# Adjoint Schr"odinger Bridge Sampler

arXiv: https://arxiv.org/abs/2506.22565 · HTML: https://mlsanity.com/paper/2506.22565/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 45 of its 82 references that are in the map.
- First posted on arXiv 27 Jun 2025.

## Placed in

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

## Builds on (most cited first, 30 of 45)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [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)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [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)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction](https://mlsanity.com/paper/2502.12147/) (2025)
- [SPICE, A Dataset of Drug-like Molecules and Peptides for Training Machine Learning Potentials](https://mlsanity.com/paper/2209.10702/) (2022)

## Built on by (most cited first, 21 of 21)

- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models](https://mlsanity.com/paper/2509.21655/) (2025)
- [MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Control](https://mlsanity.com/paper/2508.10684/) (2025)
- [Discrete Adjoint Matching](https://mlsanity.com/paper/2602.07132/) (2026)
- [EVODiff: Entropy-aware Variance Optimized Diffusion Inference](https://mlsanity.com/paper/2509.26096/) (2025)
- [Adjoint Matching through the Lens of the Stochastic Maximum Principle in Optimal Control](https://mlsanity.com/paper/2604.08580/) (2026)
- [MolCrystalFlow: Molecular Crystal Structure Prediction via Flow Matching](https://mlsanity.com/paper/2602.16020/) (2026)
- [Scalable Maximum Entropy Reinforcement Learning for Diffusion Policies via Adjoint Matching](https://mlsanity.com/paper/2606.22630/) (2026)
- [MolGuidance: Advanced Guidance Strategies for Conditional Molecular Generation with Flow Matching](https://mlsanity.com/paper/2512.12198/) (2025)
- [Efficient Adjoint Matching for Fine-tuning Diffusion Models](https://mlsanity.com/paper/2605.11480/) (2026)
- [Simple Approximation and Derivative Free Inference-Time Scaling for Diffusion Models via Sequential Monte Carlo on Path Measures](https://mlsanity.com/paper/2605.17850/) (2026)
- [Exact Global MCMC with Denoising Diffusion](https://mlsanity.com/paper/2609.00279/) (2026)
- [Entropy Across the Bridge: Conditional-Marginal Discretization for Flow and Schr"odinger Samplers](https://mlsanity.com/paper/2605.16126/) (2026)
- [ATLAS: A Foundation Neural Sampler for Amorphous Materials](https://mlsanity.com/paper/2607.19198/) (2026)
- [Sampling via Stochastic Interpolants by Langevin-based Velocity and Initialization Estimation in Flow ODEs](https://mlsanity.com/paper/2601.08527/) (2026)
- [Towards Intrinsically Calibrated Uncertainty Quantification in Industrial Data-Driven Models via Diffusion Sampler](https://mlsanity.com/paper/2604.01870/) (2026)
- [Neural Transport Nested Sampling](https://mlsanity.com/paper/2609.29413/) (2026)
- [Averaged Mirror Descent and Dual Gradient Methods: Convergent Algorithms for Entropic Gromov-Wasserstein Problems](https://mlsanity.com/paper/2609.31848/) (2026)
- [Fixed-point neural samplers on discrete spaces](https://mlsanity.com/paper/2610.01739/) (2026)
- [Distributionally Robust Schr"odinger Bridge](https://mlsanity.com/paper/2610.02043/) (2026)
- [Diverse Motion Customization via Control-based Dynamic Optimization](https://mlsanity.com/paper/2610.07911/) (2026)

## Abstract (arXiv)

Computational methods for learning to sample from the Boltzmann distribution -- where the target distribution is known only up to an unnormalized energy function -- have advanced significantly recently. Due to the lack of explicit target samples, however, prior diffusion-based methods, known as diffusion samplers, often require importance-weighted estimation or complicated learning processes. Both trade off scalability with extensive evaluations of the energy and model, thereby limiting their practical usage. In this work, we propose Adjoint Schr"odinger Bridge Sampler (ASBS), a new diffusion sampler that employs simple and scalable matching-based objectives yet without the need to estimate target samples during training. ASBS is grounded on a mathematical model -- the Schr"odinger Bridge -- which enhances sampling efficiency via kinetic-optimal transportation. Through a new lens of stochastic optimal control theory, we demonstrate how SB-based diffusion samplers can be learned at scale via Adjoint Matching and prove convergence to the global solution. Notably, ASBS generalizes the recent Adjoint Sampling (Havens et al., 2025) to arbitrary source distributions by relaxing the so-called memoryless condition that largely restricts the design space. Through extensive experiments, we demonstrate the effectiveness of ASBS on sampling from classical energy functions, amortized conformer generation, and molecular Boltzmann distributions. Code available at https://github.com/facebookresearch/adjointsamplers