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

TL;DR

The paper analyzes FEONet, a finite-element operator network that learns solution operators for parametric second-order elliptic PDEs without supervision from input-output pairs. By embedding FEM within a neural framework, it derives convergence results and explicit error estimates, showing that the overall accuracy depends critically on the finite-element matrix condition number $\kappa(A)$. For self-adjoint problems, it obtains a complete error bound via Barron-space regularity, demonstrating how neural approximation and generalization error scale with mesh size $h$, network size $n$, and samples $M$. Numerical experiments on 2D Poisson and convection-diffusion problems validate the theory and show that preconditioning can substantially improve convergence and training efficiency. The work provides a rigorous, topology-aware foundation for data-free operator learning in complex geometries and paves the way for real-time predictions in parametric PDE settings.

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.

Figures (5)

• Figure 1: Examples of complex domains where FEONet can predict solutions.
• Figure 2: Schematic illustration of the decomposition of the FEONet into the encoder $\Psi$, approximator $\mathcal{N}$ and reconstructor $\Phi$.
• Figure 3: The relative $L^2$ errors resulting from varying the number of training samples and the model size.
• Figure 4: The relative $L^2$ errors against the number of elements.
• Figure 5: The FEONet prediction for 2D Poisson equation without or with preconditioning.

Theorems & Definitions (23)

• Theorem 2.2
• Lemma 3.1
• Theorem 3.2: Approximation error
• proof
• Definition 3.3
• Theorem 3.4
• Theorem 3.5: Generalization error
• proof
• Theorem 3.6
• proof