# FOCUS: Optimal Control for Multi-Entity World Modeling in Text-to-Image Generation

arXiv: https://arxiv.org/abs/2510.02315 · HTML: https://mlsanity.com/paper/2510.02315/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Video-generation world models and simulators, one of 11 directions in World models.
- Builds on 33 of its 34 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Placed in

- World models / [Video-generation world models and simulators](https://mlsanity.com/wm/video-generation-world-models-and-simulators/)
- Diffusion models / [Image generation, text-to-image and editing](https://mlsanity.com/diffusion/image-generation-text-to-image-and-editing/)

## Builds on (most cited first, 30 of 33)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [BLIP: Bootstrapping Language-Image Pre-training for Unified Vision-Language Understanding and Generation](https://mlsanity.com/paper/2201.12086/) (2022)
- [Qwen2-VL: Enhancing Vision-Language Model's Perception of the World at Any Resolution](https://mlsanity.com/paper/2409.12191/) (2024)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Prompt-to-Prompt Image Editing with Cross Attention Control](https://mlsanity.com/paper/2208.01626/) (2022)
- [Pick-a-Pic: An Open Dataset of User Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2305.01569/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [GLIGEN: Open-Set Grounded Text-to-Image Generation](https://mlsanity.com/paper/2301.07093/) (2023)
- [Attend-and-Excite: Attention-Based Semantic Guidance for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2301.13826/) (2023)
- [Compositional Visual Generation with Composable Diffusion Models](https://mlsanity.com/paper/2206.01714/) (2022)
- [MultiDiffusion: Fusing Diffusion Paths for Controlled Image Generation](https://mlsanity.com/paper/2302.08113/) (2023)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [Training-Free Structured Diffusion Guidance for Compositional Text-to-Image Synthesis](https://mlsanity.com/paper/2212.05032/) (2022)
- [Scaling Open-Vocabulary Object Detection](https://mlsanity.com/paper/2306.09683/) (2023)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Divide & Bind Your Attention for Improved Generative Semantic Nursing](https://mlsanity.com/paper/2307.10864/) (2023)
- [Be Yourself: Bounded Attention for Multi-Subject Text-to-Image Generation](https://mlsanity.com/paper/2403.16990/) (2024)
- [CONFORM: Contrast is All You Need For High-Fidelity Text-to-Image Diffusion Models](https://mlsanity.com/paper/2312.06059/) (2023)

## Abstract (arXiv)

Text-to-image (T2I) models excel on single-entity prompts but struggle with multi-entity scenes, often exhibiting attribute leakage, identity entanglement, and subject omissions. We present a principled theoretical framework that steers sampling toward multi-subject fidelity by casting flow matching (FM) as stochastic optimal control (SOC), yielding a single hyperparameter controlled trade-off between fidelity and object-centric state separation / binding consistency. Within this framework, we derive two architecture-agnostic algorithms: (i) a training-free test-time controller that perturbs the base velocity with a single-pass update, and (ii) Adjoint Matching, a lightweight fine-tuning rule that regresses a control network to a backward adjoint signal. The same formulation unifies prior attention heuristics, extends to diffusion models via a flow--diffusion correspondence, and provides the first fine-tuning route explicitly designed for multi-subject fidelity. In addition, we also introduce FOCUS (Flow Optimal Control for Unentangled Subjects), a probabilistic attention-binding objective compatible with both algorithms. Empirically, on Stable Diffusion 3.5 and FLUX.1, both algorithms consistently improve multi-subject alignment while maintaining base-model style; test-time control runs efficiently on commodity GPUs, and fine-tuned models generalize to unseen prompts.