Hunan Key Laboratory of Smart Roadway and Cooperative Vehicle-Infrastructure Systems, Changsha University of Science and Technology, Changsha, 410205, China. School of Traffic & Transportation Engineering, Changsha University of Science and Technology, Changsha, 410004, China. Department of Civil and Environmental Engineering, University of Tennessee, 319 John D. Tickle Building, Knoxville, TN 37996-2313, USA.
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.
