The lowest-order neural approximated virtual element method on polygonal elements

The lowest-order neural approximated virtual element method on polygonal elements is proposed here. This method employs a neural network to locally approximate the virtual element basis functions, thereby eliminating issues concerning stabilization and projection operators, which are the key components of the standard virtual element method. By employing neural networks, the computational burden of approximating the virtual basis functions is shifted to the offline phase, aligning the novel method with the finite element method in the online assembling phase. We enhance the original approach, mainly designed for quadrilateral elements, by refining the local approximation space with additional harmonic functions to improve the neural network’s accuracy on polygonal elements. Furthermore, we propose and analyze different training strategies, each offering varying levels of accuracy and supported by theoretical justifications. Several numerical experiments are conducted to validate our procedure on quite general polygonal meshes and demonstrate the advantages of the proposed method across different problem formulations, particularly in cases where the heavy usage of projection and stabilization terms may represent challenges for the standard version of the method. Particular attention is reserved for triangular meshes with hanging nodes which assume a central role in many virtual element applications.