论文标题
通过深度学习的数据驱动的孤子映射，用于与傅立叶神经操作员深度学习
Data-driven soliton mappings for integrable fractional nonlinear wave equations via deep learning with Fourier neural operator
论文作者
论文摘要
在本文中，我们首先将傅立叶神经操作员（FNO）扩展到两个函数空间之间的孤子映射，其中一个是分数 - 订单索引空间$ \ {ε|ε\ in（0，1，1）\}在（0，1）\} $中，在（0，1）\} $中，在级数的非线性波动中，而另一个表示solitonic solitonic solutonic solititone solution solution solution solution solution selution space space。具体而言，最近在本文中研究了分数非线性schrödinger（FNL），分数Korteweg-de Vries（FKDV），分数修改的Korteweg-De Vries（FMKDV）（FMKDV）和分数Sine-Gordon（FSINEG）方程（FSINEG）方程。我们提出火车并通过记录火车和测试损失来评估进度。为了说明精度，还将数据驱动的孤子与确切的解决方案进行了比较。此外，我们考虑了几个关键因素的影响（例如，包含Relu $（x）$，Sigmoid $（x）$，swish $（x）$和$ x \ tanh（x）$的激活功能，完全连接层的深度）对FNO算法的性能。我们还使用新的激活功能，即$ x \ tanh（x）$，该功能在深度学习领域不使用。本文获得的结果可能有助于进一步了解分数积分非线性波系统中的神经网络以及两个空间之间的映射。
In this paper, we firstly extend the Fourier neural operator (FNO) to discovery the soliton mapping between two function spaces, where one is the fractional-order index space $\{ε|ε\in (0, 1)\}$ in the fractional integrable nonlinear wave equations while another denotes the solitonic solution function space. To be specific, the fractional nonlinear Schrödinger (fNLS), fractional Korteweg-de Vries (fKdV), fractional modified Korteweg-de Vries (fmKdV) and fractional sine-Gordon (fsineG) equations proposed recently are studied in this paper. We present the train and evaluate progress by recording the train and test loss. To illustrate the accuracies, the data-driven solitons are also compared to the exact solutions. Moreover, we consider the influences of several critical factors (e.g., activation functions containing Relu$(x)$, Sigmoid$(x)$, Swish$(x)$ and $x\tanh(x)$, depths of fully connected layer) on the performance of the FNO algorithm. We also use a new activation function, namely, $x\tanh(x)$, which is not used in the field of deep learning. The results obtained in this paper may be useful to further understand the neural networks in the fractional integrable nonlinear wave systems and the mappings between two spaces.
