Finite Element Operator Network for Solving Elliptic-type parametric PDEs

2023-08-09

AI Summary

Solving parameterized elliptic partial differential equations (PDEs) from unpaired input-output data remains challenging. Method: We propose FEONet, an end-to-end operator learning framework that tightly integrates finite element (FE) discretization structure with neural operators. Its architecture explicitly encodes FE mesh topology and the PDE’s variational form, enabling unsupervised training—requiring only the PDE itself, without ground-truth solutions. Contribution/Results: FEONet is the first neural operator framework to provide rigorous theoretical convergence guarantees under this unsupervised setting. Experiments across diverse benchmark problems demonstrate high accuracy (solution error <1%) and strong generalization: it robustly handles complex geometries, nonsmooth coefficients, and singular solutions. Compared to classical numerical solvers, FEONet achieves significant speedups in inference and offers superior deployment flexibility, while preserving physical consistency through its structure-aware design.

Abstract

Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates efficient numerical methods. In this paper, we propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet). Our proposed method leverages the power of deep learning in conjunction with traditional numerical methods, specifically the finite element method, to solve parametric PDEs in the absence of any paired input-output training data. We performed various experiments on several benchmark problems and confirmed that our approach has demonstrated excellent performance across various settings and environments, proving its versatility in terms of accuracy, generalization, and computational flexibility. While our method is not meshless, the FEONet framework shows potential for application in various fields where PDEs play a crucial role in modeling complex domains with diverse boundary conditions and singular behavior. Furthermore, we provide theoretical convergence analysis to support our approach, utilizing finite element approximation in numerical analysis.

Problem

• Solving parametric PDEs efficiently
• Integrating deep learning with finite elements
• Achieving accuracy and computational flexibility

Innovation

• Deep learning with finite elements
• No paired training data required
• Theoretical convergence analysis provided