# Probability-Flow Distillation: Distribution Matching in Parameter Space

arXiv: https://arxiv.org/abs/2605.09071 · HTML: https://mlsanity.com/paper/2605.09071/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 28 of its 37 references that are in the map.
- First posted on arXiv 9 May 2026.

## Placed in

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

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

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [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)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Magic3D: High-Resolution Text-to-3D Content Creation](https://mlsanity.com/paper/2211.10440/) (2022)
- [ProlificDreamer: High-Fidelity and Diverse Text-to-3D Generation with Variational Score Distillation](https://mlsanity.com/paper/2305.16213/) (2023)
- [Null-text Inversion for Editing Real Images using Guided Diffusion Models](https://mlsanity.com/paper/2211.09794/) (2022)
- [ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2304.05977/) (2023)
- [One-step Diffusion with Distribution Matching Distillation](https://mlsanity.com/paper/2311.18828/) (2023)
- [Fantasia3D: Disentangling Geometry and Appearance for High-quality Text-to-3D Content Creation](https://mlsanity.com/paper/2303.13873/) (2023)
- [Deep Marching Tetrahedra: a Hybrid Representation for High-Resolution 3D Shape Synthesis](https://mlsanity.com/paper/2111.04276/) (2021)
- [Rethinking FID: Towards a Better Evaluation Metric for Image Generation](https://mlsanity.com/paper/2401.09603/) (2023)
- [LucidDreamer: Towards High-Fidelity Text-to-3D Generation via Interval Score Matching](https://mlsanity.com/paper/2311.11284/) (2023)
- [Dual Diffusion Implicit Bridges for Image-to-Image Translation](https://mlsanity.com/paper/2203.08382/) (2022)
- [Text-to-3D with Classifier Score Distillation](https://mlsanity.com/paper/2310.19415/) (2023)
- [HiFA: High-fidelity Text-to-3D Generation with Advanced Diffusion Guidance](https://mlsanity.com/paper/2305.18766/) (2023)
- [Score Distillation via Reparametrized DDIM](https://mlsanity.com/paper/2405.15891/) (2024)
- [Rethinking Score Distillation as a Bridge Between Image Distributions](https://mlsanity.com/paper/2406.09417/) (2024)
- [Score Distillation Sampling for Audio: Source Separation, Synthesis, and Beyond](https://mlsanity.com/paper/2505.04621/) (2025)

## Abstract (arXiv)

Score distillation methods use pretrained diffusion models as priors for optimizing parameters through differentiable forward models, most notably in text-to-3D generation. Yet the distribution they induce over those parameters is not well understood. Observing that existing distillation methods reduce to one of three: Score Distillation Sampling (SDS), Score Distillation via Inversion (SDI), and Variational Score Distillation (VSD), we extend the particle variational inference view of VSD to the other two. We show that SDS collapses onto the modes of the target, while SDI converges to a contracted version of it, and explain why SDI needs a negative classifier-free guidance scale. Next, we observe that the DDIM posterior mean equals a single Euler step of the probability-flow ODE (PF-ODE). Replacing this step in SDI with a full reverse solve makes the target a fixed point, but it requires solving two concatenated PF-ODEs. Dropping a Jacobian from the resulting gradient gives Probability-Flow Distillation (PFD), which requires solving only the forward PF-ODE. Experiments on synthetic targets, the CelebA dataset, and text-to-3D generation support our analysis and demonstrate the practical effectiveness of PFD.