ADLGM: An efficient adaptive sampling deep learning Galerkin method

In this paper, we devise an adaptive sampling technique applied to the deep Galerkin method (DGM), aimed to improve and speed up the training of the deep neural network when learning the solution of partial differential equations (PDEs). The proposed adaptive algorithm is inspired by the mesh adaptivity techniques used in the classical numerical PDE field. Its implementation to the DGM paradigm is natural, is done efficiently, and it is shown to improve the DGM algorithm. We demonstrate that our adaptive sampling DGM scheme is convergent and more accurate than DGM - as long as the residual mirrors the local error - at the same number of training steps and using the same or less number of training points. We present a multitude of tests applied to selected PDEs discussing the robustness of our scheme.(c) 2023 Elsevier Inc. All rights reserved.