  We propose a new machine-learning approach for fiber-optic communication
systems whose signal propagation is governed by the nonlinear Schr\"odinger
equation (NLSE). Our main observation is that the popular split-step method
(SSM) for numerically solving the NLSE has essentially the same functional form
as a deep multi-layer neural network; in both cases, one alternates linear
steps and pointwise nonlinearities. We exploit this connection by
parameterizing the SSM and viewing the linear steps as general linear
functions, similar to the weight matrices in a neural network. The resulting
physics-based machine-learning model has several advantages over "black-box"
function approximators. For example, it allows us to examine and interpret the
learned solutions in order to understand why they perform well. As an
application, low-complexity nonlinear equalization is considered, where the
task is to efficiently invert the NLSE. This is commonly referred to as digital
backpropagation (DBP). Rather than employing neural networks, the proposed
algorithm, dubbed learned DBP (LDBP), uses the physics-based model with
trainable filters in each step and its complexity is reduced by progressively
pruning filter taps during gradient descent. Our main finding is that the
filters can be pruned to remarkably short lengths-as few as 3 taps/step-without
sacrificing performance. As a result, the complexity can be reduced by orders
of magnitude in comparison to prior work. By inspecting the filter responses,
an additional theoretical justification for the learned parameter
configurations is provided. Our work illustrates that combining data-driven
optimization with existing domain knowledge can generate new insights into old
communications problems.
