# Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching

arXiv: https://arxiv.org/abs/2504.11713 · HTML: https://mlsanity.com/paper/2504.11713/

## Key facts

- Cited by 85 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 58 of its 85 references that are in the map.
- First posted on arXiv 16 Apr 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, 30 of 58)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [GFlowNet Foundations](https://mlsanity.com/paper/2111.09266/) (2021)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Trajectory balance: Improved credit assignment in GFlowNets](https://mlsanity.com/paper/2201.13259/) (2022)
- [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)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [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)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)

## Built on by (most cited first, 30 of 85)

- [Q-learning with Adjoint Matching](https://mlsanity.com/paper/2601.14234/) (2026)
- [Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2506.16471/) (2025)
- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [Tilt Matching for Scalable Sampling and Fine-Tuning](https://mlsanity.com/paper/2512.21829/) (2025)
- [Adjoint Schr"odinger Bridge Sampler](https://mlsanity.com/paper/2506.22565/) (2025)
- [MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Control](https://mlsanity.com/paper/2508.10684/) (2025)
- [Reinforce Adjoint Matching: Scaling RL Post-Training of Diffusion and Flow-Matching Models](https://mlsanity.com/paper/2605.10759/) (2026)
- [Proximal Diffusion Neural Sampler](https://mlsanity.com/paper/2510.03824/) (2025)
- [Flow Policy Gradients for Robot Control](https://mlsanity.com/paper/2602.02481/) (2026)
- [Value Gradient Guidance for Flow Matching Alignment](https://mlsanity.com/paper/2512.05116/) (2025)
- [Progressive Tempering Sampler with Diffusion](https://mlsanity.com/paper/2506.05231/) (2025)
- [FALCON: Few-step Accurate Likelihoods for Continuous Flows](https://mlsanity.com/paper/2512.09914/) (2025)
- [Enhanced Diffusion Sampling: Efficient Rare Event Sampling and Free Energy Calculation with Diffusion Models](https://mlsanity.com/paper/2602.16634/) (2026)
- [Amortized Sampling with Transferable Normalizing Flows](https://mlsanity.com/paper/2508.18175/) (2025)
- [Scalable Boltzmann Generators for equilibrium sampling of large-scale materials](https://mlsanity.com/paper/2509.25486/) (2025)
- [HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization](https://mlsanity.com/paper/2511.08425/) (2025)
- [Discrete Adjoint Matching](https://mlsanity.com/paper/2602.07132/) (2026)
- [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)
- [Learning Boltzmann Generators via Constrained Mass Transport](https://mlsanity.com/paper/2510.18460/) (2025)
- [Non-equilibrium Annealed Adjoint Sampler](https://mlsanity.com/paper/2506.18165/) (2025)
- [Energy-Weighted Flow Matching: Unlocking Continuous Normalizing Flows for Efficient and Scalable Boltzmann Sampling](https://mlsanity.com/paper/2509.03726/) (2025)
- [Sublinear iterations can suffice even for DDPMs](https://mlsanity.com/paper/2511.04844/) (2025)
- [Adaptive Destruction Processes for Diffusion Samplers](https://mlsanity.com/paper/2506.01541/) (2025)
- [Reinforced sequential Monte Carlo for amortised sampling](https://mlsanity.com/paper/2510.11711/) (2025)
- [Control Consistency Losses for Diffusion Bridges](https://mlsanity.com/paper/2512.05070/) (2025)
- [Control Variate Score Matching for Diffusion Models](https://mlsanity.com/paper/2512.20003/) (2025)
- [Adjoint Matching through the Lens of the Stochastic Maximum Principle in Optimal Control](https://mlsanity.com/paper/2604.08580/) (2026)
- [Posterior Inference in Latent Space for Scalable Constrained Black-box Optimization](https://mlsanity.com/paper/2507.00480/) (2025)
- [Flow Sampling: Learning to Sample from Unnormalized Densities via Denoising Conditional Processes](https://mlsanity.com/paper/2605.03984/) (2026)

## Abstract (arXiv)

We introduce Adjoint Sampling, a highly scalable and efficient algorithm for learning diffusion processes that sample from unnormalized densities, or energy functions. It is the first on-policy approach that allows significantly more gradient updates than the number of energy evaluations and model samples, allowing us to scale to much larger problem settings than previously explored by similar methods. Our framework is theoretically grounded in stochastic optimal control and shares the same theoretical guarantees as Adjoint Matching, being able to train without the need for corrective measures that push samples towards the target distribution. We show how to incorporate key symmetries, as well as periodic boundary conditions, for modeling molecules in both cartesian and torsional coordinates. We demonstrate the effectiveness of our approach through extensive experiments on classical energy functions, and further scale up to neural network-based energy models where we perform amortized conformer generation across many molecular systems. To encourage further research in developing highly scalable sampling methods, we plan to open source these challenging benchmarks, where successful methods can directly impact progress in computational chemistry.