# Training Free Guided Flow Matching with Optimal Control

arXiv: https://arxiv.org/abs/2410.18070 · HTML: https://mlsanity.com/paper/2410.18070/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 31 of its 52 references that are in the map.
- First posted on arXiv 23 Oct 2024.

## 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 31)

- [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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [StyleCLIP: Text-Driven Manipulation of StyleGAN Imagery](https://mlsanity.com/paper/2103.17249/) (2021)
- [Training Diffusion Models with Reinforcement Learning](https://mlsanity.com/paper/2305.13301/) (2023)
- [High Fidelity Neural Audio Compression](https://mlsanity.com/paper/2210.13438/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Regularisation of Neural Networks by Enforcing Lipschitz Continuity](https://mlsanity.com/paper/1804.04368/) (2018)
- [Geometric Latent Diffusion Models for 3D Molecule Generation](https://mlsanity.com/paper/2305.01140/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Models](https://mlsanity.com/paper/2406.08070/) (2024)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [A Survey on Audio Diffusion Models: Text To Speech Synthesis and Enhancement in Generative AI](https://mlsanity.com/paper/2303.13336/) (2023)
- [Equivariant Flow Matching with Hybrid Probability Transport](https://mlsanity.com/paper/2312.07168/) (2023)
- [Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation](https://mlsanity.com/paper/2305.18470/) (2023)
- [Categorical Flow Matching on Statistical Manifolds](https://mlsanity.com/paper/2405.16441/) (2024)
- [Improved motif-scaffolding with SE(3) flow matching](https://mlsanity.com/paper/2401.04082/) (2024)
- [Sequence-Augmented SE(3)-Flow Matching For Conditional Protein Backbone Generation](https://mlsanity.com/paper/2405.20313/) (2024)
- [Full-Atom Peptide Design based on Multi-modal Flow Matching](https://mlsanity.com/paper/2406.00735/) (2024)
- [Enhancing Diffusion-Based Image Synthesis with Robust Classifier Guidance](https://mlsanity.com/paper/2208.08664/) (2022)
- [Some Fundamental Aspects about Lipschitz Continuity of Neural Networks](https://mlsanity.com/paper/2302.10886/) (2023)
- [AdjointDPM: Adjoint Sensitivity Method for Gradient Backpropagation of Diffusion Probabilistic Models](https://mlsanity.com/paper/2307.10711/) (2023)

## Built on by (most cited first, 25 of 25)

- [On the Guidance of Flow Matching](https://mlsanity.com/paper/2502.02150/) (2025)
- [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025)
- [Source-Guided Flow Matching](https://mlsanity.com/paper/2508.14807/) (2025)
- [HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization](https://mlsanity.com/paper/2511.08425/) (2025)
- [Efficient Tail-Aware Generative Optimization via Flow Model Fine-Tuning](https://mlsanity.com/paper/2602.16796/) (2026)
- [Variational Supervised Contrastive Learning](https://mlsanity.com/paper/2506.07413/) (2025)
- [Generative Models for Crystalline Materials](https://mlsanity.com/paper/2511.22652/) (2025)
- [Greed is Good: A Unifying Perspective on Guided Generation](https://mlsanity.com/paper/2502.08006/) (2025)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Trajectory Stitching for Solving Inverse Problems with Flow-Based Models](https://mlsanity.com/paper/2602.08538/) (2026)
- [Inference-Time Attribute Distribution Alignment for Unconditional Diffusion](https://mlsanity.com/paper/2605.07456/) (2026)
- [Solving Inverse Problems with Flow-based Models via Model Predictive Control](https://mlsanity.com/paper/2601.23231/) (2026)
- [TFTF: Training-Free Targeted Flow for Conditional Sampling](https://mlsanity.com/paper/2602.12932/) (2026)
- [Minimizing Structural Vibrations via Guided Flow Matching Design Optimization](https://mlsanity.com/paper/2506.15263/) (2025)
- [Semantic Editing with Coupled Stochastic Differential Equations](https://mlsanity.com/paper/2509.24223/) (2025)
- [Trust-Region Noise Search for Black-Box Alignment of Diffusion and Flow Models](https://mlsanity.com/paper/2603.14504/) (2026)
- [Physically Grounded 3D Generative Reconstruction under Hand Occlusion using Proprioception and Multi-Contact Touch](https://mlsanity.com/paper/2604.09100/) (2026)
- [Scaffold-Conditioned Preference Triplets for Controllable Molecular Optimization with Large Language Models](https://mlsanity.com/paper/2604.12350/) (2026)
- [Guided Streaming Stochastic Interpolant Policy](https://mlsanity.com/paper/2605.10051/) (2026)
- [Pepti-drift: Scalable Safe-Active Peptide Generation Without Inference-Time Guidance](https://mlsanity.com/paper/2606.27824/) (2026)
- [Not All Prediction Targets Keep Training-Free Diffusion Guidance on the Manifold](https://mlsanity.com/paper/2607.00647/) (2026)
- [Diffeomorphic Optimization](https://mlsanity.com/paper/2607.00947/) (2026)
- [ParetoTransport: Generative Optimization by Mass Transport Toward The Pareto Front](https://mlsanity.com/paper/2609.07706/) (2026)
- [LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems](https://mlsanity.com/paper/2610.04326/) (2026)
- [BridgeTwist: Twisting Schr"odinger Bridges for Training-Free Conditional Sampling](https://mlsanity.com/paper/2610.18658/) (2026)

## Abstract (arXiv)

Controlled generation with pre-trained Diffusion and Flow Matching models has vast applications. One strategy for guiding ODE-based generative models is through optimizing a target loss R(x1) while staying close to the prior distribution. Along this line, some recent work showed the effectiveness of guiding flow model by differentiating through its ODE sampling process. Despite the superior performance, the theoretical understanding of this line of methods is still preliminary, leaving space for algorithm improvement. Moreover, existing methods predominately focus on Euclidean data manifold, and there is a compelling need for guided flow methods on complex geometries such as SO(3), which prevails in high-stake scientific applications like protein design. We present OC-Flow, a general and theoretically grounded training-free framework for guided flow matching using optimal control. Building upon advances in optimal control theory, we develop effective and practical algorithms for solving optimal control in guided ODE-based generation and provide a systematic theoretical analysis of the convergence guarantee in both Euclidean and SO(3). We show that existing backprop-through-ODE methods can be interpreted as special cases of Euclidean OC-Flow. OC-Flow achieved superior performance in extensive experiments on text-guided image manipulation, conditional molecule generation, and all-atom peptide design.