Deep Ritz method with Fourier feature mapping: A deep learning approach for solving variational models of microstructure
This work addresses a problem relevant to researchers and engineers dealing with complex non-convex energy minimization problems, particularly in the context of microstructure modeling, and offers an incremental yet significant improvement over existing methods.
The authors tackled the problem of solving minimization problems with multi-well, non-convex energy potentials using the Deep Ritz Method, and achieved improved results by introducing Fourier feature mapping, enabling the generation of high-frequency, multiscale solutions. The approach was tested on three benchmark problems in both 1D and 2D.
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.