mlsanityData to 7 Oct 2026

Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling

arXiv 2404.12940 ↗First posted 19 Apr 2024stat.ML cs.CV cs.LGcited by 43

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Conventional diffusion models typically relies on a fixed forward process, which implicitly defines complex marginal distributions over latent variables. This can often complicate the reverse process' task in learning generative trajectories, and results in costly inference for diffusion models. To address these limitations, we introduce Neural Flow Diffusion Models (NFDM), a novel framework that enhances diffusion models by supporting a broader range of forward processes beyond the standard Gaussian. We also propose a novel parameterization technique for learning the forward process. Our framework provides an end-to-end, simulation-free optimization objective, effectively minimizing a variational upper bound on the negative log-likelihood. Experimental results demonstrate NFDM's strong performance, evidenced by state-of-the-art likelihood estimation. Furthermore, we investigate NFDM's capacity for learning generative dynamics with specific characteristics, such as deterministic straight lines trajectories, and demonstrate how the framework may be adopted for learning bridges between two distributions. The results underscores NFDM's versatility and its potential for a wide range of applications.

Key facts. Cited by 43 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 65 of its 84 references that are in the map. First posted on arXiv 19 Apr 2024.

Builds on

65 of 84 references.

Closest ancestors · 35

Same direction, newest first

  1. 2023Diffusion Models With Learned Adaptive Noise20 Dec
  2. 2023One-step Diffusion with Distribution Matching Distillation30 Nov
  3. 2023DiffEnc: Variational Diffusion with a Learned Encoder30 Oct
  4. 2023Neural Diffusion Models12 Oct
  5. 2023Improved Techniques for Maximum Likelihood Estimation for Diffusion ODEs6 May
  6. 2023Diffusion Schr"odinger Bridge Matching29 Mar
  7. 2023Consistency Models2 Mar
  8. 2023Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation1 Mar
  9. 2023Where to Diffuse, How to Diffuse, and How to Get Back: Automated Learning for Multivariate Diffusions14 Feb
  10. 2023Star-Shaped Denoising Diffusion Probabilistic Models10 Feb
  11. 2023ShiftDDPMs: Exploring Conditional Diffusion Models by Shifting Diffusion Trajectories5 Feb
  12. 2023Minimizing Trajectory Curvature of ODE-based Generative Models27 Jan
23 more
  1. 2022Wavelet Diffusion Models are fast and scalable Image Generators29 Nov
  2. 2022Fast Sampling of Diffusion Models via Operator Learning24 Nov
  3. 2022f-DM: A Multi-stage Diffusion Model via Progressive Signal Transformation10 Oct
  4. 2022Blurring Diffusion Models12 Sep
  5. 2022Soft Diffusion: Score Matching for General Corruptions12 Sep
  6. 2022Diffusion Models: A Comprehensive Survey of Methods and Applications2 Sep
  7. 2022A Flexible Diffusion Model17 Jun
  8. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps2 Jun
  9. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  10. 2022Maximum Likelihood Training of Implicit Nonlinear Diffusion Models27 May
  11. 2022Pseudo Numerical Methods for Diffusion Models on Manifolds20 Feb
  12. 2021High-Resolution Image Synthesis with Latent Diffusion Models20 Dec
  13. 2021Diffusion Normalizing Flow14 Oct
  14. 2021Variational Diffusion Models1 Jul
  15. 2021Score-based Generative Modeling in Latent Space10 Jun
  16. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  17. 2021Cascaded Diffusion Models for High Fidelity Image Generation30 May
  18. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  19. 2021Improved Denoising Diffusion Probabilistic Models18 Feb
  20. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  21. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  22. 2020Denoising Diffusion Implicit Models6 Oct
  23. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 30

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  3. 2022Flow Matching for Generative Modelingcited by 5,426
  4. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  5. 2018Neural Ordinary Differential Equationscited by 3,733
  6. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  7. 2016Density estimation using Real NVPcited by 2,029
  8. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  9. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  10. 2019StarGAN v2: Diverse Image Synthesis for Multiple Domainscited by 1,095
  11. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  12. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
18 more
  1. 2023Improving and generalizing flow-based generative models with minibatch optimal transportcited by 835
  2. 2016Pixel Recurrent Neural Networkscited by 814
  3. 2017Generative Adversarial Networks: An Overviewcited by 588
  4. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  5. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 277
  6. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 267
  7. 2022Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problemcited by 229
  8. 2019Input complexity and out-of-distribution detection with likelihood-based generative modelscited by 186
  9. 2022Lossy Image Compression with Conditional Diffusion Modelscited by 165
  10. 2022Generative Modelling With Inverse Heat Dissipationcited by 164
  11. 2022NaturalSpeech: End-to-End Text to Speech Synthesis with Human-Level Qualitycited by 151
  12. 2020Likelihood Regret: An Out-of-Distribution Detection Score For Variational Auto-encodercited by 145
  13. 2020Learning Differential Equations that are Easy to Solvecited by 81
  14. 2023On Kinetic Optimal Probability Paths for Generative Modelscited by 42
  15. 2023Generalized Schr"odinger Bridge Matchingcited by 40
  16. 2023Bespoke Solvers for Generative Flow Modelscited by 36
  17. 2021ANFIC: Image Compression Using Augmented Normalizing Flowscited by 20
  18. 2020Anomaly Detection in Trajectory Data with Normalizing Flowscited by 11

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 6; 2025: 22; 2026: 15.

