mlsanityData to 7 Oct 2026

Minimizing Trajectory Curvature of ODE-based Generative Models

arXiv 2301.12003 ↗First posted 27 Jan 2023cs.LG cs.AI cs.CVcited by 87

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Recent ODE/SDE-based generative models, such as diffusion models, rectified flows, and flow matching, define a generative process as a time reversal of a fixed forward process. Even though these models show impressive performance on large-scale datasets, numerical simulation requires multiple evaluations of a neural network, leading to a slow sampling speed. We attribute the reason to the high curvature of the learned generative trajectories, as it is directly related to the truncation error of a numerical solver. Based on the relationship between the forward process and the curvature, here we present an efficient method of training the forward process to minimize the curvature of generative trajectories without any ODE/SDE simulation. Experiments show that our method achieves a lower curvature than previous models and, therefore, decreased sampling costs while maintaining competitive performance. Code is available at https://github.com/sangyun884/fast-ode.

Key facts. Cited by 87 papers in the map (data through 7 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 40 of its 52 references that are in the map. First posted on arXiv 27 Jan 2023.

Builds on

40 of 52 references.

Closest ancestors · 17

Same direction, newest first

  1. 2022Fast Sampling of Diffusion Models via Operator Learning24 Nov
  2. 2022f-DM: A Multi-stage Diffusion Model via Progressive Signal Transformation10 Oct
  3. 2022Blurring Diffusion Models12 Sep
  4. 2022Soft Diffusion: Score Matching for General Corruptions12 Sep
  5. 2022Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise19 Aug
  6. 2022Progressive Deblurring of Diffusion Models for Coarse-to-Fine Image Synthesis16 Jul
  7. 2022DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps2 Jun
  8. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  9. 2022Fast Sampling of Diffusion Models with Exponential Integrator29 Apr
  10. 2021Tackling the Generative Learning Trilemma with Denoising Diffusion GANs15 Dec
  11. 2021Diffusion Normalizing Flow14 Oct
  12. 2021Variational Diffusion Models1 Jul
5 more
  1. 2021Score-based Generative Modeling in Latent Space10 Jun
  2. 2021Diffusion Models Beat GANs on Image Synthesis11 May
  3. 2021Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed7 Jan
  4. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  5. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 23

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  2. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,539
  3. 2022Photorealistic Text-to-Image Diffusion Models with Deep Language Understandingcited by 6,322
  4. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 5,986
  5. 2022Generative Adversarial Networkscited by 5,603
  6. 2022Flow Matching for Generative Modelingcited by 5,426
  7. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
  8. 2013Auto-Encoding Variational Bayescited by 3,980
  9. 2018Neural Ordinary Differential Equationscited by 3,733
  10. 2019Analyzing and Improving the Image Quality of StyleGANcited by 3,580
  11. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,550
  12. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,609
11 more
  1. 2015Variational Inference with Normalizing Flowscited by 2,161
  2. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  3. 2020Training Generative Adversarial Networks with Limited Datacited by 1,184
  4. 2022Building Normalizing Flows with Stochastic Interpolantscited by 964
  5. 2020NVAE: A Deep Hierarchical Variational Autoencodercited by 635
  6. 2020Differentiable Augmentation for Data-Efficient GAN Trainingcited by 361
  7. 2022Rectified Flow: A Marginal Preserving Approach to Optimal Transportcited by 277
  8. 2020Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Imagescited by 261
  9. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  10. 2022Generative Modelling With Inverse Heat Dissipationcited by 164
  11. 2020Learning Differential Equations that are Easy to Solvecited by 81

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 15; 2024: 29; 2025: 24; 2026: 19.

87 papers.

