Deep Ritz Physics-Informed Neural Network Method for Solving the Variational Inequality

TL;DR

This paper introduces a Deep Ritz method utilizing Physics-Informed Neural Networks combined with Bayesian optimization and adaptive data strategies to efficiently solve elliptic variational inequalities with high accuracy.

Contribution

It presents a novel Deep Ritz approach that integrates PINNs, Bayesian optimization, and adaptive data updates for improved solutions of variational inequalities.

Findings

Effectively approximates analytical solutions.

Enhances convergence and accuracy through adaptive strategies.

Demonstrates superior performance in numerical experiments.

Abstract

Variational inequalities are widely applied in mechanical engineering, fluid penetration, transportation, and other fields. In this paper, a Deep Ritz method based on Physics-Informed Neural Networks (PINNs) is proposed to enhance the accuracy and efficiency of solving elliptic variational inequalities. The Ritz variational method is firstly utilized to transform the variational inequality problem into an optimization problem. Then Bayesian optimization is employed to tune the weights of the loss function, and a residual-based adaptive dataset update strategy is introduced to improve the convergence and accuracy of the model. Numerical experiments show that the proposed method can effectively approximate the analytical solution.