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

Area of Science:

• Computational Mathematics
• Scientific Machine Learning
• Numerical Analysis

Background:

• The finite element method (FEM) is a standard numerical technique for solving partial differential equations (PDEs).
• FEM's mesh-based approach incurs high computational costs, particularly for complex multiscale problems.
• Machine learning (ML) methods offer data-driven PDE solutions but face challenges like high training costs and limited reusability.

Purpose of the Study:

• To introduce the neural-operator element method (NOEM), a hybrid approach combining FEM with operator learning.
• To address the computational expense and reusability issues associated with traditional FEM and ML-based PDE solvers.
• To develop a novel numerical method for efficient and accurate simulations of PDEs.

Main Methods:

• NOEM integrates neural operators within the FEM framework to create 'neural-operator elements' (NOEs).
• Neural operators are employed to simulate subdomains requiring fine meshing in FEM.
• NOEs are incorporated into the variational framework alongside standard finite elements for comprehensive solution representation.

Main Results:

• NOEM significantly reduces the need for dense meshing, leading to more efficient simulations.
• The method demonstrates accuracy, efficiency, and scalability across various complex problems.
• Numerical experiments include nonlinear PDEs, multiscale simulations, complex geometries, and discontinuous coefficient fields.

Conclusions:

• NOEM offers a synergistic combination of FEM and operator learning for superior PDE solving capabilities.
• The method provides a computationally efficient and reusable alternative for complex simulations.
• NOEM shows promise for advancing numerical methods in computational science and engineering.