NOEM: efficient and scalable finite element method enabled by reusable neural operators

The finite element method (FEM) is a well-established numerical method for solving partial differential equations (PDEs). However, its mesh-based nature gives rise to substantial computational costs, especially for complex multiscale simulations. Emerging machine learning-based methods provide data-driven solutions to PDEs, yet they present challenges, including high training cost and low model reusability. Here we propose the neural-operator element method (NOEM) by synergistically combining FEM with operator learning to address these challenges. NOEM leverages neural operators to simulate subdomains that require fine meshes in FEM. In each subdomain, a neural operator is used to build a single element, namely, a neural-operator element (NOE). NOEs are then integrated with standard finite elements to represent the entire solution through the variational framework. Thereby, NOEM does not necessitate dense meshing and offers efficient simulations. We demonstrate the accuracy, efficiency and scalability of NOEM by performing systematic theoretical analysis and numerical experiments, such as nonlinear PDEs, multiscale problems, PDEs on complex geometries and discontinuous coefficient fields.