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）的一种特定类型。它为偏微分方程（PDEs）的求解提供了一种基于神经网络的稳健方法。DFR策略基于逼近PDE弱残差的对偶范数，这等价于最小化误差的能量范数。为了计算弱残差范数的对偶，DFR方法采用了测试空间的一组标准正交谱基，该基在矩形或长方体域中对多个函数空间已知。本文通过引入传统区域分解（DD）的思想对DFR方法进行了扩展。这带来了两项改进：（a）可在更一般的多边形域中求解问题；（b）利用多弗勒标记算法开发了一种测试空间自适应细化技术。对于前者，我们证明，在非严格假设下，所采用的损失函数与H1误差之间的理想等价性得以保持，并通过L形域问题数值验证了其符合显式边界。对于后者，我们展示了细化策略如何相较于基准的传统DFR实现（其测试函数空间维度显著较低）带来潜在的显著改进，从而以更合理的计算成本更好地逼近奇异解。