  Photoacoustic tomography (PAT) is a hybrid medical imaging technique that
offer high contrast and a high spatial resolution. One challenging mathematical
problem associated with PAT is reconstructing the initial pressure of the wave
equation from data collected at the specific surface where the detectors are
positioned. The study addresses this problem when PAT is modeled by a wave
equation with unknown sound speed $c$, which is a function of spatial
variables, and under the assumption that both the Dirichlet and Neumann
boundary values on the detector surface are measured. In practical, we
introduce a novel implicit learning framework to simultaneously estimate the
unknown $c$ and the reconstruction operator using only Dirichlet and Neumann
boundary measurement data. The experimental results confirm the success of our
proposed framework, demonstrating its ability to accurately estimate variable
sound speed and the reconstruction operator in PAT.
