Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs
In this paper, we provide a theoretical analysis of a type of operator learning method without data reliance based on the classical finite element approximation, which is called the finite element operator network (FEONet). We first establish the convergence of this method for general second-order linear elliptic PDEs with respect to the parameters for neural network approximation. In this regard, we address the role of the condition number of the finite element matrix in the convergence of the method. Secondly, we derive an explicit error estimate for the self-adjoint case. For this, we investigate some regularity properties of the solution in certain function classes for a neural network approximation, verifying the sufficient condition for the solution to have the desired regularity. Finally, we will also conduct some numerical experiments that support the theoretical findings, confirming the role of the condition number of the finite element matrix in the overall convergence.
Code (1)
Tasks
Operator learningSimilar Papers 제목 키워드 기반
An Introductory Guide to Koopman Learning
Koopman operators provide a linear framework for data-driven analyses of nonlinear dynamical systems, but their infinite-dimensional nature presents major computational challenges. In this article, we offer an introducto…
Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs
In this paper, we study the finite element operator network (FEONet), an operator-learning method for parametric problems, originally introduced in J. Y. Lee, S. Ko, and Y. Hong, Finite Element Operator Network for Solvi…
Finite Element Operator Network for Solving Elliptic-type parametric PDEs
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…
Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation
Minimizing PDE-residual losses is a common strategy to promote physical consistency in neural operators. However, standard formulations often lack variational correctness, meaning that small residuals do not guarantee sm…
Tackling multiphysics problems via finite element-guided physics-informed operator learning
This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an …