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In long-haul optical communication systems, compensating nonlinear effects through digital signal processing (DSP) is difficult due to intractable interactions between Kerr nonlinearity, chromatic dispersion (CD) and amplified spontaneous emission (ASE) noise from inline amplifiers. Optimizing the standard digital back propagation (DBP) as a deep neural network (DNN) with interleaving linear and nonlinear operations for fiber nonlinearity compensation was shown to improve transmission performance in idealized simulation environments. Here, we extend such concepts to practical single-channel and polarization division multiplexed wavelength division multiplexed experiments. We show improved performance compared to state-of-the-art DSP algorithms and additionally, the optimized DNNbased DBP parameters exhibit a mathematical structure which guides us to further analyze the noise statistics of fiber nonlinearity compensation. This machine learning-inspired analysis reveals that ASE noise and incomplete CD compensation of the Kerr nonlinear term produce extra distortions that accumulates along the DBP stages. Therefore, the best DSP should balance between suppressing these distortions and inverting the fiber propagation effects, and such trade-off shifts across different DBP stages in a quantifiable manner. Instead of the common 'black-box' approach to intractable problems, our work shows how machine learning can be a complementary tool to human analytical thinking and help advance theoretical understandings in disciplines such as optics.

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Nonlinear interactions between neighboring pulses has always been a fundamental bottleneck in soliton transmission systems. Recently, coherent transceivers, digital signal processing (DSP) and the new nonlinear Fourier transform (NFT) theoretical framework has revived and generalized the field of soliton transmissions into nonlinear frequency division multiplexing (NFDM). We hereby demonstrate analytically and experimentally that one can considerably improve soliton transmission performance by intentionally allowing neighboring solitons to interact and collide during propagation and exchange positions at the receiver followed by standard NFT processing. This can be achieved by designing neighboring solitons' eigenvalues 𝝀 to have opposite signs in the real part while the magnitude |𝕽(𝝀)| is optimized for a given transmission distance so that neighboring transmitted pulses would have swapped their timing positions at the receiver. Experimental results for 6.13 Gbaud 1-soliton systems demonstrate a transmission reach improvement of 100% for 16APSK and 60% for 8PSK modulated on the b-coefficients. The proposed scheme eliminated a long-standing fundamental limitation in soliton transmissions, opened up new dimensions in transmitter signal design and receiver signal processing for nonlinear optical communication systems.