Deep learning driven self-adaptive hp finite element method
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The fi nite element method (FEM) is a popular tool for solving engineering problems governed by Partial Di fferential Equations (PDEs). The accuracy of the numerical solution depends on the quality of the computational mesh. We consider the self-adaptive hp-FEM, which generates optimal mesh refi nements and delivers exponential convergence of the numerical error with respect to the mesh size. Thus, it enables solving di ficult engineering problems with the highest possible numerical accuracy. We replace the computationally expensive kernel of the refi nement algorithm with a deep neural network in this work. The network learns how to optimally re fine the elements and modify the orders of the polynomials. In this way, the deterministic algorithm is replaced by a neural network that selects similar quality refi nements in a fraction of the time needed by the original algorithm.