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.

翻译：深度傅里叶残差（DFR）方法是变分物理信息神经网络（VPINNs）的一种特殊形式。它为偏微分方程（PDEs）提供了一种基于神经网络的稳健求解方案。DFR策略的核心思想是逼近PDE弱残差的对偶范数，这等价于最小化误差的能量范数。为计算弱残差范数的对偶量，DFR方法采用了测试空间的完备正交谱基函数，该基函数在矩形或长方体区域上对于多种函数空间是已知的。在本工作中，我们将传统区域分解（DD）的思想引入DFR方法，实现了两方面的改进：(a) 能够在更一般的多边形区域上求解问题；(b) 利用Dofler标记算法开发测试空间的自适应细化技术。针对前者，我们证明在非限制性假设下，所采用的损失函数与H1-误差之间仍保持理想等价性，并通过L形区域问题数值验证了显式误差界的满足性。针对后者，我们展示了相较于采用显著低维测试函数空间的经典DFR参考实现，细化策略如何带来潜在重大改进——使我们能够在更合理的计算成本下更精确地逼近奇异解。