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Analysing next-generation cosmological data requires balancing accuratemodeling of non-linear gravitational structure formation and computationaldemands. We propose a solution by introducing a machine learning-basedfield-level emulator, within the Hamiltonian Monte Carlo-based Bayesian OriginReconstruction from Galaxies (BORG) inference algorithm. Built on a V-netneural network architecture, the emulator enhances the predictions byfirst-order Lagrangian perturbation theory to be accurately aligned with full$N$-body simulations while significantly reducing evaluation time. We test itsincorporation in BORG for sampling cosmic initial conditions using mock databased on non-linear large-scale structures from $N$-body simulations andGaussian noise. The method efficiently and accurately explores thehigh-dimensional parameter space of initial conditions, fully extracting thecross-correlation information of the data field binned at a resolution of$1.95h^{-1}$ Mpc. Percent-level agreement with the ground truth in the powerspectrum and bispectrum is achieved up to the Nyquist frequency $k_\mathrm{N}\approx 2.79h \; \mathrm{Mpc}^{-1}$. Posterior resimulations - using theinferred initial conditions for $N$-body simulations - show that the recoveryof information in the initial conditions is sufficient to accurately reproducehalo properties. In particular, we show highly accurate $M_{200\mathrm{c}}$halo mass function and stacked density profiles of haloes in different massbins $[0.853,16]\times 10^{14}M_{\odot}h^{-1}$. As all availablecross-correlation information is extracted, we acknowledge that limitations inrecovering the initial conditions stem from the noise level and data gridresolution. This is promising as it underscores the significance of accuratenon-linear modeling, indicating the potential for extracting additionalinformation at smaller scales.

Bayesian Inference of Initial Conditions from Non-Linear Cosmic Structures using Field-Level Emulators