# Weighted Adjoint Matching: Finetuning Flow Matching Models with Time-Dependent Control Cost

arXiv: https://arxiv.org/abs/2610.20035 · HTML: https://mlsanity.com/paper/2610.20035/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 35 of its 46 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 35)

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [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)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2304.05977/) (2023)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Training Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2305.13301/) (2023)
- [DPOK: Reinforcement Learning for Fine-tuning Text-to-Image Diffusion Models](https://mlsanity.com/paper/2305.16381/) (2023)
- [Directly Fine-Tuning Diffusion Models on Differentiable Rewards](https://mlsanity.com/paper/2309.17400/) (2023)
- [Certifying Some Distributional Robustness with Principled Adversarial Training](https://mlsanity.com/paper/1710.10571/) (2017)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)
- [Fine-Tuning of Continuous-Time Diffusion Models as Entropy-Regularized Control](https://mlsanity.com/paper/2402.15194/) (2024)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Review](https://mlsanity.com/paper/2407.13734/) (2024)
- [TempFlow-GRPO: When Timing Matters for GRPO in Flow Models](https://mlsanity.com/paper/2508.04324/) (2025)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [Spontaneous Symmetry Breaking in Generative Diffusion Models](https://mlsanity.com/paper/2305.19693/) (2023)
- [Diffusion Models Generate Images Like Painters: an Analytical Theory of Outline First, Details Later](https://mlsanity.com/paper/2303.02490/) (2023)
- [Critical windows: non-asymptotic theory for feature emergence in diffusion models](https://mlsanity.com/paper/2403.01633/) (2024)
- [Stochastic Optimal Control Matching](https://mlsanity.com/paper/2312.02027/) (2023)
- [Implicit Diffusion: Efficient Optimization through Stochastic Sampling](https://mlsanity.com/paper/2402.05468/) (2024)
- [Tilt Matching for Scalable Sampling and Fine-Tuning](https://mlsanity.com/paper/2512.21829/) (2025)
- [Stochastic Control for Fine-tuning Diffusion Models: Optimality, Regularity, and Convergence](https://mlsanity.com/paper/2412.18164/) (2024)
- [DiverseGRPO: Mitigating Mode Collapse in Image Generation via Diversity-Aware GRPO](https://mlsanity.com/paper/2512.21514/) (2025)
- [Avoiding mode collapse in diffusion models fine-tuned with reinforcement learning](https://mlsanity.com/paper/2410.08315/) (2024)

## Abstract (arXiv)

We introduce Weighted Adjoint Matching, a method to finetune a pretrained diffusion or flow matching model given a differentiable reward. We derive it from Adjoint Matching, which reformulates finetuning as a stochastic optimal control problem with a quadratic control cost. Building on the observation that these models generate coarse features before fine details, we place a time-dependent weight on the control cost. A high cost early in the generative process preserves balance and diversity in the coarse features, which carry most of the semantics, while a low cost at the end lets the model increase the reward by refining details. We prove that our method is equivalent to maximizing the reward under time-varying Kullback--Leibler regularization on the generative paths. Experiments on Gaussian mixtures and on MNIST show a better tradeoff between reward and mode balance than Adjoint Matching.