Neural enrichment finite element method: A hybrid framework for problems with strong oscillations or interface problems

TL;DR

The paper introduces NEFEM, a hybrid neural network-enhanced finite element method for efficiently solving problems with oscillations or interfaces, reducing degrees of freedom and requiring minimal prior knowledge.

Contribution

It develops a neural network-based enrichment framework within SGFEM, improving approximation and adaptivity with minimal a priori information and providing theoretical error analysis.

Findings

Significantly reduces degrees of freedom needed for accurate solutions.

Provides a reliable and efficient residual-based error estimator.

Validates effectiveness through numerical experiments.

Abstract

We propose a hybrid method, the Neural Enrichment Finite Element Method (NEFEM), designed for problems involving strong oscillations or interface problems with weak discontinuities. This method is based on the stable generalized finite element method (SGFEM) framework, wherein neural networks (NNs) are introduced as enrichment functions for adaptivity, and the Ritz functional is applied for the training process. This works makes two main contributions. First, the method constructs local subspaces with superior approximation properties, significantly reducing the required number of degrees of freedom (DoFs). Second, minimal \emph{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 SGFEM is studied, notably without imposing additional regularity assumptions. These analytic results guide the network architecture design and training strategies. The performance and effectiveness of the proposed method is validated through several numerical experiments.