D. K. Shin

Kyoungjin Jung, Jae Yong Lee, Dongwook Shin

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.

Seungchan Ko, Jiyeon Kim, Dongwook Shin

In this paper, we study the finite element operator network (FEONet), an operator-learning method for parametric problems, originally introduced in J. Y. Lee, S. Ko, and Y. Hong, Finite Element Operator Network for Solving Elliptic-Type Parametric PDEs, SIAM J. Sci. Comput., 47(2), C501-C528, 2025. FEONet realizes the parameter-to-solution map on a finite element space and admits a training procedure that does not require training data, while exhibiting high accuracy and robustness across a broad class of problems. However, its computational cost increases and accuracy may deteriorate as the number of elements grows, posing notable challenges for large-scale problems. In this paper, we propose a new sparse network architecture motivated by the structure of the finite elements to address this issue. Throughout extensive numerical experiments, we show that the proposed sparse network achieves substantial improvements in computational cost and efficiency while maintaining comparable accuracy. We also establish theoretical results demonstrating that the sparse architecture can approximate the target operator effectively and provide a stability analysis ensuring reliable training and prediction.