# Generative diffusion models from a PDE perspective

arXiv: https://arxiv.org/abs/2501.17054 · HTML: https://mlsanity.com/paper/2501.17054/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 8 of its 41 references that are in the map.
- First posted on arXiv 28 Jan 2025.

## Placed in

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

## Builds on (most cited first, 8 of 8)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models](https://mlsanity.com/paper/2210.08933/) (2022)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [On the Generalization of Diffusion Model](https://mlsanity.com/paper/2305.14712/) (2023)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Score-based generative models are provably robust: an uncertainty quantification perspective](https://mlsanity.com/paper/2405.15754/) (2024)

## Built on by (most cited first, 2 of 2)

- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Coarse-Grained Kullback--Leibler Control of Diffusion-Based Generative AI](https://mlsanity.com/paper/2601.01045/) (2026)

## Abstract (arXiv)

Diffusion models have become the de facto framework for generating new datasets. The core of these models lies in the ability to reverse a diffusion process in time. The goal of this manuscript is to explain, from a PDE perspective, how this method works and how to derive the PDE governing the reverse dynamics as well as to study its solution analytically. By linking forward and reverse dynamics, we show that the reverse process's distribution has its support contained within the original distribution. Consequently, diffusion methods, in their analytical formulation, do not inherently regularize the original distribution, and thus, there is no generalization principle. This raises a question: where does generalization arise, given that in practice it does occur? Moreover, we derive an explicit solution to the reverse process's SDE under the assumption that the starting point of the forward process is fixed. This provides a new derivation that links two popular approaches to generative diffusion models: stable diffusion (discrete dynamics) and the score-based approach (continuous dynamics). Finally, we explore the case where the original distribution consists of a finite set of data points. In this scenario, the reverse dynamics are explicit (i.e., the loss function has a clear minimizer), and solving the dynamics fails to generate new samples: the dynamics converge to the original samples. In a sense, solving the minimization problem exactly is "too good for its own good" (i.e., an overfitting regime).