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

Abstract We provide a theoretical analysis of an operator learning method that does not rely on data, based on the classical finite element approximation, which is called the finite element operator network. We first establish the convergence of this method for general second-order linear elliptic partial differential equations with respect to the parameters used in the neural network approximation, highlighting 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 conditions for the solution to exhibit the desired regularity. Finally, we conduct numerical experiments that support our theoretical findings, confirming the significant impact of the condition number of the finite element matrix on overall convergence, which can be greatly improved using preconditioning techniques.