mlsanityData to 14 Oct 2026

Primal-Dual Flow Matching for Sample-Wise Constrained Generation

arXiv 2610.12699 ↗First posted 8 Oct 2026cs.LGcited by 0

On the map: Diffusion models → Flow matching, time series and science

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.

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.

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23 of 37 references.

Closest ancestors · 8

Same direction, newest first

  1. 2025On Flow Matching KL Divergence7 Nov
  2. 2025Constraint-Aware Flow Matching via Randomized Exploration18 Aug
  3. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints4 Jun
  4. 2024Reflected Flow Matching26 May
  5. 2023Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation29 May
  6. 2023Improving and generalizing flow-based generative models with minibatch optimal transport1 Feb
  7. 2022Flow Matching for Generative Modeling6 Oct
  8. 2022Equivariant Diffusion for Molecule Generation in 3D31 Mar

Other references · 15

  1. 2021Convex optimizationcited by 2,515
  2. 2021E(n) Equivariant Graph Neural Networkscited by 965
  3. 2024Constrained Synthesis with Projected Diffusion Modelscited by 89
  4. 2023Reflected Diffusion Modelscited by 87
  5. 2023Diffusion Models for Constrained Domainscited by 63
  6. 2024APEBench: A Benchmark for Autoregressive Neural Emulators of PDEscited by 60
  7. 2024O(d/T) Convergence Theory for Diffusion Probabilistic Models under Minimal Assumptionscited by 50
  8. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 49
  9. 2023Training neural operators to preserve invariant measures of chaotic attractorscited by 46
  10. 2023Mirror Diffusion Models for Constrained and Watermarked Generationcited by 42
  11. 2024Constrained Diffusion Models via Dual Trainingcited by 20
  12. 2025HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimizationcited by 11
3 more
  1. 2026Training-Free Adaptation of Diffusion Models via Doob's h-Transformcited by 11
  2. 2024Neural Approximate Mirror Maps for Constrained Diffusion Modelscited by 9
  3. 2026Learning to Emulate Chaos: Adversarial Optimal Transport Regularizationcited by 1

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  2. 2024Flow matching achieves almost minimax optimal convergence
  3. 2026When do explicit penalties improve flow matching?
  4. 2026Constrained Flow Matching via Lagrangian Dual Flows
  5. 2026Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions
  6. 2024Optimal Flow Matching: Learning Straight Trajectories in Just One Step
  7. 2024Gradient-Free Generation for Hard-Constrained Systems
  8. 2024Minibatch Optimal Transport and Perplexity Bound Estimation in Discrete Flow Matching