mlsanityData to 7 Oct 2026

Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching

arXiv 2504.11713 ↗First posted 16 Apr 2025cs.LG cs.AIcited by 78

On the map: Diffusion models → Flow matching, time series and science

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.

Key facts. Cited by 78 papers in the map (data through 7 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.

Builds on

58 of 85 references.

Closest ancestors · 3

Same direction, newest first

  1. 2023Equivariant flow matching26 Jun
  2. 2022Flow Matching for Generative Modeling6 Oct
  3. 2022GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation6 Mar

Other references · 55

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  3. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  4. 2018Neural Ordinary Differential Equationscited by 3,733
  5. 2022Elucidating the Design Space of Diffusion-Based Generative Modelscited by 2,994
  6. 2015Variational Inference with Normalizing Flowscited by 2,161
  7. 2021E(n) Equivariant Graph Neural Networkscited by 951
  8. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  9. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  10. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  11. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
  12. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
38 more
  1. 2021GFlowNet Foundationscited by 286
  2. 2022Torsional Diffusion for Molecular Conformer Generationcited by 262
  3. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 248
  4. 2022Trajectory balance: Improved credit assignment in GFlowNetscited by 244
  5. 2020GEOM: Energy-annotated molecular conformations for property prediction and molecular generationcited by 238
  6. 2024Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Controlcited by 222
  7. 2022Riemannian Score-Based Generative Modellingcited by 216
  8. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 168
  9. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 168
  10. 2020Stochastic Normalizing Flowscited by 156
  11. 2023Diffusion Schr"odinger Bridge Matchingcited by 151
  12. 2019Flow-based generative models for Markov chain Monte Carlo in lattice field theorycited by 147
  13. 2022An optimal control perspective on diffusion-based generative modelingcited by 145
  14. 2023Denoising Diffusion Samplerscited by 144
  15. 2022Flow Annealed Importance Sampling Bootstrapcited by 124
  16. 2022Riemannian Diffusion Modelscited by 120
  17. 2024Iterated Denoising Energy Matching for Sampling from Boltzmann Densitiescited by 113
  18. 2021GeoMol: Torsional Geometric Generation of Molecular 3D Conformer Ensemblescited by 106
  19. 2023Improved sampling via learned diffusionscited by 103
  20. 2014On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpointcited by 99
  21. 2023A theory of continuous generative flow networkscited by 98
  22. 2019Theoretical guarantees for sampling and inference in generative models with latent diffusionscited by 97
  23. 2021Annealed Flow Transport Monte Carlocited by 90
  24. 2022SPICE, A Dataset of Drug-like Molecules and Peptides for Training Machine Learning Potentialscited by 80
  25. 2023Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modelingcited by 80
  26. 2025Learning Smooth and Expressive Interatomic Potentials for Physical Property Predictioncited by 79
  27. 2023Non-Denoising Forward-Time Diffusionscited by 75
  28. 2024NETS: A Non-Equilibrium Transport Samplercited by 67
  29. 2022Continual Repeated Annealed Flow Transport Monte Carlocited by 66
  30. 2024Particle Denoising Diffusion Samplercited by 63
  31. 2023Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamicscited by 62
  32. 2024Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processescited by 61
  33. 2023Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimizationcited by 57
  34. 2020Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Propertiescited by 51
  35. 2024Sequential Controlled Langevin Diffusionscited by 47
  36. 2024Target Score Matchingcited by 46
  37. 2024ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generationcited by 41
  38. 2023Generalized Schr"odinger Bridge Matchingcited by 40

50 most cited of 55 All 55 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 29; 2026: 49.

78 papers.

