Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs

TL;DR

This paper provides a theoretical analysis of the FEONet, a finite element-based operator learning method for second-order elliptic PDEs, establishing convergence, error estimates, and numerical validation.

Contribution

It introduces a convergence analysis and explicit error estimates for FEONet, linking finite element matrix condition numbers to the method's accuracy without relying on data.

Findings

Convergence of FEONet is established for general second-order linear elliptic PDEs.

Explicit error estimates are derived for the self-adjoint case.

Numerical experiments confirm the theoretical role of the finite element matrix condition number.

Abstract

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.