Randomized neural network methods for solving obstacle problems
Volume 127 / 2024
Abstract
We explore the use of randomized neural networks (RNNs) for solving obstacle problems, which are elliptic variational inequalities of the first kind that arise in various fields. Obstacle problems can be regarded as free boundary problems, and they are difficult to solve numerically because of their nonsmooth and nonlinear nature. Our first method, RNN with Petrov-Galerkin (RNN-PG), applies RNN to the Petrov-Galerkin variational formulation and solves the resulting linearized subproblems by Picard or Newton iterations. Our second method, RNN with PDE (RNN-PDE), directly solves the partial differential equations (PDEs) associated with the obstacle problems using RNN. We conduct numerical experiments to demonstrate the effectiveness and efficiency of our methods, and show that RNN-PDE outperforms RNN-PG in terms of speed and accuracy.
