Field-level inference (FLI) directly models the full density field rather than compressed summary statistics. Two distinct approaches emerge: explicit FLI, which jointly infers cosmological parameters and initial conditions p(δ_ICs, θ|δ_Obs) to create true "digital twins" of our Universe, and implicit FLI, which directly constrains cosmology p(θ|δ_Obs) without reconstructing initial conditions and thus cannot provide a complete digital twin.
Explicit FLI enables reconstruction of dark matter distributions and formation histories but requires sampling in O(10^9) dimensional spaces. Implicit FLI extracts optimal summaries from the data, reducing the problem to O(10-100) dimensions while potentially achieving similar constraining power.
Machine learning advances in generative modeling now make high-dimensional cosmological inference tractable. The choice between approaches involves fundamental trade-offs: explicit FLI provides complete physical reconstruction at a high computational cost, while implicit FLI offers efficiency but sacrifices reconstruction capabilities. Key challenges include developing robust neural validation frameworks and scalable sampling methods for realistic forward models.