# Hybrid Physical-Neural ODEs for Fast N-body Simulations

arXiv: https://arxiv.org/abs/2207.05509 · HTML: https://mlsanity.com/paper/2207.05509/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 11 references that are in the map.
- First posted on arXiv 12 Jul 2022.

## Builds on (most cited first, 4 of 4)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Learning to Predict the Cosmological Structure Formation](https://mlsanity.com/paper/1811.06533/) (2018)
- [AI-assisted super-resolution cosmological simulations](https://mlsanity.com/paper/2010.06608/) (2020)

## Built on by (most cited first, 5 of 5)

- [From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems](https://mlsanity.com/paper/2405.11542/) (2024)
- [COmoving Computer Acceleration (COCA): N-body simulations in an emulated frame of reference](https://mlsanity.com/paper/2409.02154/) (2024)
- [Bridging Simulators with Conditional Optimal Transport](https://mlsanity.com/paper/2510.24631/) (2025)
- [Hybrid Physical-Neural Simulator for Fast Cosmological Hydrodynamics](https://mlsanity.com/paper/2510.26593/) (2025)
- [Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equations](https://mlsanity.com/paper/2603.18331/) (2026)

## Abstract (arXiv)

We present a new scheme to compensate for the small-scales approximations resulting from Particle-Mesh (PM) schemes for cosmological N-body simulations. This kind of simulations are fast and low computational cost realizations of the large scale structures, but lack resolution on small scales. To improve their accuracy, we introduce an additional effective force within the differential equations of the simulation, parameterized by a Fourier-space Neural Network acting on the PM-estimated gravitational potential. We compare the results for the matter power spectrum obtained to the ones obtained by the PGD scheme (Potential gradient descent scheme). We notice a similar improvement in term of power spectrum, but we find that our approach outperforms PGD for the cross-correlation coefficients, and is more robust to changes in simulation settings (different resolutions, different cosmologies).