A Unified Density Operator View of Flow Control and Merging

2026-02-08

AI Summary

Existing approaches typically treat reward-guided control and multi-model fusion in pretrained flow models as separate problems. This work proposes a unified probabilistic framework that interprets both as special cases within a space of density operators, enabling task-aware flow composition for the first time and supporting logical operations—such as intersection, union, and interpolation—under reward guidance, all with theoretical guarantees. Leveraging mirror descent, we design the Reward-guided Flow Merging (RFM) algorithm, which reduces complex fusion tasks to a sequence of standard fine-tuning steps. The method demonstrates superior performance and strong interpretability across high-dimensional tasks, including image generation, de novo molecular design, and low-energy conformation sampling.

Technology Category

• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Multimodal Learning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

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.

Problem

• flow merging
• reward adaptation
• generative models
• density operators
• control-based generation

Innovation

• Reward-Guided Flow Merging
• Density Operator Framework
• Generative Model Fusion
• Mirror Descent
• Generative Circuits