Deep Domain Decomposition Method for Solving the Variational Inequality Problems

TL;DR

This paper introduces a deep domain decomposition method combining PINN and domain decomposition techniques to efficiently solve elliptic variational inequality problems, demonstrating high accuracy and iteration independence from grid size.

Contribution

It presents a novel approach integrating PINN with domain decomposition for variational inequalities, including residual-adaptive training and analysis of overlap effects.

Findings

Mean square error reached 1.0e-07

Iteration count independent of grid length h

Effective in complex regions

Abstract

By integrating physics-informed neural network (PINN) techniques with domain decomposition method, a deep domain decomposition method is presented for solving elliptic variational inequality problems. Based on the Ritz variation method, the elliptic variational inequality problem is firstly reformulated as an optimization problem, and then the subproblem in each subdomain is solved by using the Ritz-PINN method, which the parameters in the network are updated by the Adam optimizer, and the residual-adaptive training by introducing a residual-adaptive dataset update strategy to gradually guide the model to learn more complex regions. Additionally, the impact of overlapping regions on the performance of the new algorithm is explored. Numerical results demonstrate the effectiveness of the proposed algorithm, the mean square error can be reached 1.0e-07, and the number of iterations is independent of grid length h under uniform overlap conditions.