Finite Element Operator Network for Solving Elliptic-type parametric PDEs
This addresses the problem of efficiently solving parametric PDEs in fields like physics and engineering, though it is an incremental hybrid approach.
The paper tackles solving parametric PDEs without paired training data by proposing a Finite Element Operator Network (FEONet) that combines deep learning with the finite element method, achieving excellent performance in accuracy, generalization, and computational flexibility across benchmark problems.
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.