# Generative models on phase space

arXiv: https://arxiv.org/abs/2604.02415 · HTML: https://mlsanity.com/paper/2604.02415/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 37 of its 71 references that are in the map.
- First posted on arXiv 2 Apr 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 37)

- [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)
- [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)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [Poisson Flow Generative Models](https://mlsanity.com/paper/2209.11178/) (2022)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [A Living Review of Machine Learning for Particle Physics](https://mlsanity.com/paper/2102.02770/) (2021)
- [The Principles of Diffusion Models](https://mlsanity.com/paper/2510.21890/) (2025)
- [PFGM++: Unlocking the Potential of Physics-Inspired Generative Models](https://mlsanity.com/paper/2302.04265/) (2023)
- [How Deep Neural Networks Learn Compositional Data: The Random Hierarchy Model](https://mlsanity.com/paper/2307.02129/) (2023)
- [On scientific understanding with artificial intelligence](https://mlsanity.com/paper/2204.01467/) (2022)
- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [A Comprehensive Survey on Data Augmentation](https://mlsanity.com/paper/2405.09591/) (2024)
- [Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics](https://mlsanity.com/paper/2405.14806/) (2024)
- [PC-JeDi: Diffusion for Particle Cloud Generation in High Energy Physics](https://mlsanity.com/paper/2303.05376/) (2023)
- [PELICAN: Permutation Equivariant and Lorentz Invariant or Covariant Aggregator Network for Particle Physics](https://mlsanity.com/paper/2211.00454/) (2022)
- [Probing the Latent Hierarchical Structure of Data via Diffusion Models](https://mlsanity.com/paper/2410.13770/) (2024)
- [How Compositional Generalization and Creativity Improve as Diffusion Models are Trained](https://mlsanity.com/paper/2502.12089/) (2025)
- [Explainable Equivariant Neural Networks for Particle Physics: PELICAN](https://mlsanity.com/paper/2307.16506/) (2023)
- [Topological Obstructions to Autoencoding](https://mlsanity.com/paper/2102.08380/) (2021)
- [EPiC-ly Fast Particle Cloud Generation with Flow-Matching and Diffusion](https://mlsanity.com/paper/2310.00049/) (2023)
- [Lorentz group equivariant autoencoders](https://mlsanity.com/paper/2212.07347/) (2022)
- [Can Diffusion Models Learn Hidden Inter-Feature Rules Behind Images?](https://mlsanity.com/paper/2502.04725/) (2025)
- [Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariant](https://mlsanity.com/paper/2505.20280/) (2025)
- [Improving new physics searches with diffusion models for event observables and jet constituents](https://mlsanity.com/paper/2312.10130/) (2023)

## Abstract (arXiv)

Deep generative models such as diffusion and flow matching are powerful machine learning tools capable of learning and sampling from high-dimensional distributions. They are particularly useful when the training data appears to be concentrated on a submanifold of the data embedding space. For high-energy physics data, consisting of collections of relativistic energy-momentum 4-vectors, this submanifold can enforce extremely strong physically-motivated priors, such as energy and momentum conservation. If these constraints are learned only approximately, rather than exactly, this can inhibit the interpretability and reliability of such generative models. To remedy this deficiency, we introduce generative models which are, by construction, confined at every step of their sampling trajectory to the manifold of massless N-particle Lorentz-invariant phase space in the center-of-momentum frame. In the case of diffusion models, the "pure noise" forward process endpoint corresponds to the uniform distribution on phase space, which provides a clear starting point from which to identify how correlations among the particles emerge during the reverse (de-noising) process. We demonstrate that our models are able to learn both few-particle and many-particle distributions with various singularity structures, paving the way for future interpretability studies using generative models trained on simulated jet data.