关于回归问题中多尺度神经网络谱偏差减少的研究
On Spectral Bias Reduction of Multi-scale Neural Networks for Regression Problems
AI总结：
本文推导了谱域扩散方程模型，用于描述两层多尺度深度神经网络（MscaleDNN）的训练误差演化，从理论上解释并数值验证了其在减少谱偏差和逼近宽频振荡函数方面的优势。
摘要：
本文推导了谱域中的扩散方程模型，用于描述两层多尺度深度神经网络（MscaleDNN）训练误差的演化，该网络旨在减少全连接深度神经网络在逼近振荡函数时的谱偏差。该扩散模型由MscaleDNN误差方程的谱形式导出，推导过程采用了神经正切核方法、梯度下降训练和正弦激活函数，并假设学习率趋于零、网络宽度和域尺寸趋于无穷大。结果表明，若在MscaleDNN中使用更多尺度，所涉及的扩散系数将具有更大的支撑集，因此，提出的频域扩散方程模型解释了MscaleDNN减少谱偏差的能力。两层MscaleDNN的扩散模型数值结果与具有合理大网络宽度的实际梯度下降训练的误差演化相匹配，从而验证了扩散模型的有效性。同时，MscaleDNN的数值结果显示在宽频率范围内的误差衰减，并证实了使用MscaleDNN逼近宽频函数的优势。
英文摘要：
In this paper, we derive diffusion equation models in the spectral domain for the evolution of training errors of two-layer multi-scale deep neural networks (MscaleDNN) \cite{caixu2019,liu2020multi}, designed to reduce the spectral bias of fully connected deep neural networks in approximating oscillatory functions. The diffusion models are obtained from the spectral form of the error equation of the MscaleDNN, derived with a neural tangent kernel approach and gradient descent training and a sine activation function, assuming a vanishing learning rate and infinite network width and domain size. The involved diffusion coefficients are shown to have larger supports if more scales are used in the MscaleDNN, and thus, the proposed diffusion equation models in the frequency domain explain the MscaleDNN's spectral bias reduction capability. Numerical results of the diffusion models for a two-layer MscaleDNN training match with the error evolution of actual gradient descent training with a reasonably large network width, thus validating the effectiveness of the diffusion models. Meanwhile, the numerical results for MscaleDNN show error decay over a wide frequency range and confirm the advantage of using the MscaleDNN in approximating functions with a wide range of frequencies.
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