N-Adaptive Ritz Method: A Neural Network Enriched Partition of Unity for
  Boundary Value Problems
Jonghyuk Baek Yanran Wang J. S. Chen
Conventional finite element methods are known to be tedious in adaptive
refinements due to their conformal regularity requirements. Further, the
enrichment functions for adaptive refinements are often not readily available
in general applications. This work introduces a novel neural network-enriched
Partition of Unity (NN-PU) approach for solving boundary value problems via
artificial neural networks with a potential energy-based loss function
minimization. The flexibility and adaptivity of the NN function space are
utilized to capture complex solution patterns that the conventional Galerkin
methods fail to capture. The NN enrichment is constructed by combining
pre-trained feature-encoded NN blocks with an additional untrained NN block.
The pre-trained NN blocks learn specific local features during the offline
stage, enabling efficient enrichment of the approximation space during the
online stage through the Ritz-type energy minimization. The NN enrichment is
introduced under the Partition of Unity (PU) framework, ensuring convergence of
the proposed method. The proposed NN-PU approximation and feature-encoded
transfer learning forms an adaptive approximation framework, termed the
neural-refinement (n-refinement), for solving boundary value problems.
Demonstrated by solving various elasticity problems, the proposed method offers
accurate solutions while notably reducing the computational cost compared to
the conventional adaptive refinement in the mesh-based methods.