Solving PDEs with machine learning techniques has become a popular alternative to conventional methods. In this context, Neural networks (NNs) are among the most commonly used machine learning tools, and in those models, the choice of an appropriate loss function is critical. In general, the main goal is to guarantee that minimizing the loss during training translates to minimizing the error in the solution at the same rate. In this work, we focus on the time-harmonic Maxwell's equations, whose weak formulation takes H(curl) as the space of test functions. We propose a NN in which the loss function is a computable approximation of the dual norm of the weak-form PDE residual. To that end, we employ the Helmholtz decomposition of the space H(curl) and construct an orthonormal basis for this space in two and three spatial dimensions. Here, we use the Discrete Sine/Cosine Transform to accurately and efficiently compute the discrete version of our proposed loss function. Moreover, in the numerical examples we show a high correlation between the proposed loss function and the H(curl)-norm of the error, even in problems with low-regularity solutions.

翻译：利用机器学习技术求解偏微分方程已成为传统方法的流行替代方案。在此背景下，神经网络是最常用的机器学习工具之一，而在这类模型中，选择合适的损失函数至关重要。通常，主要目标是确保训练过程中损失函数的减小能够以相同速率转化为求解误差的减小。本文聚焦于时谐麦克斯韦方程组，其弱形式以 H(curl) 作为测试函数空间。我们提出一种神经网络，其损失函数是对弱形式偏微分方程残差对偶范数的可计算近似。为此，我们利用 H(curl) 空间的亥姆霍兹分解，并在二维和三维空间中为该空间构造标准正交基。我们采用离散正弦/余弦变换来精确高效地计算所提损失函数的离散版本。此外，数值算例表明，即使在低正则性解的问题中，所提损失函数与 H(curl) 范数误差之间也具有高度相关性。