  Accurate measurement of optical absorption coefficients from photoacoustic
imaging (PAI) data would enable direct mapping of molecular concentrations,
providing vital clinical insight. The ill-posed nature of the problem of
absorption coefficient recovery has prohibited PAI from achieving this goal in
living systems due to the domain gap between simulation and experiment. To
bridge this gap, we introduce a collection of experimentally well-characterised
imaging phantoms and their digital twins. This first-of-a-kind phantom data set
enables supervised training of a U-Net on experimental data for pixel-wise
estimation of absorption coefficients. We show that training on simulated data
results in artefacts and biases in the estimates, reinforcing the existence of
a domain gap between simulation and experiment. Training on experimentally
acquired data, however, yielded more accurate and robust estimates of optical
absorption coefficients. We compare the results to fluence correction with a
Monte Carlo model from reference optical properties of the materials, which
yields a quantification error of approximately 20%. Application of the trained
U-Nets to a blood flow phantom demonstrated spectral biases when training on
simulated data, while application to a mouse model highlighted the ability of
both learning-based approaches to recover the depth-dependent loss of signal
intensity. We demonstrate that training on experimental phantoms can restore
the correlation of signal amplitudes measured in depth. While the absolute
quantification error remains high and further improvements are needed, our
results highlight the promise of deep learning to advance quantitative PAI.
