In this work, we solve inverse problems of nonlinear Schr\"{o}dinger equations that can be formulated as a learning process of a special convolutional neural network. Instead of attempting to approximate functions in the inverse problems, we embed a library as a low dimensional manifold in the network such that unknowns can be reduced to some scalars. The nonlinear Schr\"{o}dinger equation (NLSE) is $i\frac{\partial \psi}{\partial t}-\beta\frac{\partial^2 \psi}{\partial x^2}+\gamma|\psi|^2\psi+V(x)\psi=0,$ where the wave function $\psi(x,t)$ is the solution to the forward problem and the potential $V(x)$ is the quantity of interest of the inverse problem. The main contributions of this work come from two aspects. First, we construct a special neural network directly from the Schr\"{o}dinger equation, which is motivated by a splitting method. The physics behind the construction enhances explainability of the neural network. The other part is using a library-search algorithm to project the solution space of the inverse problem to a lower-dimensional space. The way to seek the solution in a reduced approximation space can be traced back to the compressed sensing theory. The motivation of this part is to alleviate the training burden in estimating functions. Instead, with a well-chosen library, one can greatly simplify the training process. A brief analysis is given, which focuses on well-possedness of some mentioned inverse problems and convergence of the neural network approximation. To show the effectiveness of the proposed method, we explore in some representative problems including simple equations and a couple equation. The results can well verify the theory part. In the future, we can further explore manifold learning to enhance the approximation effect of the library-search algorithm.

翻译：在本工作中，我们求解了非线性薛定谔方程的反问题，该问题可表述为一种特殊卷积神经网络的学习过程。不同于试图近似反问题中的函数，我们将一个库作为低维流形嵌入网络，从而将未知量简化为若干标量。非线性薛定谔方程（NLSE）为 $i\frac{\partial \psi}{\partial t}-\beta\frac{\partial^2 \psi}{\partial x^2}+\gamma|\psi|^2\psi+V(x)\psi=0$，其中波函数 $\psi(x,t)$ 是正问题的解，势函数 $V(x)$ 是反问题的关注量。本工作的主要贡献体现在两个方面：首先，我们基于分裂方法直接从薛定谔方程构造了一种特殊神经网络，其背后的物理机制增强了网络的可解释性；其次，利用库搜索算法将反问题的解空间投影到更低维空间，这种在约化逼近空间中寻求解的方法可追溯至压缩感知理论。该部分的动机在于减轻函数估计的训练负担，通过精心选择的库可大幅简化训练过程。我们给出了简要分析，重点讨论了若干提及反问题的适定性及神经网络逼近的收敛性。为验证所提方法的有效性，我们在包括简单方程和耦合方程在内的一些代表性问题上进行了探索，结果充分验证了理论部分。未来可进一步探索流形学习以增强库搜索算法的逼近效果。