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.

翻译：本文对一类基于经典有限元逼近、无需数据依赖的算子学习方法——有限元算子网络（FEONet）进行了理论分析。首先，针对一般二阶线性椭圆型偏微分方程，我们建立了该方法关于神经网络逼近参数的收敛性，并讨论了有限元矩阵条件数在该方法收敛性中的作用。其次，针对自伴情形，我们推导了显式误差估计。为此，我们研究了神经网络逼近中解在某些函数类中的正则性性质，验证了该解具有所需正则性的充分条件。最后，我们还将开展若干数值实验以支持理论发现，确认有限元矩阵条件数在整体收敛性中的作用。