In this work, we demonstrate that the high-accuracy computation of the continuous nonlinear spectrum can be performed by using artificial neural networks. We propose the artificial neural network (NN) architecture that can efficiently perform the nonlinear Fourier (NF) optical signal processing. The NN consists of sequential convolution layers and fully connected output layers. This NN predicts only one component of the continuous NF spectrum, such that two identical NNs have to be used to predict the real and imaginary parts of the reflection coefficient. To train the NN, we precomputed 94035 optical signals. 9403 signals were used for validation and excluded from training. The final value of the relative error for the entire validation dataset was less than 0.3%. Our findings highlight the fundamental possibility of using the NNs to analyze and process complex optical signals when the conventional algorithms can fail to deliver an acceptable result.