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We develop the Randomized Neural Networks with Petrov-Galerkin Methods(RNN-PG methods) to solve linear elasticity problems. RNN-PG methods usePetrov-Galerkin variational framework, where the solution is approximated byrandomized neural networks and the test functions are piecewise polynomials.Unlike conventional neural networks, the parameters of the hidden layers of therandomized neural networks are fixed randomly, while the parameters of theoutput layer are determined by the least square method, which can effectivelyapproximate the solution. We also develop mixed RNN-PG methods for linearelasticity problems, which ensure the symmetry of the stress tensor and avoidlocking effects. We compare RNN-PG methods with the finite element method, themixed discontinuous Galerkin method, and the physics-informed neural network onseveral examples, and the numerical results demonstrate that RNN-PG methodsachieve higher accuracy and efficiency.

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