A residual-based finite element surrogate solver for elliptic partial differential equations
For researchers in scientific computing, this work provides a theoretically grounded, data-free neural solver for elliptic PDEs that bridges finite element theory and deep learning.
The paper introduces a data-free, residual-based finite element surrogate solver for elliptic PDEs that combines CNNs with finite element discretization, achieving stable convergence under grid refinement and robustness on complex geometries and oscillatory solutions.
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.