One photo pins only one direction: a shared transport-map likelihood cuts hundreds of forward solves per posterior in hierarchical inversion

Posterior evaluation in hierarchical inversion no longer calls the forward model cell by cell: one triangular transport-map likelihood, trained once per design, is shared by every cell, and a new photograph yields a posterior directly. Previously the forward model sat inside the likelihood, so each posterior evaluation of the hierarchical model cost one forward solve per cell, and inference cost grew with forward-model cost. At the synthetic validation design the map's posteriors agree with direct sampling in the wetting properties, and simulation-based calibration detects no departure for any parameter; a single posterior needs several hundred times fewer forward-model evaluations than direct sampling with a gradient-free ensemble sampler, and a calibration study of 200 posteriors needs about five orders of magnitude fewer.

The construction orders one cell's parameters and datum into a triangular map and writes the datum's conditional density given the parameters from the map's last block, which serves as the surrogate likelihood. That block carries the noise scale as an exact coordinate in normalized units with a learned correction, rather than asking a fixed-order polynomial to match a target that diverges exponentially in the noise scale. Training rows come from reduced-order forward solves reused under fresh noise, with one map trained per design; on the six cells of real photographs the map's posterior agrees with direct sampling and is consistent with the measured radii.

The demonstrated setting is photographs of a stable hole in a thin liquid film: six cells from three substrates at two volume settings, with a reduced-order model replacing the boundary-value solve and direct sampling as the reference; synthetic validation and scaling designs test the map, and real photographs provide the end-to-end demonstration.

On the larger scaling design the map and its three replicate maps shift the posterior mainly along one combination of contact angles and capillary lengths, agreeing less closely than at the validation design; the noise scale sits high at the synthetic designs, and learning the noise width and shape accounts for most of that shift. In the two calibration studies at the real-data design, departures appear in the knife-edge and volume parameters, which the author attributes to surrogate posterior mass placed outside the forward model's domain after out-of-domain truths are redrawn.