43 papers.

Sort:

By area:

  1. 2024Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Controlcited by 222
  2. 2024Normalizing Flows are Capable Generative Modelscited by 89
  3. 2024Metric Flow Matching for Smooth Interpolations on the Data Manifoldcited by 74
  4. 2025Fractal Generative Modelscited by 44
  5. 2025Noise Hypernetworks: Amortizing Test-Time Compute in Diffusion Modelscited by 35
  6. 2025No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplerscited by 25
  7. 2024Diffusion Models as Cartoonists: The Curious Case of High Density Regionscited by 19
  8. 2025SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equationscited by 19
  9. 2025Learning normalized image densities via dual score matchingcited by 17
  10. 2025FARMER: Flow AutoRegressive Transformer over Pixelscited by 17
  11. 2025Curly Flow Matching for Learning Non-gradient Field Dynamicscited by 17
  12. 2025VCT: Training Consistency Models with Variational Noise Couplingcited by 15
31 more
  1. 2025Equivariant Neural Diffusion for Molecule Generationcited by 13
  2. 2025Straight-Line Diffusion Model for Efficient 3D Molecular Generationcited by 8
  3. 2024What's the score? Automated Denoising Score Matching for Nonlinear Diffusionscited by 7
  4. 2025Adaptive Destruction Processes for Diffusion Samplerscited by 6
  5. 2026Faster Inference of Flow-Based Generative Models via Improved Data-Noise Couplingcited by 6
  6. 2025Training Latent Diffusion Models with Interacting Particle Algorithmscited by 4
  7. 2026Cascaded Flow Matching for Heterogeneous Tabular Data with Mixed-Type Featurescited by 4
  8. 2025CoVAE: Consistency Training of Variational Autoencoderscited by 3
  9. 2025Fractional Diffusion Bridge Modelscited by 3
  10. 2026FlowCast: Trajectory Forecasting for Scalable Zero-Cost Speculative Flow Matchingcited by 3
  11. 2026FastFlow: Accelerating The Generative Flow Matching Models with Bandit Inferencecited by 3
  12. 2026Forward-Learned Discrete Diffusion: Learning how to noise to denoise fastercited by 3
  13. 2025Flow Diverse and Efficient: Learning Momentum Flow Matching via Stochastic Velocity Field Samplingcited by 2
  14. 2026GeodesicNVS: Probability Density Geodesic Flow Matching for Novel View Synthesiscited by 2
  15. 2026Learning Energy-Based Models from Stochastic Interpolants using Spatiotemporal Differencescited by 2
  16. 2025Learning Straight Flows by Learning Curved Interpolantscited by 1
  17. 2025On Designing Diffusion Autoencoders for Efficient Generation and Representation Learningcited by 1
  18. 2025HOTA: Hamiltonian framework for Optimal Transport Advectioncited by 1
  19. 2025Restora-Flow: Mask-Guided Image Restoration with Flow Matchingcited by 1
  20. 2026Towards Latent Diffusion Suitable For Textcited by 1
  21. 2026Insertion Based Sequence Generation with Learnable Order Dynamicscited by 1
  22. 2026VQ-SAD: Vector Quantized Structure Aware Diffusion For Molecule Generationcited by 1
  23. 2026Flow Map Learning via Nongradient Vector Flowcited by 1
  24. 2024Multidimensional Adaptive Coefficient for Inference Trajectory Optimization in Flow and Diffusioncited by 0
  25. 2025Fractal Flow: Hierarchical and Interpretable Normalizing Flow via Topic Modeling and Recursive Strategycited by 0
  26. 2025Adapting Noise to Data: Generative Flows from 1D Processescited by 0
  27. 2025Breaking the Likelihood-Quality Trade-off in Diffusion Models by Merging Pretrained Expertscited by 0
  28. 2026Learning Normalized Energy Models for Linear Inverse Problemscited by 0
  29. 2026Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspectivecited by 0
  30. 2026Simulation-free and finite-time diffusion modelcited by 0
  31. 2026Retrieval-Augmented Diffusion Modeling for Stochastic Discount Factor Portfolioscited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2025Diffusion Models: A Mathematical Introduction
  2. 2022Understanding Diffusion Models: A Unified Perspective
  3. 2025Diff2Flow: Training Flow Matching Models via Diffusion Model Alignment
  4. 2022Let us Build Bridges: Understanding and Extending Diffusion Generative Models
  5. 2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization
  6. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  7. 2023Improving Efficiency of Diffusion Models via Multi-Stage Framework and Tailored Multi-Decoder Architectures
  8. 2025From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity