An AI super-resolution field emulator for cosmological hydrodynamics: the Lyman-{\alpha} forest

We extend our super-resolution and emulation framework for cosmological dark matter simulations to include hydrodynamics. We present a two-stage deep learning model to emulate high-resolution (HR-HydroSim) baryonic fields from low-resolution (LR-HydroSim) simulations at redshift $z = 3$. The method takes as inputs an LR-HydroSim and the high-resolution initial conditions (HR-HydroICs). First, the model stochastically generates high-resolution baryonic fields from the LR-HydroSim. Second, a deterministic emulator refines these fields using HR-HydroICs to reconstruct small-scale structures including displacement, velocity, internal energy, and gas/star classification. Trained on paired low- and high-resolution simulations produced with \texttt{MP-Gadget}, the model captures small-scale structures of the intergalactic medium and %Lyman-$\alpha$ forest observables down to the 100 kpc pressure smoothing scale relevant to the Lyman-$\alpha$ forest. The model achieves subpercent error for overdensity, temperature, velocity, and optical depth fields, a mean relative error of 1.07\% in the large-scale flux power spectrum ($k < 3 \times 10^{-2}\ \mathrm{s/km}$), and less than 10\% error in the flux probability distribution function. Notably, the two-stage model reduces the compute time by a factor of $\sim$450 compared to full smoothed particle hydrodynamics at the same resolution. This work demonstrates the potential of this framework as a powerful and efficient tool for generating high-resolution fields offering fast and accurate alternatives to traditional cosmological hydrodynamic simulations and enabling large-volume mock datasets for next-generation cosmological surveys.