Finite Element Operator Learning for Solving Parametric PDEs without Labeled Data
Keywords: Scientific machine learning, finite element methods, physics-informed operator learning, parametric partial differential equations
TL;DR: We proposed a novel approach for solving parametric PDEs based on the finite element methods, which is called Finite Element Operator Network (FEONet).
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 demonstrate the effectiveness of our approach on several benchmark problems and show that it outperforms existing state-of-the-art methods in terms of accuracy, generalization, and computational flexibility. Our 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.
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Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
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Submission Number: 1251
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