Deep Ritz method with Fourier feature mapping: A deep learning approach for solving variational models of microstructure

Abstract: This paper presents a novel approach that combines the Deep Ritz Method (DRM) with Fourier feature mapping to solve minimization problems comprised of multi-well, non-convex energy potentials. These problems present computational challenges as they lack a global minimum. Through an investigation of three benchmark problems in both 1D and 2D, we observe that DRM suffers from spectral bias pathology, limiting its ability to learn solutions with high frequencies. To overcome this limitation, we modify the method by introducing Fourier feature mapping. This modification involves applying a Fourier mapping to the input layer before it passes through the hidden and output layers. Our results demonstrate that Fourier feature mapping enables DRM to generate high-frequency, multiscale solutions for the benchmark problems in both 1D and 2D, offering a promising advancement in tackling complex non-convex energy minimization problems.