A Novel Method for Solving Nonlinear Schrödinger Equation with a Potential by Deep Learning

The improved physical information neural network algorithm has been proven to be used to study integrable systems. In this paper, the improved physical information neural network algorithm is used to study the defocusing nonlinear Schrödinger (NLS) equation with time-varying potential, and the rogue wave solution of the equation is obtained. At the same time, the influence of the number of network layers, neurons and the number of sampling points on the network performance is studied. Experiments show that the number of hidden layers and the number of neurons in each hidden layer affect the relative 2  -norm error. With fixed configuration points, the relative norm error does not decrease with the increase in the number of boundary data points, which indicates that in this case, the number of boundary data points has no obvious influence on the error. Through the experiment, the rogue wave solution of the defocusing NLS equation is successfully captured by IPINN method for the first time. The experimental results of this paper are also com-pared with the results obtained by the physical information neural network method and show that the improved algorithm has higher accuracy. The results of this paper will be contributed to the generalization of deep learning algorithms for solving defocusing NLS equations with time-varying potential.


Introduction
It is well known that differential systems are nonlinear phenomena that describe The peaks and troughs of these waves correspond to the inflection points of stock prices in the financial market. Obviously, it provides a theoretical mechanism for people to understand the phenomenon of the financial crisis. In recent years, many classical analytical and numerical methods have been developed in the field of computing. Although these methods give numerical solutions or analytical solutions of partial differential systems to a certain extent, such as Lie symmetry in analytical methods, this method first calculates the characteristic sequence set, then calculates the infinitesimal generator, and finally transforms the partial differential system into an ordinary differential system solution. The amount of calculation involved in this process is very large. Although there is the help of a computer only reduces the computational burden to a certain extent, and some problems must be calculated manually. Especially when the partial differential problem is transformed into an ordinary differential problem, although the order of the differential equation is reduced, some ordinary differential problems cannot be solved, and the final problem is still unsolved. Another example is the finite difference method in the classical numerical method, which divides the solution domain into differential grids, but with the increase of the complexity of the boundary, the grid intersections cannot all be guaranteed to fall on the boundary conditions; this method lacks flexibility on complex boundaries. Therefore, the solution of differential systems has always been a research hotspot in the scientific community.
In recent years, with the development and application of artificial intelligence technology in various neighborhoods, experts have turned their attention to artificial intelligence, and they have considered using neural networks to study differential systems. For example, I.E. Lagaris and A.C. Likas studied the solutions of ordinary and partial differential equations using artificial neural networks in 1998 [1]. In 2000, S. He and K. Reif investigated the solution of partial differential equations by multilayer neural networks [2]. In the same year, I.E. Lagaris et al. studied differential equations with irregular boundaries by neural networks [3]. In 2006, A. Malek and R.S. Beidokhti obtained numerical solutions for high-order differential equations by a hybrid neural network method [4]. In Legendre neural network methods [10]. The breakthrough in using a neural network to solve differential equations is the physical information neural network(PINN) algorithm proposed by professor Raissi from Brown University in 2019 [11]. The physical information neural network algorithm does not need to assume the expression of the solution. It embeds the physical information and the initial-boundary value conditions in the neural network. Only a small number of random sample points on the initial-boundary value conditions can obtain the numerical solution of the differential equation. In view of these advantages, many scholars have studied them. Such as Z. Yan et al. studied forward and inverse problems of the Schrödinger equation with  -symmetric harmonic potential [12], the team of Y. Chen used the PINN method to study integrable systems [13] [14] [15] [16]. With the research on PINN, it is found that it does not converge on some complex problems, so many improvement methods are proposed. Such as Bayesian physics-informed neural networks (B-PINNs) [17], fractional Physics-Informed Neural Networks(fPINNs) [18], Parareal physics-informed neural network (PPINN) [19], Conservative physics-informed neural networks(CPINN) [20], nonlocal Physics-Informed Neural Networks(nPINNs) [21] and so on. Jagtap A. D. et al. added an adaptive activation function [22] to PINN and studied the inverse problem of differential equations. Subsequently, based on the work of Jagtap A. D., the team of Chen Yong used the PINN algorithm with an adaptive activation function to study the problem of solving differential equations and named it IPINN [23]. We also studied the rogue wave solution [24] and the soliton solution [25] using by IPINN method. Although the PINN algorithm and its improvement have made some achievements, the research of these new algorithms on differential systems is much more than that.
Recently, some scholars discovered an interesting thing in their research. It is found that the nonlinear wave phenomenon that occurs in the focused NLS equation also appears stably in the generalized defocusing NLS equation with an external potential [26], and they used PINN method to study the rogue wave solutions of the defocusing NLS equation with spatio-temporal potential. In this paper, we will use the IPINN algorithm to study defocusing NLS equation. It is known that the IPINN algorithm converges faster and has higher accuracy by comparing the results obtained by the IPINN algorithm and the PINN algorithm.
The numerical experiments in this paper are performed on a computer with an 11th generation Intel(R) Core(TM) i7-11800H @ 2.30 GHz processor and 16.0 GB memory.

