Randomized Neural Networks with Petrov-Galerkin Methods for Solving Linear Elasticity Problems

We develop the Randomized Neural Networks with Petrov-Galerkin Methods (RNN-PG methods) to solve linear elasticity problems. RNN-PG methods use Petrov-Galerkin variational framework, where the solution is approximated by randomized neural networks and the test functions are piecewise polynomials. Unlike conventional neural networks, the parameters of the hidden layers of the randomized neural networks are fixed randomly, while the parameters of the output layer are determined by the least square method, which can effectively approximate the solution. We also develop mixed RNN-PG methods for linear elasticity problems, which ensure the symmetry of the stress tensor and avoid locking effects. We compare RNN-PG methods with the finite element method, the mixed discontinuous Galerkin method, and the physics-informed neural network on several examples, and the numerical results demonstrate that RNN-PG methods achieve higher accuracy and efficiency.