用傅里叶特征映射改进深度瑞兹法，解决微结构能量极小化的高频多尺度难题
Deep Ritz method with Fourier feature mapping: A deep learning approach for solving variational models of microstructure
- 在输入层引入傅里叶特征映射，缓解深度瑞兹法的频谱偏差问题
- 在1D和2D基准问题上成功生成高频率、多尺度解，突破原方法局限
- 适合研究材料微结构演化、非凸能量泛函求解的科研人员
本文提出一种新方法，将深度瑞兹法（Deep Ritz Method, DRM）与傅里叶特征映射结合，用于求解包含多阱非凸能势的变分模型。这类问题因缺乏全局最小值而计算困难。通过在1D和2D三个基准问题上的实验发现，传统DRM存在频谱偏差，难以学习高频解。为此，我们在输入层引入傅里叶特征映射，即在数据进入网络前进行傅里叶变换。结果表明，该改进使DRM能够有效生成1D和2D基准问题的高频、多尺度解，为复杂非凸能量极小化问题提供了有前景的新解决方案。
原文摘要 · Abstract (English)
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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