Rogue Wave Solution for Defocusing NLS Equation
The defocusing NLS equation with spatio-temporal potential [27] is written as V t x denotes the spatio-temporal potential and it can be written as , q is the complex solution with the independent variables x, t of Equation (1). Li Wang and Zhenya Yan [27] proved for the first time that there is also an analytical rogue wave solution for the defocusing NLS equation. We assume , the real part and imaginary part of ( ) In this paper, we focus on solving the rogue wave solution of Equation (1) Loss function is defined as

Loss Loss Loss Loss Loss
are given by initial-boundary value conditions. ( ) u t x , ( ) the Latin hypercube sampling strategy [28]. The optimal weights W , bias b and activation function slopes a are obtained by automatic differentiation Journal of Applied Mathematics and Physics technology, Adam [29] and L-BFGS [30] algorithm to minimize the loss function Loss , and the numerical solution of Equation (1) is finally determined. We also give the deep neural network in Figure 1 and schematic of IPINN for the defocusing NLS equation in Figure 2.
The rogue wave solution [27] of Equation (1) is   In order to study the influence of the number of hidden layers and the neurons in each layer on the relative 2  norm errors of ( )  Table 1. According to the data from Table 1, relative 2  norm errors of ( ) , q t x of neurons with fixed hidden layers do not decrease monotonically with the increase in the number of hidden layers. When the hidden layer is fixed, the norm error does not decrease monotonically with the increase of neurons. Therefore, we know that the norm error is affected by the number of hidden layers and neurons. At the same time, we study the influence of sampling points  Table 2. When q N is fixed, the relative 2  norm error of ( )  norm error of ( ) , q t x decreases with the increase of q N , which also shows that the boundary sampling points and regional configuration points jointly affect the relative 2  norm error of ( ) In order to understand the influence of regional configuration points f N on the predicted rogue wave solution, we fixed the improved physical information neural network, which has 8 hidden layers and each layer contains 10 neurons. We set the initial-boundary value sampling point 300. From Figure 4, we can see that with the increase in regional configuration points 1000, 2000,3000, 4000

Conclusion
In this paper, we mainly use the IPINN algorithm to solve the rogue wave solution of the defocusing NLS equation and study the influence of the network's hidden layer and its neurons on the relative norm error value of ( ) ( ) ( ) , , , , , u t x v t x q t x .
At the same time, in order to illustrate that IPINN has better network performance than PINN, such as faster convergence and higher precision. The experimental results show that IPINN has better potential in solving high-order and nonlinear differential equations. Whether to apply it to more complex high-order differential equations or adding conservation laws or more integrable system properties to the IPINN algorithm to expand IPINN algorithm are our research content in the future.