Randomized neural network methods for solving obstacle problems (Q6621121)

The aim of the work is to investigate the use of randomized neural networks (RNNs), to solving obstacle problems. Two RNN-based methods are considered. The RNN-Petrov-Galerkin (RNN-PG) method applies RNN to the Petrov-Galerkin variational formulation of the obstacle problem and solves the regularized and linearized subproblems by Newton's method. The second method, the RNN-PDE method, applies RNN to solve PDE associated to the obstacle problem without any numerical integration. The obstacle problem and different equivalent formulations are described in the second section of the article.\N\NThe considered obstacle problem reads:\N\Nfind $u\in K$, such that $a(u,v-u)\geq (f, v-u)$, for $v\in K$, $K=v\in H^1(\Omega)$; $v\geq\psi$ a.e.in $\Omega$, $v=g$ on $\Gamma$,\N\Nwhere $\Omega\subseteq R^d$, $d=2,3$, $f\in L^2(\Omega)$, $g\in H^{1/2}(\Gamma)$, $\psi\in H^1(\Omega)$ and $a(u,v)=\int_{\Omega}{\nabla u}{\nabla v}dx$, $(f,v)=\int _{\Omega}fv dx$.\N\NThe RNN-PG method is then shortly described in the third section and is applied to the following formulation of the obstacle problem: find $\phi\in H^1(\Omega)$, satisfying $\psi + |\phi|-\phi =0$ on $\Gamma$ such that $a(\phi,v) + (\phi,v) = a(|\phi|,v)-(|\phi|,v)-(f,v) + a(\psi,v)$. The RNN-PDE is presented in the fourth section and is inspired from algorithms published in previous papers of one of the authors, \textit{L. Xue} and \textit{X.-L. Cheng} [Comput. Math. Appl. 48, No. 10--11, 1651--1657 (2004; Zbl 1082.74038)].\N\NIn order to prove the performance of both methods, the algorithms are applied to some obstacle problems and the results are presented and discussed.\N\NFor the entire collection see [Zbl 07926078].