Adaptive Deep Fourier Residual method via overlapping domain decomposition

TL;DR

This work extends the Deep Fourier Residual (DFR) method, a variational PINN approach, to polygonal domains by introducing overlapping domain decomposition (DD) of the computational domain and a $\ abla$-based local refinement strategy guided by a $\star$-norm error indicator. It defines a local, non-inverting DD framework that preserves an equivalence between the discrete loss and the $H^1$ error under mild assumptions, and extends the spectral test-space basis to non-rectangular geometries via partitions of unity. The authors develop an adaptive procedure (including Dofler-like marking) that selectively enriches the test-space in regions with high error, enabling accurate resolution of singularities and sharp gradients with localized Fourier modes. Numerical experiments in 1D and 2D, including L-shaped and pentagonal domains, validate strong loss–error correlations and demonstrate substantial accuracy gains with modest computational cost, highlighting the method’s practical potential for robust VPINN solutions on polygonal domains. The work also provides a framework for extending DD-based DFR to more complex geometries and adaptive strategies, outlining open questions on learning-rate choice and quadrature for low-regularity problems.

Abstract

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.

Figures (20)

• Figure 1: NN architecture sketch. Non-trainable layers are: the loss function calculation, the homogeneous Dirichlet boundary condition (BC) imposition, and the least-squares (LS) solution.
• Figure 2: Example of a convex polygon $\Omega$ (left), a simple cover of $\Omega$ with five rotated rectangles $\Omega_i$, and the superposition of all the subdomains (right).
• Figure 3: Example of an initial partition of the domain $\Omega=(a,b)$, the hat functions $\nu_i$ and the resulting subdomains $\Omega_{i}$.
• Figure 5: Example of an initial partition of a rectilinear polygon, the hat functions $\nu_i$ in 2D and the resulting subdomain decomposition.
• Figure 6: The cover of the pentagonal domain.

Theorems & Definitions (37)

• Remark 4.1: Numerical integration
• Definition 6.3
• Proposition 6.4
• Remark 7.1
• Definition A.1
• Proposition A.2
• proof
• Definition A.3
• Proposition A.4
• proof