Deep Ritz method with Fourier feature mapping: A deep learning approach for solving variational models of microstructure
Journal:
arXiv
Published Date:
Feb 8, 2025
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.
