mlsanityData to 7 Oct 2026

Bayesian Flow Networks

arXiv 2308.07037 ↗First posted 14 Aug 2023cs.LG cs.AIcited by 62

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

Abstract (arXiv)

This paper introduces Bayesian Flow Networks (BFNs), a new class of generative model in which the parameters of a set of independent distributions are modified with Bayesian inference in the light of noisy data samples, then passed as input to a neural network that outputs a second, interdependent distribution. Starting from a simple prior and iteratively updating the two distributions yields a generative procedure similar to the reverse process of diffusion models; however it is conceptually simpler in that no forward process is required. Discrete and continuous-time loss functions are derived for continuous, discretised and discrete data, along with sample generation procedures. Notably, the network inputs for discrete data lie on the probability simplex, and are therefore natively differentiable, paving the way for gradient-based sample guidance and few-step generation in discrete domains such as language modelling. The loss function directly optimises data compression and places no restrictions on the network architecture. In our experiments BFNs achieve competitive log-likelihoods for image modelling on dynamically binarized MNIST and CIFAR-10, and outperform all known discrete diffusion models on the text8 character-level language modelling task.

Key facts. Cited by 62 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 37 of its 50 references that are in the map. First posted on arXiv 14 Aug 2023.

Builds on

37 of 50 references.

References · 37

  1. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  2. 2021High-Resolution Image Synthesis with Latent Diffusion Modelscited by 18,694
  3. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  4. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  5. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  6. 2021Improved Denoising Diffusion Probabilistic Modelscited by 3,736
  7. 2016Gaussian Error Linear Units (GELUs)cited by 3,431
  8. 2015Variational Inference with Normalizing Flowscited by 2,161
  9. 2021Structured Denoising Diffusion Models in Discrete State-Spacescited by 1,665
  10. 2019Generating Long Sequences with Sparse Transformerscited by 1,523
  11. 2021Variational Diffusion Modelscited by 1,145
  12. 2022Diffusion-LM Improves Controllable Text Generationcited by 1,006
25 more
  1. 2018Image Transformercited by 770
  2. 2020NVAE: A Deep Hierarchical Variational Autoencodercited by 635
  3. 2021Score-based Generative Modeling in Latent Spacecited by 634
  4. 2021Argmax Flows and Multinomial Diffusion: Learning Categorical Distributionscited by 566
  5. 2013Generating Sequences With Recurrent Neural Networkscited by 549
  6. 2017PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modificationscited by 415
  7. 2022Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioningcited by 362
  8. 2020Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Imagescited by 261
  9. 2021Autoregressive Diffusion Modelscited by 208
  10. 2022Continuous diffusion for categorical datacited by 202
  11. 2022Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Qualitycited by 181
  12. 2019Adaptive Attention Span in Transformerscited by 148
  13. 2022Self-conditioned Embedding Diffusion for Text Generationcited by 114
  14. 2021Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimationcited by 95
  15. 2019Practical Lossless Compression with Latent Variables using Bits Back Codingcited by 85
  16. 2023Reflected Diffusion Modelscited by 81
  17. 2018Generating High Fidelity Images with Subscale Pixel Networks and Multidimensional Upscalingcited by 74
  18. 2019Discrete Flows: Invertible Generative Models of Discrete Datacited by 68
  19. 2019Latent Normalizing Flows for Discrete Sequencescited by 66
  20. 2022Training and Inference on Any-Order Autoregressive Models the Right Waycited by 64
  21. 2023TESS: Text-to-Text Self-Conditioned Simplex Diffusioncited by 63
  22. 2021Consistency Regularization for Variational Auto-Encoderscited by 50
  23. 2022Categorical SDEs with Simplex Diffusioncited by 42
  24. 2020Locally Masked Convolution for Autoregressive Modelscited by 22
  25. 2019PixelVAE++: Improved PixelVAE with Discrete Priorcited by 15

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 5; 2024: 16; 2025: 27; 2026: 14.

62 papers.

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By area:

