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 s...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.展开更多
文摘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.
