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.

翻译：我们发展了基于Petrov-Galerkin方法的随机神经网络（RNN-PG方法）用于求解线弹性问题。RNN-PG方法采用Petrov-Galerkin变分框架，其中解由随机神经网络近似，而测试函数为分片多项式。与传统神经网络不同，随机神经网络隐藏层参数固定随机选取，输出层参数则通过最小二乘法确定，从而能够有效逼近解。我们还针对线弹性问题发展了混合RNN-PG方法，确保了应力张量的对称性并避免了闭锁效应。通过多个算例将RNN-PG方法与有限元法、混合间断Galerkin方法及物理信息神经网络进行对比，数值结果表明RNN-PG方法具有更高的精度和效率。