# Diffusion Probabilistic Fields

arXiv: https://arxiv.org/abs/2303.00165 · HTML: https://mlsanity.com/paper/2303.00165/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 49 of its 71 references that are in the map.
- First posted on arXiv 1 Mar 2023.

## Placed in

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

## Builds on (most cited first, 30 of 49)

- [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)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Progressive Growing of GANs for Improved Quality, Stability, and Variation](https://mlsanity.com/paper/1710.10196/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [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)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Least Squares Generative Adversarial Networks](https://mlsanity.com/paper/1611.04076/) (2016)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)

## Built on by (most cited first, 28 of 28)

- [HyperDiffusion: Generating Implicit Neural Fields with Weight-Space Diffusion](https://mlsanity.com/paper/2303.17015/) (2023)
- [HiFA: High-fidelity Text-to-3D Generation with Advanced Diffusion Guidance](https://mlsanity.com/paper/2305.18766/) (2023)
- [On the Design Fundamentals of Diffusion Models: A Survey](https://mlsanity.com/paper/2306.04542/) (2023)
- [Swallowing the Bitter Pill: Simplified Scalable Conformer Generation](https://mlsanity.com/paper/2311.17932/) (2023)
- [ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generation](https://mlsanity.com/paper/2410.22388/) (2024)
- [∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified States](https://mlsanity.com/paper/2303.18242/) (2023)
- [Grounding Continuous Representations in Geometry: Equivariant Neural Fields](https://mlsanity.com/paper/2406.05753/) (2024)
- [DDMI: Domain-Agnostic Latent Diffusion Models for Synthesizing High-Quality Implicit Neural Representations](https://mlsanity.com/paper/2401.12517/) (2024)
- [Stochastic Optimal Control for Diffusion Bridges in Function Spaces](https://mlsanity.com/paper/2405.20630/) (2024)
- [Functional Diffusion](https://mlsanity.com/paper/2311.15435/) (2023)
- [Manifold Diffusion Fields](https://mlsanity.com/paper/2305.15586/) (2023)
- [Single Mesh Diffusion Models with Field Latents for Texture Generation](https://mlsanity.com/paper/2312.09250/) (2023)
- [Generative Neural Fields by Mixtures of Neural Implicit Functions](https://mlsanity.com/paper/2310.19464/) (2023)
- [3D Shape Tokenization via Latent Flow Matching](https://mlsanity.com/paper/2412.15618/) (2024)
- [Generative Hyperelasticity with Physics-Informed Probabilistic Diffusion Fields](https://mlsanity.com/paper/2310.03745/) (2023)
- [Latent Field Discovery In Interacting Dynamical Systems With Neural Fields](https://mlsanity.com/paper/2310.20679/) (2023)
- [MedFuncta: A Unified Framework for Learning Efficient Medical Neural Fields](https://mlsanity.com/paper/2502.14401/) (2025)
- [E(3)-equivariant models cannot learn chirality: Field-based molecular generation](https://mlsanity.com/paper/2402.15864/) (2024)
- [Flow Straight and Fast in Hilbert Space: Functional Rectified Flow](https://mlsanity.com/paper/2509.10384/) (2025)
- [Functional Mean Flow in Hilbert Space](https://mlsanity.com/paper/2511.12898/) (2025)
- [Incomplete Data, Complete Dynamics: A Diffusion Approach](https://mlsanity.com/paper/2509.20098/) (2025)
- [LIFT: Latent Implicit Functions for Task- and Data-Agnostic Encoding](https://mlsanity.com/paper/2503.15420/) (2025)
- [Trajectory Consistency for One-Step Generation on Euler Mean Flows](https://mlsanity.com/paper/2602.02571/) (2026)
- [T1: Scaling Diffusion Probabilistic Fields to High-Resolution on Unified Visual Modalities](https://mlsanity.com/paper/2305.14674/) (2023)
- [DoubleDiffusion: Combining Heat Diffusion with Denoising Diffusion for Texture Generation on 3D Meshes](https://mlsanity.com/paper/2501.03397/) (2025)
- [Neural Field Tokenizations with Hierarchy and Spatial Locality Priors](https://mlsanity.com/paper/2606.08204/) (2026)
- [MING: A Functional Approach to Learning Molecular Generative Models](https://mlsanity.com/paper/2410.12522/) (2024)
- [A standard transformer and attention with linear biases for molecular conformer generation](https://mlsanity.com/paper/2506.19834/) (2025)

## Abstract (arXiv)

Diffusion probabilistic models have quickly become a major approach for generative modeling of images, 3D geometry, video and other domains. However, to adapt diffusion generative modeling to these domains the denoising network needs to be carefully designed for each domain independently, oftentimes under the assumption that data lives in a Euclidean grid. In this paper we introduce Diffusion Probabilistic Fields (DPF), a diffusion model that can learn distributions over continuous functions defined over metric spaces, commonly known as fields. We extend the formulation of diffusion probabilistic models to deal with this field parametrization in an explicit way, enabling us to define an end-to-end learning algorithm that side-steps the requirement of representing fields with latent vectors as in previous approaches (Dupont et al., 2022a; Du et al., 2021). We empirically show that, while using the same denoising network, DPF effectively deals with different modalities like 2D images and 3D geometry, in addition to modeling distributions over fields defined on non-Euclidean metric spaces.