# Poisson Flow Generative Models

arXiv: https://arxiv.org/abs/2209.11178 · HTML: https://mlsanity.com/paper/2209.11178/

## Key facts

- Cited by 95 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 43 references that are in the map.
- First posted on arXiv 22 Sep 2022.

## Builds on (most cited first, 28 of 28)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (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)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [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)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Training Generative Adversarial Networks with Limited Data](https://mlsanity.com/paper/2006.06676/) (2020)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Semantic Segmentation with Generative Models: Semi-Supervised Learning and Strong Out-of-Domain Generalization](https://mlsanity.com/paper/2104.05833/) (2021)
- [ViTGAN: Training GANs with Vision Transformers](https://mlsanity.com/paper/2107.04589/) (2021)
- [From Facial Parts Responses to Face Detection: A Deep Learning Approach](https://mlsanity.com/paper/1509.06451/) (2015)

## Built on by (most cited first, 30 of 95)

- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [Diff-Instruct: A Universal Approach for Transferring Knowledge From Pre-trained Diffusion Models](https://mlsanity.com/paper/2305.18455/) (2023)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [Elucidating the Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2308.15321/) (2023)
- [Input Perturbation Reduces Exposure Bias in Diffusion Models](https://mlsanity.com/paper/2301.11706/) (2023)
- [Restart Sampling for Improving Generative Processes](https://mlsanity.com/paper/2306.14878/) (2023)
- [PFGM++: Unlocking the Potential of Physics-Inspired Generative Models](https://mlsanity.com/paper/2302.04265/) (2023)
- [Particle Guidance: non-I.I.D. Diverse Sampling with Diffusion Models](https://mlsanity.com/paper/2310.13102/) (2023)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [One-Step Diffusion Distillation via Deep Equilibrium Models](https://mlsanity.com/paper/2401.08639/) (2023)
- [CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems](https://mlsanity.com/paper/2312.10527/) (2023)
- [Diffusion Models Generate Images Like Painters: an Analytical Theory of Outline First, Details Later](https://mlsanity.com/paper/2303.02490/) (2023)
- [Improved Techniques for Maximum Likelihood Estimation for Diffusion ODEs](https://mlsanity.com/paper/2305.03935/) (2023)
- [Stable Target Field for Reduced Variance Score Estimation in Diffusion Models](https://mlsanity.com/paper/2302.00670/) (2023)
- [Video-T1: Test-Time Scaling for Video Generation](https://mlsanity.com/paper/2503.18942/) (2025)
- [DCTdiff: Intriguing Properties of Image Generative Modeling in the DCT Space](https://mlsanity.com/paper/2412.15032/) (2024)
- [Contractive Diffusion Probabilistic Models](https://mlsanity.com/paper/2401.13115/) (2024)
- [Deep Generative Models for Offline Policy Learning: Tutorial, Survey, and Perspectives on Future Directions](https://mlsanity.com/paper/2402.13777/) (2024)
- [DisCo-Diff: Enhancing Continuous Diffusion Models with Discrete Latents](https://mlsanity.com/paper/2407.03300/) (2024)
- [GenPhys: From Physical Processes to Generative Models](https://mlsanity.com/paper/2304.02637/) (2023)
- [Score-based Denoising Diffusion with Non-Isotropic Gaussian Noise Models](https://mlsanity.com/paper/2210.12254/) (2022)
- [On Investigating the Conservative Property of Score-Based Generative Models](https://mlsanity.com/paper/2209.12753/) (2022)
- [Score dynamics: scaling molecular dynamics with picoseconds timestep via conditional diffusion model](https://mlsanity.com/paper/2310.01678/) (2023)
- [Convergence Analysis of Flow Matching in Latent Space with Transformers](https://mlsanity.com/paper/2404.02538/) (2024)
- [Prompt Sliders for Fine-Grained Control, Editing and Erasing of Concepts in Diffusion Models](https://mlsanity.com/paper/2409.16535/) (2024)
- [Flow Map Distillation Without Data](https://mlsanity.com/paper/2511.19428/) (2025)
- [RectifID: Personalizing Rectified Flow with Anchored Classifier Guidance](https://mlsanity.com/paper/2405.14677/) (2024)
- [Diffusion Rejection Sampling](https://mlsanity.com/paper/2405.17880/) (2024)

## Abstract (arXiv)

We propose a new "Poisson flow" generative model (PFGM) that maps a uniform distribution on a high-dimensional hemisphere into any data distribution. We interpret the data points as electrical charges on the z=0 hyperplane in a space augmented with an additional dimension z, generating a high-dimensional electric field (the gradient of the solution to Poisson equation). We prove that if these charges flow upward along electric field lines, their initial distribution in the z=0 plane transforms into a distribution on the hemisphere of radius r that becomes uniform in the r →∞ limit. To learn the bijective transformation, we estimate the normalized field in the augmented space. For sampling, we devise a backward ODE that is anchored by the physically meaningful additional dimension: the samples hit the unaugmented data manifold when the z reaches zero. Experimentally, PFGM achieves current state-of-the-art performance among the normalizing flow models on CIFAR-10, with an Inception score of 9.68 and a FID score of 2.35. It also performs on par with the state-of-the-art SDE approaches while offering 10×  to 20 × acceleration on image generation tasks. Additionally, PFGM appears more tolerant of estimation errors on a weaker network architecture and robust to the step size in the Euler method. The code is available at https://github.com/Newbeeer/poissonflow .