Qijia Zhou

Yiyang Wang, Qijia Zhou, Shengyuan Deng et al.

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.

Qijia Zhou, Yiyang Wang, Shengyuan Deng et al.

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.