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Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling
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Proposes Neural Flow Diffusion Models (NFDM), which learn the forward process of a diffusion model via a learnable invertible transformation, trained with a simulation-free variational bound, and extends it to bridges (NFBM) and straight-trajectory dynamics (NFDM-OT).
Problem
Standard diffusion models use a fixed, typically linear Gaussian forward process, which yields complex marginals that complicate learning the reverse process and make sampling costly.
Approach
Make the forward process learnable by defining marginals implicitly through an invertible transformation F_phi(eps, t, x) of Gaussian noise, jointly optimize forward and reverse processes with a simulation-free variational bound on NLL, and add curvature penalties to learn straight deterministic trajectories.
Key results
- NFDM achieves 2.48 BPD on CIFAR-10 (non-Gaussian forward process), beating the best baseline i-DODE at 2.56 and MuLAN at 2.55.
- NFDM-OT reaches FID 4.11 on ImageNet 32 with 12 NFE, and 5.20 on CIFAR-10 with 12 NFE, outperforming trajectory-curvature baselines at comparable NFE.
- NFBM reaches FID 12.06 on AFHQ dog-to-cat 64 at 1000 steps, versus 12.39 for GSBM and 14.16 for DSBM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a learnable forward process (NFDM) instead of the strongest listed baseline | Diffusion modeling · CIFAR-10 test | BPD: 2.55 -> 2.48 | Helped |
| Use a learnable forward process (NFDM) instead of the strongest listed baseline | Diffusion modeling · ImageNet 32 test | BPD: 3.43 -> 3.34 | Helped |
| Use a learnable forward process (NFDM) instead of the strongest listed baseline | Diffusion modeling · ImageNet 64 test | BPD: 3.35 -> 3.20 | Helped |
| Replace Gaussian with non-Gaussian forward marginals (ablation) | NFDM · CIFAR-10 test | BPD: 2.49 -> 2.48 | Helped |
| Replace Gaussian with non-Gaussian forward marginals (ablation) | NFDM · ImageNet 32 test | BPD: 3.36 -> 3.34 | Helped |
| Replace Gaussian with non-Gaussian forward marginals (ablation) | NFDM · ImageNet 64 test | BPD: 3.20 -> 3.20 | No clear effect |
| Add a curvature penalty to learn straighter generative trajectories (ablation) | NFDM-OT · CIFAR-10 | FID | Helped |
| Use NFDM-OT rather than Multisample Flow Matching at NFE 12 | Few-step generation · ImageNet 32, NFE 12 | FID: 7.18 -> 4.11 | Helped |
| Use NFDM-OT rather than Multisample Flow Matching at NFE 12 | Few-step generation · ImageNet 64, NFE 12 | FID: 17.6 -> 11.58 | Helped |
| Use NFBM rather than GSBM for dog-to-cat translation | Image bridge modeling · AFHQ 64 dog → cat | FID: 12.39 -> 12.06 | Helped |
| Parameterize the forward process with a neural network rather than a conventional diffusion forward process | NFDM · Training optimization iteration | iteration duration: 2.2 times | Hurt |
Limitations
- Optimization iterations about 2.2x slower than conventional diffusion models.
- Forward process parameterizations must have tractable log-determinants, restricting the design space.
- NFDM-OT straightness is an example rather than the best method for straight-line dynamics; higher curvature penalty slows convergence.
Experimental setup
Datasets: CIFAR-10 ImageNet 32 ImageNet 64 AFHQ
Benchmarks: CIFAR-10 ImageNet 32 ImageNet 64 AFHQ 64
Metrics: bits per dimension FID NFE
Baselines: DDPM Score SDE Improved DDPM VDM Score Flow Flow Matching Stochastic Interpolants i-DODE NDM MuLAN DDIM DPM Solver
Builds on: score-based SDE DDPM Improved DDPM variational diffusion models continuous normalizing flows
Closest papers by meaning
- Neural Diffusion Models (ICML 2024 · similarity 0.93)
- Diff2Flow: Training Flow Matching Models via Diffusion Model Alignment (CVPR 2025 · similarity 0.90)
- Revisiting Diffusion Models: From Generative Pre-training to One-Step Generation (ICML 2025 · similarity 0.89)
- Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization (ICLR 2024 · similarity 0.89)
- A solvable model of learning generative diffusion: theory and insights (NeurIPS 2025 · similarity 0.89)
- When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Nets (ICML 2025 · similarity 0.89)
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