  Derived from the regular perturbation treatment of the nonlinear Schrodinger
equation, a machine learning-based scheme to mitigate the intra-channel optical
fiber nonlinearity is proposed. Referred to as the perturbation theory-aided
(PA) learned digital back-propagation (LDBP), the proposed scheme constructs a
deep neural network (DNN) in a way similar to the split-step Fourier method:
linear and nonlinear operations alternate. Inspired by the perturbation
analysis, the intra-channel cross-phase modulation term is conveniently
represented by matrix operations in the DNN. The introduction of this term in
each nonlinear operation considerably improves the performance, as well as
enables the flexibility of PA-LDBP by adjusting the numbers of spans per step.
The proposed scheme is evaluated by numerical simulations of a single carrier
optical fiber communication system operating at 32 Gbaud with 64-quadrature
amplitude modulation and 20*80 km transmission distance. The results show that
the proposed scheme achieves approximately 3.5 dB, 1.8 dB, 1.4 dB, and 0.5 dB
performance gain in terms of Q2 factor over the linear compensation, when the
numbers of spans per step are 1, 2, 4, and 10, respectively. Two methods are
proposed to reduce the complexity of PALDBP, i.e., pruning the number of
perturbation coefficients and chromatic dispersion compensation in the
frequency domain for multi-span per step cases. Investigation of the
performance and complexity suggests that PA-LDBP attains improved performance
gains with reduced complexity when compared to LDBP in the cases of 4 and 10
spans per step.
