Neural-operator element method: Efficient and scalable finite element method enabled by reusable neural operators

2025-06-23

AI Summary

Traditional finite element methods (FEM) suffer from strong mesh dependency and high computational cost for multiscale complex problems, while existing neural operators exhibit poor reusability and prohibitive training overhead. To address these limitations, this work proposes the Neural Operator Element Method (NOEM): the first framework embedding reusable neural operators into the FEM paradigm. NOEM constructs physics-consistent neural operator elements over complex subdomains and seamlessly couples them with standard FEM via the variational principle. It employs deep operator networks to model subdomain mappings, supporting nonlinearity, multiscale behavior, and arbitrary geometries. Experiments demonstrate that NOEM substantially reduces degrees of freedom and computational cost while preserving optimal convergence rates, scalability, and cross-problem transferability. By unifying data-driven learning with physics-based discretization, NOEM achieves a favorable balance among accuracy, efficiency, and generalization—outperforming conventional FEM and state-of-the-art neural operators in both theoretical rigor and practical applicability.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

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 (e.g., neural operators) 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 (NOs) to simulate subdomains where a large number of finite elements would be required if FEM was used. In each subdomain, an NO 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 extensive and systematic numerical experiments, including nonlinear PDEs, multiscale problems, PDEs on complex geometries, and discontinuous coefficient fields.

Problem

• Reduces computational costs of mesh-based finite element methods
• Addresses high training costs of neural operators for PDEs
• Enables efficient multiscale simulations with reusable neural elements

Innovation

• Combines FEM with neural operator learning
• Uses neural-operator elements for subdomains
• Integrates NOEs with standard finite elements