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  1. 2024Scaling Rectified Flow Transformers for High-Resolution Image Synthesiscited by 4,094
  2. 2024SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformerscited by 707
  3. 2023Multisample Flow Matching: Straightening Flows with Minibatch Couplingscited by 267
  4. 2023Flow Matching in Latent Spacecited by 155
  5. 2024CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Modelscited by 138
  6. 2024Improving the Training of Rectified Flowscited by 106
  7. 2024Consistency Flow Matching: Defining Straight Flows with Velocity Consistencycited by 101
  8. 2024PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Acceleratorcited by 95
  9. 2023Stochastic interpolants with data-dependent couplingscited by 94
  10. 2023Global Structure-Aware Diffusion Process for Low-Light Image Enhancementcited by 92
  11. 2024PnP-Flow: Plug-and-Play Image Restoration with Flow Matchingcited by 92
  12. 2024ZigMa: A DiT-style Zigzag Mamba Diffusion Modelcited by 84
38 more
  1. 2024DepthFM: Fast Monocular Depth Estimation with Flow Matchingcited by 76
  2. 2024Rectified Diffusion: Straightness Is Not Your Need in Rectified Flowcited by 59
  3. 2023Diffusion Models With Learned Adaptive Noisecited by 50
  4. 2023A Comprehensive Survey on Knowledge Distillation of Diffusion Modelscited by 45
  5. 2023Boosting Latent Diffusion with Flow Matchingcited by 44
  6. 2024Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modellingcited by 43
  7. 2024Imagine Flash: Accelerating Emu Diffusion Models with Backward Distillationcited by 37
  8. 2024Simple ReFlow: Improved Techniques for Fast Flow Modelscited by 34
  9. 2023Latent Space Editing in Transformer-Based Flow Matchingcited by 32
  10. 2025FlowTok: Flowing Seamlessly Across Text and Image Tokenscited by 31
  11. 2023Adaptive Guidance: Training-free Acceleration of Conditional Diffusion Modelscited by 29
  12. 2024Dynamic Conditional Optimal Transport through Simulation-Free Flowscited by 24
  13. 2023Neural Diffusion Modelscited by 20
  14. 2023AdaDiff: Adaptive Step Selection for Fast Diffusion Modelscited by 18
  15. 2024Epsilon-VAE: Denoising as Visual Decodingcited by 16
  16. 2025The Curse of Conditions: Analyzing and Improving Optimal Transport for Conditional Flow-Based Generationcited by 16
  17. 2025VCT: Training Consistency Models with Variational Noise Couplingcited by 15
  18. 2025On the Relation between Rectified Flows and Optimal Transportcited by 15
  19. 2024Immiscible Diffusion: Accelerating Diffusion Training with Noise Assignmentcited by 14
  20. 2023Catch-Up Distillation: You Only Need to Train Once for Accelerating Samplingcited by 13
  21. 2025Improved Training Technique for Latent Consistency Modelscited by 13
  22. 2025Flow Matching with Semidiscrete Couplingscited by 12
  23. 2025On Fitting Flow Models with Large Sinkhorn Couplingscited by 10
  24. 2023Guided Diffusion from Self-Supervised Diffusion Featurescited by 9
  25. 2025Autoregressive Distillation of Diffusion Transformerscited by 9
  26. 2025Bidirectional Diffusion Bridge Modelscited by 8
  27. 2024Soft Masked Mamba Diffusion Model for CT to MRI Conversioncited by 7
  28. 2024Bag of Design Choices for Inference of High-Resolution Masked Generative Transformercited by 7
  29. 2024Denoising Plane Wave Ultrasound Images Using Diffusion Probabilistic Modelscited by 6
  30. 2025Improved Training Technique for Shortcut Modelscited by 6
  31. 2026Faster Inference of Flow-Based Generative Models via Improved Data-Noise Couplingcited by 6
  32. 2025ProReflow: Progressive Reflow with Decomposed Velocitycited by 5
  33. 2026FlowLet: Conditional 3D Brain MRI Synthesis using Wavelet Flow Matchingcited by 5
  34. 2024Switched Flow Matching: Eliminating Singularities via Switching ODEscited by 4
  35. 2024Improving Consistency Models with Generator-Augmented Flowscited by 4
  36. 2024SF-Speech: Straightened Flow for Zero-Shot Voice Clonecited by 4
  37. 2025Improved Immiscible Diffusion: Accelerate Diffusion Training by Reducing Its Miscibilitycited by 4
  38. 2025Balanced conic rectified flowcited by 4

50 most cited of 87 All 87 (JSON)

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  1. 2025Overcoming the Curvature Bottleneck in MeanFlow
  2. 2024On the Trajectory Regularity of ODE-based Diffusion Sampling
  3. 2023Exploring the Optimal Choice for Generative Processes in Diffusion Models: Ordinary vs Stochastic Differential Equations
  4. 2025Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows
  5. 2025Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Models
  6. 2023Noise in the reverse process improves the approximation capabilities of diffusion models
  7. 2026Look-Ahead and Look-Back Flows: Training-Free Image Generation with Trajectory Smoothing
  8. 2026Learning to Solve Generative ODEs Beyond the Linear Span