Many problems can be solved with the simulation of dynamic processes in deformable media. They are the simulation of elastic wave propagation in rocks including ice formations, and wave scattering on rock-fracture zones. Such studies are important for solving inverse problems of seismic exploration and seismic data processing to get a better estimation of hydrocarbon reserves, locate hydrocarbons and other minerals. Therefore, it is necessary to develop high-precision numerical methods used to simulate elastic waves in deformable media. One of such methods is the grid-characteristic approach used in this work. It is suitable for solving direct problems, i.e., to analyze the propagation of elastic waves in a medium with known properties. Neural networks can be applied to solve the inverse problem: reconstructing the geology from seismic survey data. Multiple solving of direct problems by the gridcharacteristic approach is used for network training. This paper contains some examples of solving a range of direct problems on the elastic wave propagation in heterogeneous rocks, also in the Arctic zone, and the problem statement for training neural networks and graphs is proposed to demonstrate the efficiency of training with two approaches.