# HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization

arXiv: https://arxiv.org/abs/2511.08425 · HTML: https://mlsanity.com/paper/2511.08425/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 49 of its 66 references that are in the map.
- First posted on arXiv 11 Nov 2025.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 30 of 49)

- [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)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [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)
- [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)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [D4RL: Datasets for Deep Data-Driven Reinforcement Learning](https://mlsanity.com/paper/2004.07219/) (2020)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Movie Gen: A Cast of Media Foundation Models](https://mlsanity.com/paper/2410.13720/) (2024)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [Universal Guidance for Diffusion Models](https://mlsanity.com/paper/2302.07121/) (2023)
- [PhysDiff: Physics-Guided Human Motion Diffusion Model](https://mlsanity.com/paper/2212.02500/) (2022)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)

## Built on by (most cited first, 11 of 11)

- [Provably Safe Generative Sampling with Constricting Barrier Functions](https://mlsanity.com/paper/2602.21429/) (2026)
- [DiRecT: Safe Diffusion-Based Planning via Receding-Horizon Denoising](https://mlsanity.com/paper/2606.15359/) (2026)
- [Solving Inverse Problems with Flow-based Models via Model Predictive Control](https://mlsanity.com/paper/2601.23231/) (2026)
- [Steering Generative Robot Policies with Lexicographic Preferences](https://mlsanity.com/paper/2609.15014/) (2026)
- [CCFM: Collision-Constrained Flow Matching for Safety-Critical Scenario Generation](https://mlsanity.com/paper/2607.04451/) (2026)
- [Constrained Flow Matching via Lagrangian Dual Flows](https://mlsanity.com/paper/2607.04513/) (2026)
- [Self-Augmented Diffusion Guidance for Physics-Informed Generation](https://mlsanity.com/paper/2608.26748/) (2026)
- [Denoising as Projection: Constrained Optimization with Gradient-Guided Diffusion](https://mlsanity.com/paper/2608.29507/) (2026)
- [ParetoTransport: Generative Optimization by Mass Transport Toward The Pareto Front](https://mlsanity.com/paper/2609.07706/) (2026)
- [Let It Go or Learn to Self-Correct: Continuous Diffusion for Constrained Discrete Tasks](https://mlsanity.com/paper/2609.09009/) (2026)
- [Primal-Dual Flow Matching for Sample-Wise Constrained Generation](https://mlsanity.com/paper/2610.12699/) (2026)

## Abstract (arXiv)

Diffusion and flow-matching have emerged as powerful methodologies for generative modeling, with remarkable success in capturing complex data distributions and enabling flexible guidance at inference time. Many downstream applications, however, demand enforcing hard constraints on generated samples (for example, robot trajectories must avoid obstacles), a requirement that goes beyond simple guidance. Prevailing projection-based approaches constrain the entire sampling path to the constraint manifold, which is overly restrictive and degrades sample quality. In this paper, we introduce a novel framework that reformulates hard-constrained sampling as a trajectory optimization problem. Our key insight is to leverage numerical optimal control to steer the sampling trajectory so that constraints are satisfied precisely at the terminal time. By exploiting the underlying structure of flow-matching models and adopting techniques from model predictive control, we transform this otherwise complex constrained optimization problem into a tractable surrogate that can be solved efficiently and effectively. Furthermore, this trajectory optimization perspective offers significant flexibility beyond mere constraint satisfaction, allowing for the inclusion of integral costs to minimize distribution shift and terminal objectives to further enhance sample quality, all within a unified framework. We provide a control-theoretic analysis of our method, establishing bounds on the approximation error between our tractable surrogate and the ideal formulation. Extensive experiments across diverse domains, including robotics (planning), partial differential equations (boundary control), and vision (text-guided image editing), demonstrate that our algorithm, which we name textitHardFlow, substantially outperforms existing methods in both constraint satisfaction and sample quality.