论文标题
使用具有物理约束的深度学习方法来求解非线性Schrödinger方程的Soliton，Greather和Rogue Wave解决方案
Soliton, Breather and Rogue Wave Solutions for Solving the Nonlinear Schrödinger Equation Using a Deep Learning Method with Physical Constraints
论文作者
论文摘要
非线性Schrodinger方程是一个经典的集成方程，它包含许多重要属性，并且发生在许多物理领域。但是，由于难以求解该方程式，尤其是在高维度中，因此提出了许多方法来有效地获得不同种类的解决方案，例如神经网络等。最近，提出了一种嵌入常规神经网络中的基本物理定律的方法，以直接从时空数据中发现方程的动力学行为。与传统的神经网络相比，该方法可以通过较少的数据获得非常准确的解决方案。同时，该方法还提供了更好的物理解释和概括。在本文中，基于上述方法，我们提出了一种改进的深度学习方法，以恢复非线性Schrodinger方程的孤子解决方案，呼吸溶液和流氓波解。特别是，关于Schrodinger方程的单阶和两阶流氓波的动力学行为和错误分析首次通过深神经网络揭示。此外，在相同的初始和边界条件下，在控制变量的帮助下，考虑了该方程的一阶流氓波动力学上每个隐藏层的不同初始点，取样的残留搭配点，网络层，每个隐藏层的神经元的影响。数值实验表明，可以通过利用这种物理约束的深度学习方法来很好地重建孤子解决方案，呼吸溶液和Rogue Wave解决方案的动力学行为。
The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas. However, due to the difficulty of solving this equation, in particular in high dimensions, lots of methods are proposed to effectively obtain different kinds of solutions, such as neural networks, among others. Recently, a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation's dynamical behaviors from spatiotemporal data directly. Compared with traditional neural networks, this method can obtain remarkably accurate solution with extraordinarily less data. Meanwhile, this method also provides a better physical explanation and generalization. In this paper, based on the above method, we present an improved deep learning method to recover the soliton solutions, breather solution and rogue wave solutions to the nonlinear Schrodinger equation. In particular, the dynamical behaviors and error analysis about the one-order and two-order rogue waves of the Schrodinger equation are revealed by the deep neural network for the first time. Moreover, the effects of different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions. Numerical experiments show that the dynamical behaviors of soliton solutions, breather solution and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.
