Deep Fourier Residual method for solving time-harmonic Maxwell's equations

Abstract

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 $H_0(\mathrm{curl},\Omega )$ 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 $H_0(\mathrm{curl},\Omega )$ 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.