A Unified Density Operator View of Flow Control and Merging

TL;DR

This paper introduces a unified probabilistic framework for controlling and merging flow-based generative models, enabling task-aware combination and reward-guided optimization with theoretical guarantees.

Contribution

It proposes a unifying framework that subsumes flow control and merging, along with a novel reward-guided merging algorithm with theoretical guarantees.

Findings

Effective reward-guided flow merging demonstrated in molecular design.

Theoretical guarantees established for the merging process.

Visual and practical insights shown in high-dimensional applications.

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.