The neural approximated virtual element method for elasticity problems

We present the Neural Approximated Virtual Element Method to numerically solve elasticity problems. This hybrid technique combines classical concepts from the Finite Element Method and the Virtual Element Method with recent advances in deep neural networks. Specifically, it is a polygonal method where the virtual basis functions are element-wise approximated by a neural network, eliminating the need for stabilization or projection operators typically required in the standard Virtual Element Method. We present the discrete formulation of the problem together with theoretical results, and we provide numerical tests on both linear and non-linear elasticity problems, demonstrating the advantages of a simple discretization, particularly in handling non-linearities.