RitzNet: A Deep Neural Network Method for Linear Stress Problems

Min Liu, Zhiqiang Cai, Karthik Ramani

Keywords: Linear Elasticity, Deep Neural Network, Ritz Method, Unsupervised Learning

Abstract: Learning based method for physics related computation has attracted significant attention recently. Effort has been devoted into learning a surrogate model which simulates system behavior from existing data. This paper presents RitzNet, an unsupervised learning method which takes any point in the computation domain as input, and learns a neural network model to output its corresponding function value satisfying the underlying governing PDEs. We focus on the linear elastic boundary value problem and formulate it as the natural minimization of its associated energy functional, whose discrete version is further utilized as the loss function of RitzNet. A standard fully connected deep neural network structure is explored in this study to model the solutions of a system of elliptic PDEs. Numerical studies on problems with analytical solutions or unknown solutions show that the proposed RitzNet is capable of approximating linear elasticity problems accurately. A parametric sensitivity study sheds light on the potential of RitzNet due to its meshless characteristics.

One-sentence Summary: We propose RitzNet, an unsupervised learning based method for numerical solving linear elasticity problems using energy functional as loss.