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The goal of generative models is to learn the intricate relations between thedata to create new simulated data, but current approaches fail in very highdimensions. When the true data generating process is based on physicalprocesses these impose symmetries and constraints, and the generative model canbe created by learning an effective description of the underlying physics,which enables scaling of the generative model to very high dimensions. In thiswork we propose Lagrangian Deep Learning (LDL) for this purpose, applying it tolearn outputs of cosmological hydrodynamical simulations. The model uses layersof Lagrangian displacements of particles describing the observables to learnthe effective physical laws. The displacements are modeled as the gradient ofan effective potential, which explicitly satisfies the translational androtational invariance. The total number of learned parameters is only of order10, and they can be viewed as effective theory parameters. We combine N-bodysolver 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. Thecomputational cost of LDL is nearly four orders of magnitude lower than thefull hydrodynamical simulations, yet it outperforms it at the same resolution.We achieve this with only of order 10 layers from the initial conditions to thefinal output, in contrast to typical cosmological simulations with thousands oftime steps. This opens up the possibility of analyzing cosmologicalobservations entirely within this framework, without the need for largedark-matter simulations.

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