Integrable Deep Learning And Localized Waves

Based on the physics-informed neural network(PINN)algorithm,and corresponding improvements have been made to it,this paper presents deep learning research of nonlin-ear systems:1,data-driven localized wave solutions for nonlinear integrable systems are studied;2,data-driven forward and inverse problems of nonlinear coupled integrable sys-tems are studied;3,a time piecewise PINNs(tp PINNs)algorithm is proposed to study the complex dynamic behaviors of low-dimensional quantum droplets.In terms of inte-grable system theory,the inverse scattering transformation and high-order poles solution of the third-order flow equation of Kaup-Newell system(TOFKN equation)are studied by means of Riemann-Hilbert(RH)approach.The research content of this paper is mainly divided into the following parts:Chapter 1 displays the relevant background and current research status,involving nonlinear localized waves,deep learning and research foundations,PINN and integrable deep learning,and RH approach.Then,the topic and main work of this paper are sum-marized.In chapter 2,based on the PINN algorithm,we study the data-driven soliton solu-tions and data-driven breather solution of the nonlinear Schr¨odinger(NLS)equation,and obtain the data-driven rogue wave solutions for the first time.We improve the PINN(IPINN)algorithm by introducing neuron-wise locally adaptive activation function and slope recovery term,then learn data-driven localized wave solutions of derivative NLS equations,including the rational solution,periodic wave,rogue wave as well as rogue wave on periodic wave background.IPINN algorithm has faster convergence speed and more stable iterative process,and obtains higher accuracy training results.In chapter 3,we extend the number of output functions and physical constraints in the IPINN algorithm,and learn the data-driven vector localized wave solutions of the Manakov system,including vector solitons,vector breather,vector rogue wave and inter-action solution.In the study of parameters discovery for inverse problems of Manakov systems,we introduce the L~2norm parameter regularization into the IPINN algorithm,then find that using clean/noisy data can accurately predict the unknown parameters of Manakov systems.Moreover,the IPINN algorithm with L~2norm parameter regulariza-tion is used to study the data-driven forward and inverse problems of the YO system,as using clean/noisy data in the data-driven inverse problems,unknown parameters can be learned with high accuracy,in particular we obtain the intermediate-bright rogue waves and dark-bright rogue waves in the data-driven forward problems.In chapter 4,a novel tp PINNs algorithm is proposed,which can study the complex dynamic behavior of nonlinear systems in a long time domain.The complex dynamics of one-dimensional quantum droplets is studied via tp PINNS algorithm in the Gross-Pitaevskii model.In the inherent modulation of single droplet,the dynamic behaviors of quantum droplet avoiding fragmentation and splitting are studied through tp PINNs algo-rithm;In the case of two droplets colliding,we learn the dynamic behavior of data-driven interference modes,which is an important feature of coherent matter wave interplay;In the excitation of breathers,initial data with cosine periodic perturbations is used to excite breathers on the droplet background.In chapter 5,the inverse scattering transformation for TOFKN equation is studied by means of RH approach,and the general form and asymptotic analysis of the localized wave solutions under zero boundary conditions(ZBCs)and non-zero boundary conditions(NZBCs)are obtained.Starting from the spectral problem of this equation,a direct scat-tering problem including analyticity,symmetry and asymptotic behavior is constructed,then an inverse scattering problem is established and solved by utilizing the matrix RH problem.Under the condition of reflectionless potential,the determinant expression for-mulas for the high-order poles solution of the TOFKN equation under ZBCs and NZBCs are systematically derived,and the asymptotic behavior of the high-order poles solution under ZBCs is analyzed in detail.Compared with the two-order flow equation of Kaup-Newell system,it is found that the third order dispersion term and the fifth order nonlinear term of the TOFKN equation affect the trajectory and velocity of the solutions.In chapter 6,a brief summary of the full text is displayed,and further prospects for the follow-up work are made.