# Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers

论文地址：

* <https://arxiv.org/abs/2111.13587>

参考资料：

* <https://www.cvmart.net/community/detail/6177>

## 整体思路以及计算方式

对于2维输入$$\mathbf X\in \mathbb R^{n\times d}$$：

$$
\mathbf Y =\mathrm{reshape}(\mathcal F(\mathbf X), n,-1, d/k) \in \mathbb R^{n\times k\times (d / k)}
$$

分块矩阵乘法：：

$$
\begin{aligned} \mathbf Y\_1 &= f(\mathbf Y \mathbf W\_1) \in \mathbb R^{n\times k \times (d/k)},\mathbf W\_1 \in \mathbb R^{k\times (d/k)\times (d/k)}\ \mathbf Y\_2 &= \mathbf Y\_1 \mathbf W\_2 \in \mathbb R^{n\times k \times (d/k)}, \mathbf W\_2 \in \mathbb R^{(d/k)\times (d/k)} \end{aligned}
$$

输出：

$$
\mathbf O=\mathcal F^{-1}(\mathrm{reshape}(\mathrm{softshrink} (\mathbf Y\_2), n , d) ) \in \mathbb R^{n\times d}
$$

FNO系列的对比图：

![](https://2520240655-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2Fb2lKDIgZ0I1cirExLedu%2Fuploads%2Fgit-blob-96c7a6153c244c6aca4045e7242a1995b11f97fd%2F1.jpg?alt=media)

## 时间复杂度

$$O(nd\log n+ n d^2 /k)$$

## 训练以及loss

不变。

## 代码

* <https://github.com/NVlabs/AFNO-transformer>

## 实验以及适用场景

适用于Encoder，效果还不错。

## 细节

softshrink操作的是因为在频域中，能量大多数集中在高频。

## 简评

这篇论文的写作是非常好的，理清楚了FNO系列的动机，改进；方法本身也值得复现。


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