# Primal-Dual Flow Matching for Sample-Wise Constrained Generation

arXiv: https://arxiv.org/abs/2610.12699 · HTML: https://mlsanity.com/paper/2610.12699/

## 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 23 of its 37 references that are in the map.
- First posted on arXiv 8 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, 23 of 23)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation](https://mlsanity.com/paper/2305.18470/) (2023)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptions](https://mlsanity.com/paper/2409.18959/) (2024)
- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [Training neural operators to preserve invariant measures of chaotic attractors](https://mlsanity.com/paper/2306.01187/) (2023)
- [Mirror Diffusion Models for Constrained and Watermarked Generation](https://mlsanity.com/paper/2310.01236/) (2023)
- [Constrained Diffusion Models via Dual Training](https://mlsanity.com/paper/2408.15094/) (2024)
- [Reflected Flow Matching](https://mlsanity.com/paper/2405.16577/) (2024)
- [HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization](https://mlsanity.com/paper/2511.08425/) (2025)
- [Training-Free Adaptation of Diffusion Models via Doob's h-Transform](https://mlsanity.com/paper/2602.16198/) (2026)
- [Neural Approximate Mirror Maps for Constrained Diffusion Models](https://mlsanity.com/paper/2406.12816/) (2024)
- [On Flow Matching KL Divergence](https://mlsanity.com/paper/2511.05480/) (2025)
- [Constraint-Aware Flow Matching via Randomized Exploration](https://mlsanity.com/paper/2508.13316/) (2025)
- [Learning to Emulate Chaos: Adversarial Optimal Transport Regularization](https://mlsanity.com/paper/2604.21097/) (2026)

## Abstract (arXiv)

We consider the problem of learning to generate data while satisfying a prescribed sample-wise constraint with a user-specified probability of constraint satisfaction. Starting with a population-level constrained negative log-likelihood minimization and using a flow-matching (FM) parameterization, under regularity assumptions and sufficient expressivity of the velocity field family, we derive guarantees on the probability of constraint satisfaction for the corresponding primal-dual optimization algorithm. We provide a practical implementation of this approach, called Primal-Dual Flow Matching (PDFM), which has direct control over constraint satisfaction rate and requires only binary membership oracle feedback for constraint satisfaction, avoiding the need for differentiable distances, projections, or special structure of the constraint set, such as convexity. We show that PDFM improves constraint satisfaction over vanilla FM and achieves competitive performance against constraint-aware baselines while maintaining distributional fidelity. Our code is available at https://github.com/ZhengyanHuan/PDFM.