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

TL;DR

This paper enhances the Deep Ritz Method by integrating Fourier feature mapping, enabling it to effectively solve complex non-convex variational problems with high-frequency solutions in multiple dimensions.

Contribution

The paper introduces Fourier feature mapping into DRM to overcome spectral bias, allowing for high-frequency, multiscale solution generation in challenging variational models.

Findings

Fourier feature mapping improves DRM's ability to learn high-frequency solutions.

The modified DRM successfully solves 1D and 2D benchmark problems.

High-frequency, multiscale solutions are achieved with the new approach.

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.