An $r$-Adaptive Finite Element Method Using Neural Networks for Parametric Self-Adjoint Elliptic Problems
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This work proposes an r-adaptive finite element method (FEM) framework that leverages neural networks (NNs) to achieve optimal mesh configurations. The method employs the Ritz energy functional as the loss function; while this currently limits applicability to symmetric and coercive problems—such as those arising from self-adjoint elliptic PDEs—it provides a rigorous variational foundation for mesh optimization. The NN is trained to determine optimal mesh node locations that minimize the Ritz energy; for simplicity, these locations are constrained to a tensor-product structure in higher dimensions. Designed specifically for parametric PDEs, the framework uses the trained NN to generate a specialized r-adapted mesh for any given parameter instance, which is then processed using standard FEM. Consequently, the approach retains the robustness and reliability guarantees of classical FEM while benefiting from the adaptive node relocation driven by the NN. The construction of FEM matrices and load vectors is implemented to ensure that their derivatives with respect to node locations—required for training—can be efficiently computed via automatic differentiation. Crucially, the linear equation solver itself does not need to be differentiable, enabling the integration of high-performance, "out-of-the-box" solvers. The method's performance is demonstrated on parametric one- and two-dimensional second-order elliptic problems.
