标题： 一种在求解薛定谔方程反问题中的智能神经网络 显示英文标题

标题： A clever neural network in solving inverse problems of Schrödinger equation

摘要： 在本工作中，我们解决了可以表述为特殊卷积神经网络学习过程的非线性薛定谔方程的反问题。 而不是尝试近似反问题中的函数，我们在网络中嵌入一个库作为低维流形，使得未知量可以减少为一些标量。 非线性薛定谔方程（NLSE）是$i\frac{\partial \psi}{\partial t}-\beta\frac{\partial^2 \psi}{\partial x^2}+\gamma|\psi|^2\psi+V(x)\psi=0,$，其中波函数$\psi(x,t)$是正问题的解，势能$V(x)$是反问题的关注量。 本工作的主要贡献来自两个方面。 首先，我们直接从薛定谔方程构建了一个特殊的神经网络，这是受到分裂方法的启发。 该构造背后的物理原理增强了神经网络的可解释性。 另一部分是使用库搜索算法将反问题的解空间投影到低维空间。 在简化近似空间中寻找解的方法可以追溯到压缩感知理论。 这一部分的动机是为了减轻估计函数时的训练负担。 相反，通过选择合适的库，可以大大简化训练过程。 给出了一项简要分析，重点在于某些提到的反问题的适定性和神经网络近似的收敛性。 为了展示所提出方法的有效性，我们在一些代表性问题中进行了探索，包括简单的方程和一对方程。 结果可以很好地验证理论部分。 在未来，我们可以进一步探索流形学习以增强库搜索算法的近似效果。 显示英文摘要

摘要： 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.