  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.
