# Unifying Bayesian Flow Networks and Diffusion Models through Stochastic Differential Equations

arXiv: https://arxiv.org/abs/2404.15766 · HTML: https://mlsanity.com/paper/2404.15766/

## Key facts

- Cited by 28 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 58 references that are in the map.
- First posted on arXiv 24 Apr 2024.

## Placed in

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

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

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [ProlificDreamer: High-Fidelity and Diverse Text-to-3D Generation with Variational Score Distillation](https://mlsanity.com/paper/2305.16213/) (2023)
- [Imagic: Text-Based Real Image Editing with Diffusion Models](https://mlsanity.com/paper/2210.09276/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Argmax Flows and Multinomial Diffusion: Learning Categorical Distributions](https://mlsanity.com/paper/2102.05379/) (2021)
- [A Continuous Time Framework for Discrete Denoising Models](https://mlsanity.com/paper/2205.14987/) (2022)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Analog Bits: Generating Discrete Data using Diffusion Models with Self-Conditioning](https://mlsanity.com/paper/2208.04202/) (2022)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [Text2LIVE: Text-Driven Layered Image and Video Editing](https://mlsanity.com/paper/2204.02491/) (2022)

## Built on by (most cited first, 28 of 28)

- [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Your Absorbing Discrete Diffusion Secretly Models the Conditional Distributions of Clean Data](https://mlsanity.com/paper/2406.03736/) (2024)
- [Scaling up Masked Diffusion Models on Text](https://mlsanity.com/paper/2410.18514/) (2024)
- [LLaDA-V: Large Language Diffusion Models with Visual Instruction Tuning](https://mlsanity.com/paper/2505.16933/) (2025)
- [Efficient Diffusion Models: A Survey](https://mlsanity.com/paper/2502.06805/) (2025)
- [Identifying and Solving Conditional Image Leakage in Image-to-Video Diffusion Model](https://mlsanity.com/paper/2406.15735/) (2024)
- [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025)
- [Straight-Line Diffusion Model for Efficient 3D Molecular Generation](https://mlsanity.com/paper/2503.02918/) (2025)
- [Empower Structure-Based Molecule Optimization with Gradient Guided Bayesian Flow Networks](https://mlsanity.com/paper/2411.13280/) (2024)
- [Context Tokens are Anchors: Understanding the Repetition Curse in dMLLMs from an Information Flow Perspective](https://mlsanity.com/paper/2601.20520/) (2026)
- [Membership Inference Attacks Against Fine-tuned Diffusion Language Models](https://mlsanity.com/paper/2601.20125/) (2026)
- [Generative Modeling with Bayesian Sample Inference](https://mlsanity.com/paper/2502.07580/) (2025)
- [Manipulating 3D Molecules in a Fixed-Dimensional E(3)-Equivariant Latent Space](https://mlsanity.com/paper/2506.00771/) (2025)
- [Incomplete Data, Complete Dynamics: A Diffusion Approach](https://mlsanity.com/paper/2509.20098/) (2025)
- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)
- [ParamReL: Learning Parameter Space Representation via Progressively Encoding Bayesian Flow Networks](https://mlsanity.com/paper/2405.15268/) (2024)
- [Target-Guided Bayesian Flow Networks for Quantitatively Constrained CAD Generation](https://mlsanity.com/paper/2510.25163/) (2025)
- [Masked Diffusion Generative Recommendation](https://mlsanity.com/paper/2601.19501/) (2026)
- [MolPIF: A Parameter Interpolation Flow Model for Molecule Generation](https://mlsanity.com/paper/2507.13762/) (2025)
- [A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models](https://mlsanity.com/paper/2605.06829/) (2026)
- [Efficient Long-Context Modeling in Diffusion Language Models via Block Approximate Sparse Attention](https://mlsanity.com/paper/2605.19726/) (2026)
- [LLaDA MoE v2: Scaling Mixture-of-Experts Diffusion Language Models](https://mlsanity.com/paper/2608.03457/) (2026)
- [Sampling Out-of-Distribution Chemical Spaces via Bayesian Flow](https://mlsanity.com/paper/2412.11439/) (2024)
- [IgCraft: A versatile sequence generation framework for antibody discovery and engineering](https://mlsanity.com/paper/2503.19821/) (2025)
- [Discrete Bayesian Sample Inference for Graph Generation](https://mlsanity.com/paper/2511.03015/) (2025)
- [Observation-Aligned Mask Priors for Learning Physical Fields from Authentic Occlusions](https://mlsanity.com/paper/2605.16818/) (2026)
- [Bayesian Flow Networks for Offline Trajectory Planning](https://mlsanity.com/paper/2608.25163/) (2026)

## Abstract (arXiv)

Bayesian flow networks (BFNs) iteratively refine the parameters, instead of the samples in diffusion models (DMs), of distributions at various noise levels through Bayesian inference. Owing to its differentiable nature, BFNs are promising in modeling both continuous and discrete data, while simultaneously maintaining fast sampling capabilities. This paper aims to understand and enhance BFNs by connecting them with DMs through stochastic differential equations (SDEs). We identify the linear SDEs corresponding to the noise-addition processes in BFNs, demonstrate that BFN's regression losses are aligned with denoise score matching, and validate the sampler in BFN as a first-order solver for the respective reverse-time SDE. Based on these findings and existing recipes of fast sampling in DMs, we propose specialized solvers for BFNs that markedly surpass the original BFN sampler in terms of sample quality with a limited number of function evaluations (e.g., 10) on both image and text datasets. Notably, our best sampler achieves an increase in speed of 5~20 times for free. Our code is available at https://github.com/ML-GSAI/BFN-Solver.