Speaker
Description
In this talk, we present a hybrid finite element/neural network method for correcting coarse finite element solutions of partial differential equations. The method works by computing fine-scale corrections for the coarse solutions using a neural network trained offline. A neural network takes as input the local values of the coarse solution, as well as other local information such as values of the source term or the velocity field, and predicts the local correction. Such a network can be trained in two ways - directly using the fine finite element solution and using the residual of the equation. The key observation is the dependence between the quality of the predictions and the size of the training set, which consists of a number of different problems, e.g. with different source terms or velocity fields and corresponding fine and coarse solutions. We also present the a priori error analysis of the method together with the stability analysis of the neural network. To support the theoretical claims, we present the results of the numerical experiments. We also illustrate the generalization of the network to other problems in terms of domain, source term, etc.
