In solving partial differential equations (PDEs), Fourier Neural Operators (FNOs) have exhibited notable effectiveness compared to Convolutional Neural Networks (CNNs). This paper presents clear empirical evidence through spectral analysis to elucidate the superiority of FNO over CNNs: FNO is significantly more capable of learning low-frequencies. This empirical evidence also unveils FNO's distinct low-frequency bias, which limits FNO's effectiveness in learning high-frequency information from PDE data. To tackle this challenge, we introduce SpecBoost, an ensemble learning framework that employs multiple FNOs to better capture high-frequency information. Specifically, a secondary FNO is utilized to learn the overlooked high-frequency information from the prediction residual of the initial FNO. Experiments demonstrate that SpecBoost noticeably enhances FNO's prediction accuracy on diverse PDE applications, achieving an up to 71% improvement.

翻译：在求解偏微分方程时，傅里叶神经算子相较于卷积神经网络展现出显著的效能。本文通过频谱分析提供了清晰的实证证据，用以阐明FNO优于CNN的原因：FNO在学习低频信息方面能力显著更强。这一实证证据还揭示了FNO独特的低频偏好，从而限制了其从偏微分方程数据中学习高频信息的有效性。为应对这一挑战，我们引入了SpecBoost，这是一种集成学习框架，采用多个FNO以更有效地捕捉高频信息。具体而言，利用二级FNO学习初始FNO预测残差中被忽略的高频信息。实验表明，SpecBoost在多种偏微分方程应用中显著提升了FNO的预测精度，最高可提升71%。