In this paper, different neural network-based methods are proposed to improve the achievable information rate in amplitude-modulated soliton communication systems. The proposed methods use simulated data to learn effective soliton detection by suppressing nonlinear impairments beyond amplifier noise, including intrinsic inter-soliton interaction, Gordon-Haus effect-induced timing jitter, and their combined impact. We first present a comprehensive study of these nonlinear impairments based on numerical simulations. Then, two neural network designs are developed based on a regression network and a classifier. We estimate the achievable information rates of the proposed learning-based soliton detection schemes as well as two model-based benchmark schemes, including the nonlinear Fourier transform eigenvalue estimation and continuous spectrum-aided eigenvalue estimation schemes. Our results demonstrate that both learning-based designs lead to substantial performance gains when compared to the benchmark schemes. Importantly, we highlight that exploiting the channel memory, introduced by solitonic interactions, can yield additional gains in the achievable information rate. Through a comparative analysis of the two neural network designs, we establish that the classifier design exhibits superior adaptability to interaction impairment and is more suitable for symbol detection tasks in the context of the investigated scenarios.