# Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

arXiv: https://arxiv.org/abs/2303.08797 · HTML: https://mlsanity.com/paper/2303.08797/

## Key facts

- Cited by 939 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 57 references that are in the map.
- First posted on arXiv 15 Mar 2023.

## Builds on (most cited first, 30 of 39)

- [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)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Simple diffusion: End-to-end diffusion for high resolution images](https://mlsanity.com/paper/2301.11093/) (2023)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Score-Based Generative Modeling with Critically-Damped Langevin Diffusion](https://mlsanity.com/paper/2112.07068/) (2021)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions](https://mlsanity.com/paper/2211.01916/) (2022)
- [Dual Diffusion Implicit Bridges for Image-to-Image Translation](https://mlsanity.com/paper/2203.08382/) (2022)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)

## Built on by (most cited first, 30 of 939)

- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [Flow-GRPO: Training Flow Matching Models via Online RL](https://mlsanity.com/paper/2505.05470/) (2025)
- [World Action Models are Zero-shot Policies](https://mlsanity.com/paper/2602.15922/) (2026)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [Diffusion Transformers with Representation Autoencoders](https://mlsanity.com/paper/2510.11690/) (2025)
- [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Adjoint Matching: Fine-tuning Flow and Diffusion Generative Models with Memoryless Stochastic Optimal Control](https://mlsanity.com/paper/2409.08861/) (2024)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [MixGRPO: Unlocking Flow-based GRPO Efficiency with Mixed ODE-SDE](https://mlsanity.com/paper/2507.21802/) (2025)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Semantic Image Inversion and Editing using Rectified Stochastic Differential Equations](https://mlsanity.com/paper/2410.10792/) (2024)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Lumina-T2X: Transforming Text into Any Modality, Resolution, and Duration via Flow-based Large Diffusion Transformers](https://mlsanity.com/paper/2405.05945/) (2024)
- [Generative Modeling via Drifting](https://mlsanity.com/paper/2602.04770/) (2026)
- [MambaAD: Exploring State Space Models for Multi-class Unsupervised Anomaly Detection](https://mlsanity.com/paper/2404.06564/) (2024)
- [Inductive Moment Matching](https://mlsanity.com/paper/2503.07565/) (2025)
- [AlphaFold Meets Flow Matching for Generating Protein Ensembles](https://mlsanity.com/paper/2402.04845/) (2024)
- [Improved Mean Flows: On the Challenges of Fastforward Generative Models](https://mlsanity.com/paper/2512.02012/) (2025)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [DiffiT: Diffusion Vision Transformers for Image Generation](https://mlsanity.com/paper/2312.02139/) (2023)
- [How to build a consistency model: Learning flow maps via self-distillation](https://mlsanity.com/paper/2505.18825/) (2025)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [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)

## Abstract (arXiv)

A class of generative models that unifies flow-based and diffusion-based methods is introduced. These models extend the framework proposed in Albergo and Vanden-Eijnden (2023), enabling the use of a broad class of continuous-time stochastic processes called stochastic interpolants to bridge any two probability density functions exactly in finite time. These interpolants are built by combining data from the two prescribed densities with an additional latent variable that shapes the bridge in a flexible way. The time-dependent density function of the interpolant is shown to satisfy a transport equation as well as a family of forward and backward Fokker-Planck equations with tunable diffusion coefficient. Upon consideration of the time evolution of an individual sample, this viewpoint leads to both deterministic and stochastic generative models based on probability flow equations or stochastic differential equations with an adjustable level of noise. The drift coefficients entering these models are time-dependent velocity fields characterized as the unique minimizers of simple quadratic objective functions, one of which is a new objective for the score. We show that minimization of these quadratic objectives leads to control of the likelihood for generative models built upon stochastic dynamics, while likelihood control for deterministic dynamics is more stringent. We also construct estimators for the likelihood and the cross entropy of interpolant-based generative models, and we discuss connections with other methods such as score-based diffusion models, stochastic localization, probabilistic denoising, and rectifying flows. In addition, we demonstrate that stochastic interpolants recover the Schr"odinger bridge between the two target densities when explicitly optimizing over the interpolant. Finally, algorithmic aspects are discussed and the approach is illustrated on numerical examples.