Artificial Neural Network Methods for the Solution of Second Order Boundary Value Problems

• 作者: Cosmin Anitescu Elena Atroshchenko Naif Alajlan Timon Rabczuk
• 来源期刊: 计算机、材料和连续体(英文)
• 学科: 数学
• 关键词: DEEP learning adaptive COLLOCATION INVERSE problems artificial NEURAL networks
• 年，卷（期）: 2019,（4）
• 页码范围: 345-359
• 页数: 15页
• 分类号: O17

摘要

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.