  An important problem in fiber-optic communications is to invert the nonlinear
Schr\"odinger equation in real time to reverse the deterministic effects of the
channel. Interestingly, the popular split-step Fourier method (SSFM) leads to a
computation graph that is reminiscent of a deep neural network. This
observation allows one to leverage tools from machine learning to reduce
complexity. In particular, the main disadvantage of the SSFM is that its
complexity using M steps is at least M times larger than a linear equalizer.
This is because the linear SSFM operator is a dense matrix. In previous work,
truncation methods such as frequency sampling, wavelets, or least-squares have
been used to obtain "cheaper" operators that can be implemented using filters.
However, a large number of filter taps are typically required to limit
truncation errors. For example, Ip and Kahn showed that for a 10 Gbaud signal
and 2000 km optical link, a truncated SSFM with 25 steps would require 70-tap
filters in each step and 100 times more operations than linear equalization. We
find that, by jointly optimizing all filters with deep learning, the complexity
can be reduced significantly for similar accuracy. Using optimized 5-tap and
3-tap filters in an alternating fashion, one requires only around 2-6 times the
complexity of linear equalization, depending on the implementation.
