Obtaining a solution of partial differential equations (PDEs) is one of the central problems of numerical analysis. Efficient methods for solving this problem are widely demanded in engineering and science. Because of this, there are various existing approaches, such as finite elements method. On other hand, due to the ongoing booming of machine learning, there was a lot of progress regarding applying neural networks to PDEs, e.g. Physics Inspired Neural Networks. Main disadvantage of such methods is the lack of mathematical guarantees for the resulting solution and the need to train the network anew every time problem parameters change. The idea of this project is to develop a hybrid method, which would combine strengths of both finite elements method and neural networks and reduce their disadvantages. Unlike existing machine learning based approaches, we are not interested in solving PDEs using the machine learning techniques, but rather in applying them in order to refine an existing finite element solution. In particular we're interested in using neural networks for learning relationship between local restrictions of coarse and fine solutions of PDEs, so that later one can use the same network to refine other solutions, thus eliminating a necessity to solve a problem using a finer mesh and hence reducing a complexity for obtaining a fine solution. We will also perform a rigorous mathematical analysis of the proposed method's performance.