  In this paper, we propose a model-based machine-learning approach for
dual-polarization systems by parameterizing the split-step Fourier method for
the Manakov-PMD equation. The resulting method combines hardware-friendly
time-domain nonlinearity mitigation via the recently proposed learned digital
backpropagation (LDBP) with distributed compensation of polarization-mode
dispersion (PMD). We refer to the resulting approach as LDBP-PMD. We train
LDBP-PMD on multiple PMD realizations and show that it converges within 1% of
its peak dB performance after 428 training iterations on average, yielding a
peak effective signal-to-noise ratio of only 0.30 dB below the PMD-free case.
Similar to state-of-the-art lumped PMD compensation algorithms in practical
systems, our approach does not assume any knowledge about the particular PMD
realization along the link, nor any knowledge about the total accumulated PMD.
This is a significant improvement compared to prior work on distributed PMD
compensation, where knowledge about the accumulated PMD is typically assumed.
We also compare different parameterization choices in terms of performance,
complexity, and convergence behavior. Lastly, we demonstrate that the learned
models can be successfully retrained after an abrupt change of the PMD
realization along the fiber.
