# Score Matching on Large Geometric Graphs for Cosmology Generation

arXiv: https://arxiv.org/abs/2508.16990 · HTML: https://mlsanity.com/paper/2508.16990/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 14 of its 18 references that are in the map.
- First posted on arXiv 23 Aug 2025.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 14 of 14)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Diffusion Probabilistic Models for 3D Point Cloud Generation](https://mlsanity.com/paper/2103.01458/) (2021)
- [Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equations](https://mlsanity.com/paper/2202.02514/) (2022)
- [Permutation Invariant Graph Generation via Score-Based Generative Modeling](https://mlsanity.com/paper/2003.00638/) (2020)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Score-Based Point Cloud Denoising](https://mlsanity.com/paper/2107.10981/) (2021)
- [The Cosmic Graph: Optimal Information Extraction from Large-Scale Structure using Catalogues](https://mlsanity.com/paper/2207.05202/) (2022)
- [A point cloud approach to generative modeling for galaxy surveys at the field level](https://mlsanity.com/paper/2311.17141/) (2023)
- [Conditional Diffusion-Flow models for generating 3D cosmic density fields: applications to f(R) cosmologies](https://mlsanity.com/paper/2502.17087/) (2025)

## Abstract (arXiv)

Generative models are a promising tool to produce cosmological simulations but face significant challenges in scalability, physical consistency, and adherence to domain symmetries, limiting their utility as alternatives to N-body simulations. To address these limitations, we introduce a score-based generative model with an equivariant graph neural network that simulates gravitational clustering of galaxies across cosmologies starting from an informed prior, respects periodic boundaries, and scales to full galaxy counts in simulations. A novel topology-aware noise schedule, crucial for large geometric graphs, is introduced. The proposed equivariant score-based model successfully generates full-scale cosmological point clouds of up to 600,000 halos, respects periodicity and a uniform prior, and outperforms existing diffusion models in capturing clustering statistics while offering significant computational advantages. This work advances cosmology by introducing a generative model designed to closely resemble the underlying gravitational clustering of structure formation, moving closer to physically realistic and efficient simulators for the evolution of large-scale structures in the universe.