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.

通过光谱分析明确展示了 Fourier 神经操作符在解决偏微分方程时相比卷积神经网络的显著有效性，并提出了 SpecBoost，一种集成学习框架，通过利用多个 Fourier 神经操作符来更好地捕捉高频信息，明显提高了在各种偏微分方程应用中的预测准确度，最高提升了 71%。

从谱的角度对傅里叶神经算子进行分析和改进的研究

Toward a Better Understanding of Fourier Neural Operators: Analysis and  Improvement from a Spectral Perspective

Machine learning force fields (MLFFs) have emerged as a promising approach to
bridge the accuracy of quantum mechanical methods and the efficiency of
classical force fields. However, the abundance of MLFF models and the challenge
of accurately predicting atomic forces pose significant obstacles in their
practical application. In this paper, we propose a novel ensemble learning
framework, EL-MLFFs, which leverages the stacking method to integrate
predictions from diverse MLFFs and enhance force prediction accuracy. By
constructing a graph representation of molecular structures and employing a
graph neural network (GNN) as the meta-model, EL-MLFFs effectively captures
atomic interactions and refines force predictions. We evaluate our approach on
two distinct datasets: methane molecules and methanol adsorbed on a Cu(100)
surface. The results demonstrate that EL-MLFFs significantly improves force
prediction accuracy compared to individual MLFFs, with the ensemble of all
eight models yielding the best performance. Moreover, our ablation study
highlights the crucial roles of the residual network and graph attention layers
in the model's architecture. The EL-MLFFs framework offers a promising solution
to the challenges of model selection and force prediction accuracy in MLFFs,
paving the way for more reliable and efficient molecular simulations.

通过构建分子结构的图表示并利用图神经网络（GNN）作为元模型，EL-MLFFs 有效地捕捉了原子间相互作用并优化了力预测，实验证明 EL-MLFFs 相对于独立的 MLFFs 显著提高了力预测准确性，最佳性能由所有八个模型组合得出。此外，剩余网络和图注意力层在模型的架构中起到了关键作用，EL-MLFFs 框架为 MLFF 的模型选择和力预测准确性的挑战提供了一个有前景的解决方案，为更可靠高效的分子模拟铺平了道路。