# Learning to Predict the Cosmological Structure Formation

arXiv: https://arxiv.org/abs/1811.06533 · HTML: https://mlsanity.com/paper/1811.06533/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2018.

## Built on by (most cited first, 30 of 38)

- [Towards Causal Representation Learning](https://mlsanity.com/paper/2102.11107/) (2021)
- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Introduction to Latent Variable Energy-Based Models: A Path Towards Autonomous Machine Intelligence](https://mlsanity.com/paper/2306.02572/) (2023)
- [The CAMELS Multifield Dataset: Learning the Universe's Fundamental Parameters with Artificial Intelligence](https://mlsanity.com/paper/2109.10915/) (2021)
- [Dimensionless machine learning: Imposing exact units equivariance](https://mlsanity.com/paper/2204.00887/) (2022)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [Fast and Credible Likelihood-Free Cosmology with Truncated Marginal Neural Ratio Estimation](https://mlsanity.com/paper/2111.08030/) (2021)
- [Field Level Neural Network Emulator for Cosmological N-body Simulations](https://mlsanity.com/paper/2206.04594/) (2022)
- [Hierarchical Implicit Neural Emulators](https://mlsanity.com/paper/2506.04528/) (2025)
- [Simulating Liquids with Graph Networks](https://mlsanity.com/paper/2203.07895/) (2022)
- [The CAMELS project: public data release](https://mlsanity.com/paper/2201.01300/) (2022)
- [Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scatter](https://mlsanity.com/paper/2201.01305/) (2022)
- [Machine Learning and Cosmology](https://mlsanity.com/paper/2203.08056/) (2022)
- [AI-assisted super-resolution cosmological simulations II: Halo substructures, velocities and higher order statistics](https://mlsanity.com/paper/2105.01016/) (2021)
- [Hybrid Physical-Neural ODEs for Fast N-body Simulations](https://mlsanity.com/paper/2207.05509/) (2022)
- [A Neural Material Point Method for Particle-based Emulation](https://mlsanity.com/paper/2408.15753/) (2024)
- [Learning from learning machines: a new generation of AI technology to meet the needs of science](https://mlsanity.com/paper/2111.13786/) (2021)
- [Learning Modular Simulations for Homogeneous Systems](https://mlsanity.com/paper/2210.16294/) (2022)
- [Non-separable Covariance Kernels for Spatiotemporal Gaussian Processes based on a Hybrid Spectral Method and the Harmonic Oscillator](https://mlsanity.com/paper/2302.09580/) (2023)
- [How to Do Machine Learning with Small Data? -- A Review from an Industrial Perspective](https://mlsanity.com/paper/2311.07126/) (2023)
- [COmoving Computer Acceleration (COCA): N-body simulations in an emulated frame of reference](https://mlsanity.com/paper/2409.02154/) (2024)
- [Simple lessons from complex learning: what a neural network model learns about cosmic structure formation](https://mlsanity.com/paper/2206.04573/) (2022)
- [Machine Learning Aided Modeling of Granular Materials: A Review](https://mlsanity.com/paper/2410.14767/) (2024)
- [ASURA-FDPS-ML: Star-by-star Galaxy Simulations Accelerated by Surrogate Modeling for Supernova Feedback](https://mlsanity.com/paper/2410.23346/) (2024)
- [Cosmological Analysis with Calibrated Neural Quantile Estimation and Approximate Simulators](https://mlsanity.com/paper/2411.14748/) (2024)
- [Are We Ready for AI-Driven Discovery? AI Verification Before the Next Fundamental Physics Breakthrough](https://mlsanity.com/paper/2607.10039/) (2026)
- [Likelihood-Free Inference in State-Space Models with Unknown Dynamics](https://mlsanity.com/paper/2111.01555/) (2021)
- [Learning Particle Dynamics Subject to Rigid Body Manipulations Using Graph Neural Networks](https://mlsanity.com/paper/2509.03446/) (2025)

## Abstract (arXiv)

Matter evolved under influence of gravity from minuscule density fluctuations. Non-perturbative structure formed hierarchically over all scales, and developed non-Gaussian features in the Universe, known as the Cosmic Web. To fully understand the structure formation of the Universe is one of the holy grails of modern astrophysics. Astrophysicists survey large volumes of the Universe and employ a large ensemble of computer simulations to compare with the observed data in order to extract the full information of our own Universe. However, to evolve trillions of galaxies over billions of years even with the simplest physics is a daunting task. We build a deep neural network, the Deep Density Displacement Model (hereafter D3M), to predict the non-linear structure formation of the Universe from simple linear perturbation theory. Our extensive analysis, demonstrates that D3M outperforms the second order perturbation theory (hereafter 2LPT), the commonly used fast approximate simulation method, in point-wise comparison, 2-point correlation, and 3-point correlation. We also show that D3M is able to accurately extrapolate far beyond its training data, and predict structure formation for significantly different cosmological parameters. Our study proves, for the first time, that deep learning is a practical and accurate alternative to approximate simulations of the gravitational structure formation of the Universe.