# Minimizing Trajectory Curvature of ODE-based Generative Models

arXiv: https://arxiv.org/abs/2301.12003 · HTML: https://mlsanity.com/paper/2301.12003/

## Key facts

- Cited by 89 papers in the map (data through 14 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.

## Placed in

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

## Builds on (most cited first, 30 of 40)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (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)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Differentiable Augmentation for Data-Efficient GAN Training](https://mlsanity.com/paper/2006.10738/) (2020)
- [Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed](https://mlsanity.com/paper/2101.02388/) (2021)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)

## Built on by (most cited first, 30 of 89)

- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Models](https://mlsanity.com/paper/2406.08070/) (2024)
- [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Accelerator](https://mlsanity.com/paper/2405.07510/) (2024)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Global Structure-Aware Diffusion Process for Low-Light Image Enhancement](https://mlsanity.com/paper/2310.17577/) (2023)
- [ZigMa: A DiT-style Zigzag Mamba Diffusion Model](https://mlsanity.com/paper/2403.13802/) (2024)
- [DepthFM: Fast Monocular Depth Estimation with Flow Matching](https://mlsanity.com/paper/2403.13788/) (2024)
- [Rectified Diffusion: Straightness Is Not Your Need in Rectified Flow](https://mlsanity.com/paper/2410.07303/) (2024)
- [Diffusion Models With Learned Adaptive Noise](https://mlsanity.com/paper/2312.13236/) (2023)
- [A Comprehensive Survey on Knowledge Distillation of Diffusion Models](https://mlsanity.com/paper/2304.04262/) (2023)
- [Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling](https://mlsanity.com/paper/2404.12940/) (2024)
- [Boosting Latent Diffusion with Flow Matching](https://mlsanity.com/paper/2312.07360/) (2023)
- [Imagine Flash: Accelerating Emu Diffusion Models with Backward Distillation](https://mlsanity.com/paper/2405.05224/) (2024)
- [Simple ReFlow: Improved Techniques for Fast Flow Models](https://mlsanity.com/paper/2410.07815/) (2024)
- [FlowTok: Flowing Seamlessly Across Text and Image Tokens](https://mlsanity.com/paper/2503.10772/) (2025)
- [Latent Space Editing in Transformer-Based Flow Matching](https://mlsanity.com/paper/2312.10825/) (2023)
- [Adaptive Guidance: Training-free Acceleration of Conditional Diffusion Models](https://mlsanity.com/paper/2312.12487/) (2023)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Neural Diffusion Models](https://mlsanity.com/paper/2310.08337/) (2023)
- [AdaDiff: Adaptive Step Selection for Fast Diffusion Models](https://mlsanity.com/paper/2311.14768/) (2023)
- [On the Relation between Rectified Flows and Optimal Transport](https://mlsanity.com/paper/2505.19712/) (2025)
- [The Curse of Conditions: Analyzing and Improving Optimal Transport for Conditional Flow-Based Generation](https://mlsanity.com/paper/2503.10636/) (2025)
- [Epsilon-VAE: Denoising as Visual Decoding](https://mlsanity.com/paper/2410.04081/) (2024)
- [VCT: Training Consistency Models with Variational Noise Coupling](https://mlsanity.com/paper/2502.18197/) (2025)

## 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.