加个对角层让傅里叶神经网络更抗噪声，自动学稀疏特征。
Robust Fourier Neural Networks
- 在傅里叶嵌入后加对角层，增强抗噪能力。
- 理论证明可学习稀疏傅里叶特征，且能处理含噪非线性混合函数。
- 适合处理带噪声信号的函数逼近任务，如物理建模与逆问题。
傅里叶嵌入在训练神经网络时展现出消除频谱偏差的巨大潜力。然而，当标签或测量值存在噪声时，仍可能出现较高泛化误差。本文表明，在傅里叶嵌入层后引入一个简单的对角层，可使网络更鲁棒地应对测量噪声，有效促使模型学习稀疏傅里叶特征。我们基于对角网络的最新进展及神经网络中的隐式正则化理论，为该傅里叶特征学习机制提供了理论支持。在特定条件下，所提方法还能学习由傅里叶特征非线性函数构成的含噪混合函数。数值实验验证了该架构的有效性，支持理论分析。
原文摘要 · Abstract (English)
Fourier embedding has shown great promise in removing spectral bias during neural network training. However, it can still suffer from high generalization errors, especially when the labels or measurements are noisy. We demonstrate that introducing a simple diagonal layer after the Fourier embedding layer makes the network more robust to measurement noise, effectively prompting it to learn sparse Fourier features. We provide theoretical justifications for this Fourier feature learning, leveraging recent developments in diagonal networks and implicit regularization in neural networks. Under certain conditions, our proposed approach can also learn functions that are noisy mixtures of nonlinear functions of Fourier features. Numerical experiments validate the effectiveness of our proposed architecture, supporting our theory.
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