# Score-based constrained generative modeling via Langevin diffusions with boundary conditions

arXiv: https://arxiv.org/abs/2510.23985 · HTML: https://mlsanity.com/paper/2510.23985/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 12 of its 43 references that are in the map.
- First posted on arXiv 28 Oct 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 12 of 12)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Constrained Diffusion Models via Dual Training](https://mlsanity.com/paper/2408.15094/) (2024)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [Constrained Generative Modeling with Manually Bridged Diffusion Models](https://mlsanity.com/paper/2502.20371/) (2025)
- [Sampling and estimation on manifolds using the Langevin diffusion](https://mlsanity.com/paper/2312.14882/) (2023)

## Built on by (most cited first, 2 of 2)

- [Picasso: Holistic Scene Reconstruction with Physics-Constrained Sampling](https://mlsanity.com/paper/2602.08058/) (2026)
- [Should the Boundary Term Be Learned in Reflected Diffusion? Conormal Trace and Reflection Masking](https://mlsanity.com/paper/2608.03469/) (2026)

## Abstract (arXiv)

Score-based generative models based on stochastic differential equations (SDEs) achieve impressive performance in sampling from unknown distributions, but often fail to satisfy underlying constraints. We propose a constrained generative model using kinetic (underdamped) Langevin dynamics with specular reflection of velocity on the boundary defining constraints. This results in piecewise continuously differentiable noising and denoising process where the latter is characterized by a time-reversed dynamics restricted to a domain with boundary due to specular boundary condition. In addition, we also contribute to existing reflected SDEs based constrained generative models, where the stochastic dynamics is restricted through an abstract local time term. By presenting efficient numerical samplers which converge with optimal rate in terms of discretizations step, we provide a comprehensive comparison of models based on confined (specularly reflected kinetic) Langevin diffusion with models based on reflected diffusion with local time.