@Article{cmc.2019.06641,
AUTHOR = {Cosmin  Anitescu, Elena  Atroshchenko, Naif  Alajlan, Timon  Rabczuk},
TITLE = {Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems},
JOURNAL = {Computers, Materials \& Continua},
VOLUME = {59},
YEAR = {2019},
NUMBER = {1},
PAGES = {345--359},
URL = {http://www.techscience.com/cmc/v59n1/27936},
ISSN = {1546-2226},
ABSTRACT = {We present a method for solving partial differential equations using artificial neural networks and an adaptive collocation strategy. In this procedure, a coarse grid of training points is used at the initial training stages, while more points are added at later stages based on the value of the residual at a larger set of evaluation points. This method increases the robustness of the neural network approximation and can result in significant computational savings, particularly when the solution is non-smooth. Numerical results are presented for benchmark problems for scalar-valued PDEs, namely Poisson and Helmholtz equations, as well as for an inverse acoustics problem.},
DOI = {10.32604/cmc.2019.06641}
}