Multigoal-oriented dual-weighted-residual error estimation using deep neural networks
This work addresses error estimation in PDEs for computational science and engineering, but it is incremental as it adapts existing dual-weighted residual methods to neural networks.
The authors tackled the problem of approximating solutions to PDEs with multiple goal functionals by developing a deep neural network-based method for a posteriori error estimation using a dual-weighted residual approach, resulting in superior approximation of quantities of interest with less training data.
Deep learning has shown successful application in visual recognition and certain artificial intelligence tasks. Deep learning is also considered as a powerful tool with high flexibility to approximate functions. In the present work, functions with desired properties are devised to approximate the solutions of PDEs. Our approach is based on a posteriori error estimation in which the adjoint problem is solved for the error localization to formulate an error estimator within the framework of neural network. An efficient and easy to implement algorithm is developed to obtain a posteriori error estimate for multiple goal functionals by employing the dual-weighted residual approach, which is followed by the computation of both primal and adjoint solutions using the neural network. The present study shows that such a data-driven model based learning has superior approximation of quantities of interest even with relatively less training data. The novel algorithmic developments are substantiated with numerical test examples. The advantages of using deep neural network over the shallow neural network are demonstrated and the convergence enhancing techniques are also presented