Analysis of hybrid finite element / neural network discretization schemes
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Analysis of hybrid finite element / neural network discretization schemes We study hybrid simulation tools that combine coarse mesh finite element solutions with fine mesh fluctuations coming from deep neural networks. In 3D Navier-Stokes simulations this approach is able to upscale the coarse solution to much finer meshes and we obtain a speed-up of more than 30 as compared to a direct solution of the equation on the fine mesh. In this talk we briefly present the idea and the potentials of the hybrid Deep Neural Network Multigrid Solver (DNN-MG) and focus on the analysis of the scheme. For a very simple Laplace problem we proof that the approach is accurate and we give a complete error analysis that decomposes the error into one that measures the quality of the fine mesh training data, into the network approximation and optimization error and into the generalization error that depends on the richness of the training data set.
