A Posteriori Error Control of Approximate Solutions to Boundary Value Problems Found by Neural Networks

The paper discusses how to verify the quality of approximate solutions to partial differential equations constructed by deep neural networks. A posterior error estimates of the functional type, that have been developed for a wide range of boundary value problems, are used to solve this problem. It is shown, that they allow one to construct guaranteed two-sided estimates of global errors and get distribution of local errors over the domain. Results of numerical experiments are presented for elliptic boundary value problems. They show that the estimates provide much more reliable information on the quality of approximate solutions generated by networks than the loss function, which is used as a quality criterion in the Deep Galerkin method. © 2023, Springer Nature Switzerland AG.