Adaptive Learning on the Grids for Elliptic Hemivariational Inequalities

2021/04/11 by Jianguo Huang, Huang, Jianguo, Chunmei Wang +3
Computer Science · Engineering · Mathematics · #65N30 #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Gear and Bearing Dynamics Analysis #Mechanical stress and fatigue analysis #Numerical Analysis (math.NA) #cs.NA #math.NA #msc:65N30

paper · pdf · doi:10.48550/arxiv.2104.04881

18 pages, 3 figures, 2 tables

arxiv created 2021/04/11 · openalex publication_date 2021/04/11 · arxiv updated 2021/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper introduces a deep learning method for solving an elliptic hemivariational inequality (HVI). In this method, an expectation minimization problem is first formulated based on the variational principle of underlying HVI, which is solved by stochastic optimization algorithms using three different training strategies for updating network parameters. The method is applied to solve two practical problems in contact mechanics, one of which is a frictional bilateral contact problem and the other of which is a frictionless normal compliance contact problem. Numerical results show that the deep learning method is efficient in solving HVIs and the adaptive mesh-free multigrid algorithm can provide the most accurate solution among the three learning methods.

Citations

• Numerical Analysis of Elliptic Hemivariational Inequalities
• Optimal approximation of continuous functions by very deep ReLU networks
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem
• Towards Understanding the Spectral Bias of Deep Learning
• Representation formulas and pointwise properties for Barron functions
• Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations
• Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations
• Reproducing Activation Function for Deep Learning

Related

• Numerical Analysis of Elliptic Hemivariational Inequalities
• Accurate adaptive deep learning method for solving elliptic problems
• Prox-PINNs: A Deep Learning Algorithmic Framework for Elliptic Variational Inequalities
• Hybrid subgradient and simulated annealing method for hemivariational inequalities
• A neural network approach to learning solutions of a class of elliptic variational inequalities