Physics Informed Neural Networks And Localized Waves Of Integrable Equations

In this paper,physics informed neural network(PINN)model is introduced,then several improved PINN algorithms are proposed and briefly analyzed due to some limitations of solving differential equations using the classical PINN algorithm.The classical PINN algorithm and these improved PINN algorithms are particularly applied to the systematic study of nonlinear localized waves,including solitons,breathers and rogue waves.This paper mainly includes three aspects: 1.The unified representations of deep feedforward neural networks are described,several basic activation functions and their recent progress are then relatively systematically analyzed and discussed.Moreover,the strict mathematical derivations of several common weight initialization strategies are given after the error backpropagation algorithm is introduced;2.The first-and second-order optimization algorithms and automatic differentiation technology required by the physics informed neural network algorithm are introduced and then the PINN algorithm is applied to solve localized waves of some famous nonlinear integrable systems.3.Several improved methods for the vanilla PINN algorithm are presented and analyzed,these improved PINN algorithms are then applied to the study of localized waves in integrable equations.Chapter 1,as the introduction of this paper,introduces the background and development of integrable system,localized wave,deep learning,physics informed neural network and its applications in science and engineering problems including mathematical physics equations,and the main work of this paper is concluded.In chapter 2,deep feedforward neural networks are introduced and then basic activation functions and their recent research progress are relatively systematically analyzed and discussed.Moreover,strict mathematical derivations of several common weight initialization strategies(e.g.,Xavier initialization and He initialization)are given after the backpropagation algorithm is briefly introduced.Besides,some common gradient descent algorithms and their recent progress are briefly compared and discussed.In chapter 3,the general framework of PINN algorithm based on the universal approximation theorem of neural network and automatic differentiation technology are introduced.The detailed process and flow chart of PINN algorithm for solving general partial differential equations(PDEs)are given and then optimization algorithms used in solving PDEs through deep neural networks are briefly discussed.The example of solving a simple dynamic system shows that compared with the standard neural network method,the PINN algorithm can achieve better solution fitting and relatively higher prediction accuracy and generalization ability with only remarkably smaller amounts of training data.Lastly,the PINN algorithm is successfully applied to the localized wave solutions of the second-order Burgers equation.In chapter 4,a PINN algorithm with sinusoidal periodic function is proposed.Compared with the classical neural network architecture,the improved PINN algorithm can learn the higher frequency parts of the network solutions.The improved PINN algorithm is then applied to solve localized wave solutions of several important nonlinear integrable equations,including the multi-soliton solution of the third-order Kd V equation,the soliton solution and breather solution of the modified Kd V equation,the kink solution of Kd V-Burgers equation and fusion and fission soliton of the Sharma-Tasso-Olver(STO)equation.Moreover,with the help of various image information,the complex dynamics of these localized wave solutions are vividly portrayed.In chapter 5,a PINN network structure with Res Net module is proposed,the residual structure with skip connection can effectively alleviate the vanishing gradient and network degeneration problems which often occur in the classical deep feedforward neural networks.A new loss function is used in this improved PINN algorithm and then the anti-kink solution of strongly nonlinear sine-Gordon equation is studied by this improved PINN algorithm.In addition,the effects of different random environment,noise injection,initial/boundary training data points,domain collocation points sampled,network layers and neurons per hidden layer on the accuracy of model results are also discussed.In chapter 6,a PINN algorithm with adaptive activation function is proposed,which greatly improves the efficiency of the algorithm and the performance of the model by endowing each neuron with adaptive learning ability.The example of nonlinear discontinuous function fitting reveals from experimental simulation and frequency analysis that the adaptive algorithm can learn the higher frequency parts of complex signals faster than the neural network method with fixed activation function.Using this improved adaptive PINN algorithm,the localized wave solution of the derivative nonlinear Schr ¨odinger equation is studied successfully,including one-order rational soliton solution,one-order genuine rational soliton solution,two-order genuine rational soliton solution and twoorder rogue wave solution.In particular,the complex dynamic behavior of the higher order rogue wave solution is discovered.Chapter 7 gives the summaries and prospects of the whole paper.