# Underdamped Diffusion Bridges with Applications to Sampling

arXiv: https://arxiv.org/abs/2503.01006 · HTML: https://mlsanity.com/paper/2503.01006/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 79 references that are in the map.
- First posted on arXiv 2 Mar 2025.

## Builds on (most cited first, 30 of 38)

- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [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)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [VarGrad: A Low-Variance Gradient Estimator for Variational Inference](https://mlsanity.com/paper/2010.10436/) (2020)
- [Continual Repeated Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2201.13117/) (2022)
- [Particle Denoising Diffusion Sampler](https://mlsanity.com/paper/2402.06320/) (2024)
- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Sampling](https://mlsanity.com/paper/2406.07423/) (2024)
- [Simulating Diffusion Bridges with Score Matching](https://mlsanity.com/paper/2111.07243/) (2021)
- [Score-Based Diffusion meets Annealed Importance Sampling](https://mlsanity.com/paper/2208.07698/) (2022)
- [Sequential Controlled Langevin Diffusions](https://mlsanity.com/paper/2412.07081/) (2024)
- [MCMC Variational Inference via Uncorrected Hamiltonian Annealing](https://mlsanity.com/paper/2107.04150/) (2021)

## Built on by (most cited first, 29 of 29)

- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Control](https://mlsanity.com/paper/2508.10684/) (2025)
- [Value Gradient Guidance for Flow Matching Alignment](https://mlsanity.com/paper/2512.05116/) (2025)
- [On scalable and efficient training of diffusion samplers](https://mlsanity.com/paper/2505.19552/) (2025)
- [Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matching](https://mlsanity.com/paper/2603.00530/) (2026)
- [Rethinking Losses for Diffusion Bridge Samplers](https://mlsanity.com/paper/2506.10982/) (2025)
- [Telegrapher's Generative Model via Kac Flows](https://mlsanity.com/paper/2506.20641/) (2025)
- [Energy-Weighted Flow Matching: Unlocking Continuous Normalizing Flows for Efficient and Scalable Boltzmann Sampling](https://mlsanity.com/paper/2509.03726/) (2025)
- [Adaptive Destruction Processes for Diffusion Samplers](https://mlsanity.com/paper/2506.01541/) (2025)
- [Data-to-Energy Stochastic Dynamics](https://mlsanity.com/paper/2509.26364/) (2025)
- [Reinforced sequential Monte Carlo for amortised sampling](https://mlsanity.com/paper/2510.11711/) (2025)
- [Scalable Maximum Entropy Reinforcement Learning for Diffusion Policies via Adjoint Matching](https://mlsanity.com/paper/2606.22630/) (2026)
- [Discrete Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2602.08243/) (2026)
- [Critically-Damped Higher-Order Langevin Dynamics for Generative Modeling](https://mlsanity.com/paper/2506.21741/) (2025)
- [Fractional Diffusion Bridge Models](https://mlsanity.com/paper/2511.01795/) (2025)
- [Harnessing the Power of Reinforcement Learning for Adaptive MCMC](https://mlsanity.com/paper/2507.00671/) (2025)
- [Discrete diffusion samplers and bridges: Off-policy algorithms and applications in latent spaces](https://mlsanity.com/paper/2602.05961/) (2026)
- [HOTA: Hamiltonian framework for Optimal Transport Advection](https://mlsanity.com/paper/2507.17513/) (2025)
- [Sampling by averaging: A multiscale approach to score estimation](https://mlsanity.com/paper/2508.15069/) (2025)
- [One-Step Diffusion Samplers via Self-Distillation and Deterministic Flow](https://mlsanity.com/paper/2512.05251/) (2025)
- [Free energy Estimation on Any State Space](https://mlsanity.com/paper/2605.31063/) (2026)
- [Analysis of kinetic Langevin Monte Carlo under the stochastic exponential Euler discretization from underdamped all the way to overdamped](https://mlsanity.com/paper/2510.03949/) (2025)
- [Adapting Noise to Data: Generative Flows from 1D Processes](https://mlsanity.com/paper/2510.12636/) (2025)
- [Contact Wasserstein Geodesics for Non-Conservative Schr"odinger Bridges](https://mlsanity.com/paper/2511.06856/) (2025)
- [Particle Monte Carlo methods for Lattice Field Theory](https://mlsanity.com/paper/2511.15196/) (2025)
- [Scalable Inference-Time Annealing with Surrogate Likelihood Estimators](https://mlsanity.com/paper/2605.31498/) (2026)
- [Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Sampling](https://mlsanity.com/paper/2607.06841/) (2026)
- [Conditioning Degenerate Diffusion Models](https://mlsanity.com/paper/2609.04090/) (2026)
- [Neural Transport Nested Sampling](https://mlsanity.com/paper/2609.29413/) (2026)

## Abstract (arXiv)

We provide a general framework for learning diffusion bridges that transport prior to target distributions. It includes existing diffusion models for generative modeling, but also underdamped versions with degenerate diffusion matrices, where the noise only acts in certain dimensions. Extending previous findings, our framework allows to rigorously show that score matching in the underdamped case is indeed equivalent to maximizing a lower bound on the likelihood. Motivated by superior convergence properties and compatibility with sophisticated numerical integration schemes of underdamped stochastic processes, we propose underdamped diffusion bridges, where a general density evolution is learned rather than prescribed by a fixed noising process. We apply our method to the challenging task of sampling from unnormalized densities without access to samples from the target distribution. Across a diverse range of sampling problems, our approach demonstrates state-of-the-art performance, notably outperforming alternative methods, while requiring significantly fewer discretization steps and no hyperparameter tuning.