unified Density Operator View of Flow Control and Merging De Santi, Riccardo Franke, Malte Hsieh, Ya-Ping Krause, Andreas Machine Learning 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. |
| title | A Unified Density Operator View of Flow Control and Merging |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2602.08012 |