Evaluation of the antigenic similarity degree between the strains of the influenza virus is highly important for vaccine production. The conventional method used to measure such a degree is related to performing the immunological assays of hemagglutinin inhibition. Namely, the antigenic distance between two strains is calculated on the basis of HI assays. Usually, such distances are visualized by using some kind of antigenic cartography method. The known drawback of the HI assay is that it is rather time-consuming and expensive. In this paper, we propose a novel approach for antigenic distance approximation based on deep learning in the feature spaces induced by hemagglutinin protein sequences and Convolutional Neural Networks (CNNs). To apply a CNN to compare the protein sequences, we utilize the encoding based on the physical and chemical characteristics of amino acids. By varying (hyper)parameters of the CNN architecture design, we find the most robust network. Further, we provide insight into the relationship between approximated antigenic distance and antigenicity by evaluating the network on the HI assay database for the H1N1 subtype. The results indicate that the best-trained network gives a high-precision approximation for the ground-truth antigenic distances, and can be used as a good exploratory tool in practical tasks.