# On the Mathematics of Diffusion Models

arXiv: https://arxiv.org/abs/2301.11108 · HTML: https://mlsanity.com/paper/2301.11108/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 5 of its 12 references that are in the map.
- First posted on arXiv 25 Jan 2023.

## Placed in

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

## Builds on (most cited first, 5 of 5)

- [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)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)

## Built on by (most cited first, 10 of 10)

- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [Interpretable Diffusion via Information Decomposition](https://mlsanity.com/paper/2310.07972/) (2023)
- [Memory in Plain Sight: Surveying the Uncanny Resemblances of Associative Memories and Diffusion Models](https://mlsanity.com/paper/2309.16750/) (2023)
- [Tackling the Singularities at the Endpoints of Time Intervals in Diffusion Models](https://mlsanity.com/paper/2403.08381/) (2024)
- [Anomaly Detection and Generation with Diffusion Models: A Survey](https://mlsanity.com/paper/2506.09368/) (2025)
- [Neural Entropy](https://mlsanity.com/paper/2409.03817/) (2024)
- [Diffusion Posterior Illumination for Ambiguity-aware Inverse Rendering](https://mlsanity.com/paper/2310.00362/) (2023)
- [Your Diffusion Model is Secretly a Noise Classifier and Benefits from Contrastive Training](https://mlsanity.com/paper/2407.08946/) (2024)
- [Comparison of Generative Learning Methods for Turbulence Surrogates](https://mlsanity.com/paper/2411.16417/) (2024)
- [Combinatorial Complex Score-based Diffusion Modelling through Stochastic Differential Equations](https://mlsanity.com/paper/2406.04916/) (2024)

## Abstract (arXiv)

This paper gives direct derivations of the differential equations and likelihood formulas of diffusion models assuming only knowledge of Gaussian distributions. A VAE analysis derives both forward and backward stochastic differential equations (SDEs) as well as non-variational integral expressions for likelihood formulas. A score-matching analysis derives the reverse diffusion ordinary differential equation (ODE) and a family of reverse-diffusion SDEs parameterized by noise level. The paper presents the mathematics directly with attributions saved for a final section.