A residual-based finite element surrogate solver for elliptic partial differential equations

Abstract

We propose a residual-based finite element surrogate solver for elliptic partial differential equations. The method combines convolutional neural networks with classical finite element discretization in a data-free setting, where the loss function is defined directly from the finite element residual. This enables the approximation of the solution operator without requiring paired input-output data. A key feature of the proposed approach is that it can be analyzed using standard finite element theory under mesh refinement. We establish a relationship between the training loss and the error in the $H^1$-seminorm, and derive training criteria that ensure optimal convergence rates. To improve efficiency, we introduce a decomposition strategy that separates the contributions of different input components. This allows the model to learn simpler sub-operators. Numerical experiments demonstrate that the proposed method achieves stable convergence under grid refinement, remains robust on complex geometries and oscillatory solutions, and extends naturally to nonlinear equations.