# Geometric Neural Diffusion Processes

arXiv: https://arxiv.org/abs/2307.05431 · HTML: https://mlsanity.com/paper/2307.05431/

## 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 55 of its 89 references that are in the map.
- First posted on arXiv 11 Jul 2023.

## Placed in

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

## Builds on (most cited first, 30 of 55)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Generative Modelling With Inverse Heat Dissipation](https://mlsanity.com/paper/2206.13397/) (2022)

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

- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [A framework for conditional diffusion modelling with applications in motif scaffolding for protein design](https://mlsanity.com/paper/2312.09236/) (2023)
- [Functional Diffusion](https://mlsanity.com/paper/2311.15435/) (2023)
- [Generating Physical Dynamics under Priors](https://mlsanity.com/paper/2409.00730/) (2024)
- [Learning multi-modal generative models with permutation-invariant encoders and tighter variational objectives](https://mlsanity.com/paper/2309.00380/) (2023)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [Structure Preserving Diffusion Models](https://mlsanity.com/paper/2402.19369/) (2024)
- [Neural Bridge Processes](https://mlsanity.com/paper/2508.07220/) (2025)
- [Revisiting Neural Processes via Fourier Transform and Volterra Series](https://mlsanity.com/paper/2606.01172/) (2026)
- [Continuous-Time Machine Learning: A Unified Mathematical Perspective](https://mlsanity.com/paper/2609.16710/) (2026)

## Abstract (arXiv)

Denoising diffusion models have proven to be a flexible and effective paradigm for generative modelling. Their recent extension to infinite dimensional Euclidean spaces has allowed for the modelling of stochastic processes. However, many problems in the natural sciences incorporate symmetries and involve data living in non-Euclidean spaces. In this work, we extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We do so by a) constructing a noising process which admits, as limiting distribution, a geometric Gaussian process that transforms under the symmetry group of interest, and b) approximating the score with a neural network that is equivariant w.r.t. this group. We show that with these conditions, the generative functional model admits the same symmetry. We demonstrate scalability and capacity of the model, using a novel Langevin-based conditional sampler, to fit complex scalar and vector fields, with Euclidean and spherical codomain, on synthetic and real-world weather data.