| Abstract: |
| We propose a hybrid method, the Neural Enrichment Finite Element Method (NEFEM), designed for problems involving strong oscillations or discontinuities. This method is based on stable generalized FEM (SGFEM) framework, wherein neural networks (NNs) are introduced as enrichment functions for adaptivity, and the Ritz functional is applied for training process. The advancements are twofolds. First, the method constructs local subspaces with superior approximation properties, significantly reducing the required number of degrees of freedom (DoFs). Second, minimal \textit{a priori} knowledge is required to define enrichment functions, as the NNs evolve heuristically during training. Furthermore, for smooth problems, we provide a residual-based error estimator and prove both its reliability and efficiency. For interface problems, a theoretical analysis on the optimal convergence of the stable GFEM is studied, notably without imposing additional regularity assumptions. These analysis results guide the network architecture design and training strategies. The performance and effectiveness of the proposed method are validated through several numerical experiments. |
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