Deep Ritz method with Fourier feature mapping: A deep learning approach for solving variational models of microstructure
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Abstract (arXiv)
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.
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References · 12
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- 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
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- 2018On Lazy Training in Differentiable Programmingcited by 675
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- 2019On Exact Computation with an Infinitely Wide Neural Netcited by 573
- 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
- 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
- 2020On the Similarity between the Laplace and Neural Tangent Kernelscited by 76
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