# Learning effective physical laws for generating cosmological hydrodynamics with Lagrangian Deep Learning

arXiv: https://arxiv.org/abs/2010.02926 · HTML: https://mlsanity.com/paper/2010.02926/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2020.

## Built on by (most cited first, 7 of 7)

- [The CAMELS Multifield Dataset: Learning the Universe's Fundamental Parameters with Artificial Intelligence](https://mlsanity.com/paper/2109.10915/) (2021)
- [On the Learning Mechanisms in Physical Reasoning](https://mlsanity.com/paper/2210.02075/) (2022)
- [Reconstruction of observed mechanical motions with Artificial Intelligence tools](https://mlsanity.com/paper/2202.11447/) (2022)
- [MadEvolve: Evolutionary Optimization of Cosmological Algorithms with Large Language Models](https://mlsanity.com/paper/2602.15951/) (2026)
- [AI-assisted super-resolution cosmological simulations II: Halo substructures, velocities and higher order statistics](https://mlsanity.com/paper/2105.01016/) (2021)
- [COmoving Computer Acceleration (COCA): N-body simulations in an emulated frame of reference](https://mlsanity.com/paper/2409.02154/) (2024)
- [Wavelet Flow For Extragalactic Foreground Simulations](https://mlsanity.com/paper/2505.21220/) (2025)

## Abstract (arXiv)

The goal of generative models is to learn the intricate relations between the data to create new simulated data, but current approaches fail in very high dimensions. When the true data generating process is based on physical processes these impose symmetries and constraints, and the generative model can be created by learning an effective description of the underlying physics, which enables scaling of the generative model to very high dimensions. In this work we propose Lagrangian Deep Learning (LDL) for this purpose, applying it to learn outputs of cosmological hydrodynamical simulations. The model uses layers of Lagrangian displacements of particles describing the observables to learn the effective physical laws. The displacements are modeled as the gradient of an effective potential, which explicitly satisfies the translational and rotational invariance. The total number of learned parameters is only of order 10, and they can be viewed as effective theory parameters. We combine N-body solver FastPM with LDL and apply them to a wide range of cosmological outputs, from the dark matter to the stellar maps, gas density and temperature. The computational cost of LDL is nearly four orders of magnitude lower than the full hydrodynamical simulations, yet it outperforms it at the same resolution. We achieve this with only of order 10 layers from the initial conditions to the final output, in contrast to typical cosmological simulations with thousands of time steps. This opens up the possibility of analyzing cosmological observations entirely within this framework, without the need for large dark-matter simulations.