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  1. 2026Q-learning with Adjoint Matchingcited by 33
  2. 2025Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densitiescited by 26
  3. 2025Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inferencecited by 22
  4. 2025Adjoint Schr"odinger Bridge Samplercited by 20
  5. 2025MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Controlcited by 20
  6. 2025Tilt Matching for Scalable Sampling and Fine-Tuningcited by 17
  7. 2026Reinforce Adjoint Matching: Scaling RL Post-Training of Diffusion and Flow-Matching Modelscited by 17
  8. 2025Proximal Diffusion Neural Samplercited by 16
  9. 2025Value Gradient Guidance for Flow Matching Alignmentcited by 14
  10. 2026Flow Policy Gradients for Robot Controlcited by 14
  11. 2025Progressive Tempering Sampler with Diffusioncited by 13
  12. 2025FALCON: Few-step Accurate Likelihoods for Continuous Flowscited by 12
38 more
  1. 2025Amortized Sampling with Transferable Normalizing Flowscited by 11
  2. 2025Scalable Boltzmann Generators for equilibrium sampling of large-scale materialscited by 11
  3. 2026Enhanced Diffusion Sampling: Efficient Rare Event Sampling and Free Energy Calculation with Diffusion Modelscited by 11
  4. 2025On scalable and efficient training of diffusion samplerscited by 10
  5. 2025HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimizationcited by 10
  6. 2026Discrete Adjoint Matchingcited by 10
  7. 2025Non-equilibrium Annealed Adjoint Samplercited by 8
  8. 2025Learning Boltzmann Generators via Constrained Mass Transportcited by 8
  9. 2025Sublinear iterations can suffice even for DDPMscited by 7
  10. 2026Bridge Matching Sampler: Scalable Sampling via Generalized Fixed-Point Diffusion Matchingcited by 7
  11. 2025Adaptive Destruction Processes for Diffusion Samplerscited by 6
  12. 2025Energy-Weighted Flow Matching: Unlocking Continuous Normalizing Flows for Efficient and Scalable Boltzmann Samplingcited by 6
  13. 2025Control Consistency Losses for Diffusion Bridgescited by 6
  14. 2025Reinforced sequential Monte Carlo for amortised samplingcited by 5
  15. 2025Posterior Inference in Latent Space for Scalable Constrained Black-box Optimizationcited by 4
  16. 2025Functional Adjoint Sampler: Scalable Sampling on Infinite Dimensional Spacescited by 4
  17. 2025Control Variate Score Matching for Diffusion Modelscited by 4
  18. 2026Discrete Adjoint Schr"odinger Bridge Samplercited by 4
  19. 2026Efficient Generative Modeling beyond Memoryless Diffusion via Adjoint Schr"odinger Bridge Matchingcited by 4
  20. 2026Adjoint Matching through the Lens of the Stochastic Maximum Principle in Optimal Controlcited by 4
  21. 2026Flow Sampling: Learning to Sample from Unnormalized Densities via Denoising Conditional Processescited by 4
  22. 2026Scalable Maximum Entropy Reinforcement Learning for Diffusion Policies via Adjoint Matchingcited by 4
  23. 2025Enhanced Sampling for Efficient Learning of Coarse-Grained Machine Learning Potentialscited by 3
  24. 2026Elign: Equivariant Diffusion Model Alignment from Foundational Machine Learning Force Fieldscited by 3
  25. 2026Improved techniques for fine-tuning flow models via adjoint matching: a deterministic control pipelinecited by 3
  26. 2026Contrastive Distribution Matching for Amortized Sequential Monte Carlo in Discrete Diffusioncited by 3
  27. 2025Sampling from Energy distributions with Target Concrete Score Identitycited by 2
  28. 2025FlexiFlow: decomposable flow matching for generation of flexible molecular ensemblecited by 2
  29. 2026Improved Constrained Generation by Bridging Pretrained Generative Modelscited by 2
  30. 2026Efficient Adjoint Matching for Fine-tuning Diffusion Modelscited by 2
  31. 2026Simple Approximation and Derivative Free Inference-Time Scaling for Diffusion Models via Sequential Monte Carlo on Path Measurescited by 2
  32. 2026SURGE: Approximation and Training Free Particle Filter for Diffusion Surrogatecited by 2
  33. 2025AI-based Methods for Simulating, Sampling, and Predicting Protein Ensemblescited by 1
  34. 2025Learning Paths for Dynamic Measure Transport: A Control Perspectivecited by 1
  35. 2026Boltzmann Generators for Condensed Matter via Riemannian Flow Matchingcited by 1
  36. 2026Counterdiabatic Hamiltonian Monte Carlocited by 1
  37. 2026A unified perspective on fine-tuning and sampling with diffusion and flow modelscited by 1
  38. 2026Conditional Diffusion Samplingcited by 1

50 most cited of 78 All 78 (JSON)

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  1. 2024AdjointDEIS: Efficient Gradients for Diffusion Models
  2. 2025Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models
  3. 2025Diffusion Models are Molecular Dynamics Simulators
  4. 2024Towards Fast Stochastic Sampling in Diffusion Generative Models
  5. 2025Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics
  6. 2023Practical and Asymptotically Exact Conditional Sampling in Diffusion Models
  7. 2024Align Your Steps: Optimizing Sampling Schedules in Diffusion Models
  8. 2023Efficient Integrators for Diffusion Generative Models