Two neural-network-based methods for solving obstacle problems

TL;DR

This paper introduces two neural network-based numerical methods for solving classical obstacle problems, demonstrating their convergence and effectiveness through theoretical proofs and numerical experiments in 1-D and 2-D.

Contribution

The paper presents two novel neural network schemes for obstacle problems, with rigorous convergence proofs and convergence rate analysis based on the number of neurons.

Findings

Convergence of the proposed schemes is theoretically established.

Numerical experiments verify the convergence rates.

Results demonstrate high-quality solutions for 1-D and 2-D problems.

Abstract

Two neural-network-based numerical schemes are proposed to solve the classical obstacle problems. The schemes are based on the universal approximation property of neural networks, and the cost functions are taken as the energy minimization of the obstacle problems. We rigorously prove the convergence of the two schemes and derive the convergence rates with the number of neurons $N$. In the simulations, we use two example problems (1-D & 2-D) to verify the convergence rate of the methods and the quality of the results.