Research On Data-driven Solutions Of Nonlinear Wave Equations Based On The PINN Framework

In this paper,the data-driven solutions and parameter discovery of nonlinear partial differential equations are studied by using deep learning method.This method is now known as the physics-informed neural network(PINN)method,which opens a new way to explore the forward and inverse problems of nonlinear partial differential equations.The key of the PINN algorithm is to use these nonlinear partial differential equations describing the laws of physics as constraints to approximate the solutions of the equations.The algorithm can approximate the solution of nonlinear partial differential equations because of the strong general approximation theorem of neural networks.Therefore,this method is applied in this paper to study two kinds of nonlinear equations,one is the(1+1)-dimensional nonlinear partial differential equation,namely,Lakshmanan-Porsezian-Daniel(LPD)equation,which is used to describe the evolution of ultra-short optical pulses in optical fibers,and the other is the(2+1)-dimensional nonlinear partial differential equation.This is a multidimensional extension model of the nonlinear Schrodinger(NLS)equation,namely,the nonlocal Davey-Stewartson(DS)I equation with parity-time(PT)symmetry.Firstly,we study some nonlinear wave solutions of the LPD model which is used to describe the evolution of ultrashort pulses in optical fibers.The data-driven converted wave solution of the LPD model is reproduced by using the PINN method.From the mathematical point of view,the formation mechanism of this kind of solution is obtained when the characteristic lines of the solitary wave and the periodic wave components are parallel.Therefore,many converted wave solutions,such as the anti-dark soliton,multi-peak soliton,non-rational W-shaped soliton,rational W-shaped soliton and periodic wave,are studied by the data-driven methods.These results are called the forward problems,or data-driven solutions.In addition,this paper also studies the inverse problem of the LPD model,that is,data-driven parameter discovery.For the inverse problem,two kinds of training data sets are constructed,one is the raw data set,the other is to apply 1%noise data to the raw data.No matter the raw data or the training data with 1%noise,the PINN method can well find the coefficients of the third-order term and fourth-order term for the LPD model.Secondly,the other content of this paper is to apply this method to the data-driven study of the high-dimensional equations.Among them,the DS I equation,which is used to describe the two-dimensional wave packet evolution model in shallow water waves,is also the model to obtain the line rogue wave solution.Therefore,the integrable multi-dimensional type of the nonlocal NLS equation introduced by Fokas,namely the nonlocal DS I equation with PT symmetry,is studied by an improved PINN method.In particular,the improved PINN algorithm is used to reproduce the line breather,kink-shaped and W-shaped line rogue wave solutions.In addition,the performance of the original PINN and improved PINN methods is quantified from several perspectives,such as relative L2 error,number of iterations,and run time.Finally,these data-driven nonlinear wave solutions can be obtained by the PINN method for both low-dimensional model(LPD equation)and high-dimensional model(DS Ⅰ equation).Therefore,this algorithm is feasible for studying low-or high-dimensional models.