# Field Level Neural Network Emulator for Cosmological N-body Simulations

arXiv: https://arxiv.org/abs/2206.04594 · HTML: https://mlsanity.com/paper/2206.04594/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 83 references that are in the map.
- First posted on arXiv 9 Jun 2022.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Learning to Predict the Cosmological Structure Formation](https://mlsanity.com/paper/1811.06533/) (2018)
- [Learning cosmology and clustering with cosmic graphs](https://mlsanity.com/paper/2204.13713/) (2022)
- [AI-assisted super-resolution cosmological simulations](https://mlsanity.com/paper/2010.06608/) (2020)
- [Translation and Rotation Equivariant Normalizing Flow (TRENF) for Optimal Cosmological Analysis](https://mlsanity.com/paper/2202.05282/) (2022)
- [A volumetric deep Convolutional Neural Network for simulation of mock dark matter halo catalogues](https://mlsanity.com/paper/1805.04537/) (2018)
- [Fast and Accurate Non-Linear Predictions of Universes with Deep Learning](https://mlsanity.com/paper/2012.00240/) (2020)
- [Robust marginalization of baryonic effects for cosmological inference at the field level](https://mlsanity.com/paper/2109.10360/) (2021)

## Built on by (most cited first, 13 of 13)

- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [Can denoising diffusion probabilistic models generate realistic astrophysical fields?](https://mlsanity.com/paper/2211.12444/) (2022)
- [Hierarchical Implicit Neural Emulators](https://mlsanity.com/paper/2506.04528/) (2025)
- [Surrogate Modeling for Computationally Expensive Simulations of Supernovae in High-Resolution Galaxy Simulations](https://mlsanity.com/paper/2311.08460/) (2023)
- [Explaining dark matter halo density profiles with neural networks](https://mlsanity.com/paper/2305.03077/) (2023)
- [COmoving Computer Acceleration (COCA): N-body simulations in an emulated frame of reference](https://mlsanity.com/paper/2409.02154/) (2024)
- [ASURA-FDPS-ML: Star-by-star Galaxy Simulations Accelerated by Surrogate Modeling for Supernova Feedback](https://mlsanity.com/paper/2410.23346/) (2024)
- [Bridging Simulators with Conditional Optimal Transport](https://mlsanity.com/paper/2510.24631/) (2025)
- [Predicting the Initial Conditions of the Universe using a Deterministic Neural Network](https://mlsanity.com/paper/2303.13056/) (2023)
- [Learning Galaxy Intrinsic Alignment Correlations](https://mlsanity.com/paper/2404.13702/) (2024)
- [Accelerating Giant Impact Simulations with Machine Learning](https://mlsanity.com/paper/2408.08873/) (2024)
- [The First Star-by-star N-body/Hydrodynamics Simulation of Our Galaxy Coupling with a Surrogate Model](https://mlsanity.com/paper/2510.23330/) (2025)
- [Emulating Cosmic Structure Formation with a Lagrangian Neural Cellular Automaton](https://mlsanity.com/paper/2607.27320/) (2026)

## Abstract (arXiv)

We build a field level emulator for cosmic structure formation that is accurate in the nonlinear regime. Our emulator consists of two convolutional neural networks trained to output the nonlinear displacements and velocities of N-body simulation particles based on their linear inputs. Cosmology dependence is encoded in the form of style parameters at each layer of the neural network, enabling the emulator to effectively interpolate the outcomes of structure formation between different flat LambdaCDM cosmologies over a wide range of background matter densities. The neural network architecture makes the model differentiable by construction, providing a powerful tool for fast field level inference. We test the accuracy of our method by considering several summary statistics, including the density power spectrum with and without redshift space distortions, the displacement power spectrum, the momentum power spectrum, the density bispectrum, halo abundances, and halo profiles with and without redshift space distortions. We compare these statistics from our emulator with the full N-body results, the COLA method, and a fiducial neural network with no cosmological dependence. We find our emulator gives accurate results down to scales of k ~ 1 Mpc-1, h, representing a considerable improvement over both COLA and the fiducial neural network. We also demonstrate that our emulator generalizes well to initial conditions containing primordial non-Gaussianity, without the need for any additional style parameters or retraining.