  1. 2025Large Language Diffusion Modelscited by 872
  2. 2023Discrete Diffusion Modeling by Estimating the Ratios of the Data Distributioncited by 712
  3. 2024Simplified and Generalized Masked Diffusion for Discrete Datacited by 606
  4. 2025Block Diffusion: Interpolating Between Autoregressive and Diffusion Language Modelscited by 385
  5. 2024Scaling up Masked Diffusion Models on Textcited by 195
  6. 2025Seed Diffusion: A Large-Scale Diffusion Language Model with High-Speed Inferencecited by 173
  7. 2024Simple Guidance Mechanisms for Discrete Diffusion Modelscited by 148
  8. 2025LLaDA-V: Large Language Diffusion Models with Visual Instruction Tuningcited by 145
  9. 2024Energy-Based Diffusion Language Models for Text Generationcited by 86
  10. 2025A Survey on Latent Reasoningcited by 71
  11. 2024MolCRAFT: Structure-Based Drug Design in Continuous Parameter Spacecited by 48
  12. 2024Categorical Flow Matching on Statistical Manifoldscited by 43
38 more
  1. 2023TokenCompose: Text-to-Image Diffusion with Token-level Supervisioncited by 40
  2. 2025Continuous Diffusion Model for Language Modelingcited by 29
  3. 2025Target Concrete Score Matching: A Holistic Framework for Discrete Diffusioncited by 25
  4. 2024UniGEM: A Unified Approach to Generation and Property Prediction for Moleculescited by 20
  5. 2024Arrows of Time for Large Language Modelscited by 18
  6. 2024Unified Generative Modeling of 3D Molecules via Bayesian Flow Networkscited by 18
  7. 2023Idempotent Generative Networkcited by 17
  8. 2025A Periodic Bayesian Flow for Material Generationcited by 15
  9. 2025Space Group Equivariant Crystal Diffusioncited by 15
  10. 2025UniMoMo: Unified Generative Modeling of 3D Molecules for De Novo Binder Designcited by 11
  11. 2025Symmetry-Aware Bayesian Flow Networks for Crystal Generationcited by 9
  12. 2025Straight-Line Diffusion Model for Efficient 3D Molecular Generationcited by 8
  13. 2025Flexible Language Modeling in Continuous Space with Transformer-based Autoregressive Flowscited by 8
  14. 2025RFG: Test-Time Scaling for Diffusion Large Language Model Reasoning with Reward-Free Guidancecited by 7
  15. 2024Empower Structure-Based Molecule Optimization with Gradient Guided Bayesian Flow Networkscited by 6
  16. 2025Piloting Structure-Based Drug Design via Modality-Specific Optimal Schedulecited by 6
  17. 2025Prior-Guided Flow Matching for Target-Aware Molecule Design with Learnable Atom Numbercited by 6
  18. 2026Membership Inference Attacks Against Fine-tuned Diffusion Language Modelscited by 5
  19. 2025Generative Modeling with Bayesian Sample Inferencecited by 4
  20. 2026Auto-Regressive Masked Diffusion Modelscited by 4
  21. 2025ShortListing Model: A Streamlined SimplexDiffusion for Discrete Variable Generationcited by 3
  22. 2025Generative Models for Crystalline Materialscited by 3
  23. 20263D Molecule Generation from Rigid Motifs via SE(3) Flowscited by 3
  24. 2026Generalized Discrete Diffusion from Snapshotscited by 3
  25. 2023Discrete Distribution Networkscited by 2
  26. 2024Weakly Augmented Variational Autoencoder in Time Series Anomaly Detectioncited by 2
  27. 2024ParamReL: Learning Parameter Space Representation via Progressively Encoding Bayesian Flow Networkscited by 2
  28. 2024Posterior Mean Matching: Generative Modeling through Online Bayesian Inferencecited by 2
  29. 2025Controllable 3D Molecular Generation for Structure-Based Drug Design Through Bayesian Flow Networks and Gradient Integrationcited by 2
  30. 2023Lecture Notes in Probabilistic Diffusion Modelscited by 1
  31. 2024Sifting through the Noise: A Survey of Diffusion Probabilistic Models and Their Applications to Biomoleculescited by 1
  32. 2025AMix-1: A Pathway to Test-Time Scalable Protein Foundation Modelcited by 1
  33. 2025MolPIF: A Parameter Interpolation Flow Model for Molecule Generationcited by 1
  34. 2025Transport-Coupled Bayesian Flows for Molecular Graph Generationcited by 1
  35. 2026Variational Bayesian Flow Network for Graph Generationcited by 1
  36. 2026A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Modelscited by 1
  37. 2026Efficient Long-Context Modeling in Diffusion Language Models via Block Approximate Sparse Attentioncited by 1
  38. 2024Marked Temporal Bayesian Flow Point Processescited by 0

50 most cited of 62 All 62 (JSON)

Similar, not cited

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  1. 2024Unifying Bayesian Flow Networks and Diffusion Models through Stochastic Differential Equations
  2. 2024Discrete Flow Matching
  3. 2025Designing a Conditional Prior Distribution for Flow-Based Generative Models
  4. 2025Generative Uncertainty in Diffusion Models
  5. 2022A Variational Perspective on Generative Flow Networks
  6. 2025Conditional Flow Matching for Bayesian Posterior Inference
  7. 2024Conditional sampling within generative diffusion models
  8. 2022Flow Matching for Generative Modeling