  Fiber nonlinearity effects cap achievable rates and ranges in long-haul
optical fiber communication links. Conventional nonlinearity compensation
methods, such as perturbation theory-based nonlinearity compensation (PB-NLC),
attempt to compensate for the nonlinearity by approximating analytical
solutions to the signal propagation over optical fibers. However, their
practical usability is limited by model mismatch and the immense computational
complexity associated with the analytical computation of perturbation triplets
and the nonlinearity distortion field. Recently, machine learning techniques
have been used to optimise parameters of PB-based approaches, which
traditionally have been determined analytically from physical models. It has
been claimed in the literature that the learned PB-NLC approaches have improved
performance and/or reduced computational complexity over their non-learned
counterparts. In this paper, we first revisit the acclaimed benefits of the
learned PB-NLC approaches by carefully carrying out a comprehensive
performance-complexity analysis utilizing state-of-the-art complexity reduction
methods. Interestingly, our results show that least squares-based PB-NLC with
clustering quantization has the best performance-complexity trade-off among the
learned PB-NLC approaches. Second, we advance the state-of-the-art of learned
PB-NLC by proposing and designing a fully learned structure. We apply a
bi-directional recurrent neural network for learning perturbation triplets that
are alike those obtained from the analytical computation and are used as input
features for the neural network to estimate the nonlinearity distortion field.
Finally, we demonstrate through numerical simulations that our proposed fully
learned approach achieves an improved performance-complexity trade-off compared
to the existing learned and non-learned PB-NLC techniques.
