# Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation

arXiv: https://arxiv.org/abs/2509.25157 · HTML: https://mlsanity.com/paper/2509.25157/

## Key facts

- Cited by 7 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 24 of its 41 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## 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, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Applying Guidance in a Limited Interval Improves Sample and Distribution Quality in Diffusion Models](https://mlsanity.com/paper/2404.07724/) (2024)
- [EquiBind: Geometric Deep Learning for Drug Binding Structure Prediction](https://mlsanity.com/paper/2202.05146/) (2022)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Improved motif-scaffolding with SE(3) flow matching](https://mlsanity.com/paper/2401.04082/) (2024)
- [Gradient-Free Generation for Hard-Constrained Systems](https://mlsanity.com/paper/2412.01786/) (2024)
- [Simultaneous Multi-Robot Motion Planning with Projected Diffusion Models](https://mlsanity.com/paper/2502.03607/) (2025)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Convergence of Continuous Normalizing Flows for Learning Probability Distributions](https://mlsanity.com/paper/2404.00551/) (2024)
- [Improving Classifier-Free Guidance in Masked Diffusion: Low-Dim Theoretical Insights with High-Dim Impact](https://mlsanity.com/paper/2507.08965/) (2025)
- [Neuro-Symbolic Generative Diffusion Models for Physically Grounded, Robust, and Safe Generation](https://mlsanity.com/paper/2506.01121/) (2025)
- [Discrete-Guided Diffusion for Scalable and Safe Multi-Robot Motion Planning](https://mlsanity.com/paper/2508.20095/) (2025)
- [Constrained Generative Modeling with Manually Bridged Diffusion Models](https://mlsanity.com/paper/2502.20371/) (2025)
- [Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling](https://mlsanity.com/paper/2505.01917/) (2025)

## Built on by (most cited first, 7 of 7)

- [Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing](https://mlsanity.com/paper/2604.07366/) (2026)
- [Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Sampling](https://mlsanity.com/paper/2605.12754/) (2026)
- [CofactVLA: Deconfounding Vision-Language-Action Models via Counterfactual Intervention](https://mlsanity.com/paper/2608.04396/) (2026)
- [Synthics: Synthetic Physics-like Datasets for Machine Learning](https://mlsanity.com/paper/2606.06724/) (2026)
- [Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields](https://mlsanity.com/paper/2608.18004/) (2026)
- [Simulator-Refined Diffusion for Radio-Frequency Inverse Design](https://mlsanity.com/paper/2609.38363/) (2026)
- [PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generation](https://mlsanity.com/paper/2609.40287/) (2026)

## Abstract (arXiv)

Generative models excel at synthesizing high-fidelity samples from complex data distributions, but they often violate hard constraints arising from physical laws or task specifications. A common remedy is to project intermediate samples onto the feasible set; however, repeated projection can distort the learned distribution and induce a mismatch with the data manifold. Thus, recent multi-stage procedures attempt to defer projection to clean samples during sampling, but they increase algorithmic complexity and accumulate errors across steps. This paper addresses these challenges by proposing a novel training-free method, Chance-constrained Flow Matching (CCFM), that integrates stochastic optimization into the sampling process, enabling effective enforcement of hard constraints while maintaining high-fidelity sample generation. Importantly, CCFM guarantees feasibility in the same manner as conventional repeated projection, yet, despite operating directly on noisy intermediate samples, it is theoretically equivalent to projecting onto the feasible set defined by clean samples. This yields a sampler that mitigates distributional distortion. Empirical experiments show that CCFM outperforms current state-of-the-art constrained generative models in modeling complex physical systems governed by partial differential equations and molecular docking problems, delivering higher feasibility and fidelity.