  We introduce for the first time the utilization of Long short-term memory
(LSTM) neural network architectures for the compensation of fiber
nonlinearities in digital coherent systems. We conduct numerical simulations
considering either C-band or O-band transmission systems for single channel and
multi-channel 16-QAM modulation format with polarization multiplexing. A
detailed analysis regarding the effect of the number of hidden units and the
length of the word of symbols that trains the LSTM algorithm and corresponds to
the considered channel memory is conducted in order to reveal the limits of
LSTM based receiver with respect to performance and complexity. The numerical
results show that LSTM Neural Networks can be very efficient as post processors
of optical receivers which classify data that have undergone non-linear
impairments in fiber and provide superior performance compared to digital back
propagation, especially in the multi-channel transmission scenario. The
complexity analysis shows that LSTM becomes more complex as the number of
hidden units and the channel memory increase can be less complex than DBP in
long distances (> 1000 km).
