  The Deep Fourier Residual (DFR) method is a specific type of variational
physics-informed neural networks (VPINNs). It provides a robust neural
network-based solution to partial differential equations (PDEs). The DFR
strategy is based on approximating the dual norm of the weak residual of a PDE.
This is equivalent to minimizing the energy norm of the error. To compute the
dual of the weak residual norm, the DFR method employs an orthonormal spectral
basis of the test space, which is known for rectangles or cuboids for multiple
function spaces.
  In this work, we extend the DFR method with ideas of traditional domain
decomposition (DD). This enables two improvements: (a) to solve problems in
more general polygonal domains, and (b) to develop an adaptive refinement
technique in the test space using a Dofler marking algorithm. In the former
case, we show that under non-restrictive assumptions we retain the desirable
equivalence between the employed loss function and the H1-error, numerically
demonstrating adherence to explicit bounds in the case of the L-shaped domain
problem. In the latter, we show how refinement strategies lead to potentially
significant improvements against a reference, classical DFR implementation with
a test function space of significantly lower dimensionality, allowing us to
better approximate singular solutions at a more reasonable computational cost.
