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Unifying Bayesian Flow Networks and Diffusion Models through Stochastic Differential Equations
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Connects Bayesian flow networks to diffusion models via linear SDEs for continuous and discrete data, shows BFN losses match denoising score matching, and proposes BFN-Solvers that give large speedups over the original BFN sampler.
Problem
The relationship between Bayesian flow networks (BFNs) and diffusion models (DMs) is unclear, and BFN sampling is slow relative to what fast DM solvers can achieve.
Approach
Shows that the BFN noise-adding process on continuous data and the latent variables for discrete data follow linear SDEs, that BFN regression losses equal denoising score matching, and that the BFN sampler is a first-order discretization of the reverse-time SDE; this lets DM fast solvers (DPM-Solver style, ODE and SDE) be applied to BFNs as BFN-Solvers.
Key results
- On CIFAR-10, BFN-Solver++2 and SDE-BFN-Solver++2 substantially outperform the original BFN sampler at NFE below 100 in FID.
- On text8 at 20 NFE, BFN-Solver2 reaches spelling accuracy 0.84 versus 0.79 for the original BFN sampler at 20 NFE.
- The best sampler gives a 5-20x speedup over the original BFN sampler for the same pre-trained model.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use BFN-Solver2 instead of the original BFN sampler | pre-trained BFN · text8, 20 NFE | spelling accuracy (SA): 0.79 -> 0.84 | Helped |
| Use BFN-Solver2 instead of the original BFN sampler | pre-trained BFN · text8, 50 NFE | spelling accuracy (SA): 0.84 -> 0.85 | Helped |
| Remove the categorical sampling step (ablation) | original BFN sampler · text8, fewer than 50 NFE | spelling accuracy (SA) | Helped |
| Remove the categorical sampling step (ablation) | original BFN sampler · text8, 50 NFE or more | spelling accuracy (SA) | No clear effect |
Surprising findings
- Removing the categorical sampling step from the BFN discrete sampler gives consistently better performance with fewer than 50 NFE.
Limitations
- Experiments use small-scale pre-trained BFN models on CIFAR-10 and text8.
- The proposed samplers cannot be directly used for exact likelihood evaluation.
- FID and spelling accuracy are only surrogate sample-quality metrics and may be biased.
Experimental setup
Datasets: CIFAR-10 text8
Benchmarks: CIFAR-10 text8
Metrics: FID spelling accuracy bits-per-character user preference (two-alternative forced choice)
Baselines: BFN original sampler D3PM OA-ARDM
Builds on: Bayesian flow networks DPM-Solver score-based SDEs D3PM
Closest papers by meaning
- A Unified Sampling Framework for Solver Searching of Diffusion Probabilistic Models (ICLR 2024 · similarity 0.87)
- Elucidating Flow Matching ODE Dynamics via Data Geometry and Denoisers (ICML 2025 · similarity 0.87)
- Divide-and-Conquer Posterior Sampling for Denoising Diffusion priors (NeurIPS 2024 · similarity 0.87)
- Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries (NeurIPS 2026 · similarity 0.87)
- Inflationary Flows: Calibrated Bayesian Inference with Diffusion-Based Models (NeurIPS 2024 · similarity 0.87)
- Provable Posterior Sampling with Denoising Oracles via Tilted Transport (NeurIPS 2024 · similarity 0.87)
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