Cosmological field-level inference requires differentiable forward models that solve the challenging dynamics of gas and dark matter under hydrodynamics and gravity. We propose a hybrid approach where gravitational forces are computed using a differentiable particle-mesh solver, while the hydrodynamics are parametrized by a neural network that maps local quantities to an effective pressure field. We demonstrate that our method improves upon alternative approaches, such as an Enthalpy Gradient Descent baseline, both at the field and summary-statistic level. The approach is furthermore highly data efficient, with a single reference simulation of cosmological structure formation being sufficient to constrain the neural pressure model. This opens the door for future applications where the model is fit directly to observational data, rather than a training set of simulations.

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