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Pre-Training Physics-Informed Neural Network with Mixed Sampling and Its Application in High-Dimensional Systems

Journal of Systems Science & Complexity 2024年第2期37卷 494-510页

作者： LIU Haiyi ZHANG Yabin WANG Lei School of Mathematics and Physics North China Electric Power UniversityBeijing 102206China School of Control and Computer Engineering North China Electric Power UniversityBeijing 102206China

Recently,the physics-informed neural network shows remarkable ability in the context of solving the low-dimensional nonlinear partial differential equations.However,for some cases of high-dimensional systems,such technique may be time-consuming and inaccurate.In this paper,the authors put forward a pre-training physics-informed neural network with mixed sampling(pPINN)to address these issues.Just based on the initial and boundary conditions,the authors design the pre-training stage to filter out the set of the misfitting points,which is regarded as part of the training points in the next stage.The authors further take the parameters of the neural network in Stage 1 as the initialization in Stage 2.The advantage of the proposed approach is that it takes less time to transfer the valuable information from the first stage to the second one to improve the calculation accuracy,especially for the high-dimensional systems.To verify the performance of the pPINN algorithm,the authors first focus on the growing-and-decaying mode of line rogue wave in the Davey-Stewartson I equation.Another case is the accelerated motion of lump in the inhomogeneous Kadomtsev-Petviashvili equation,which admits a more complex evolution than the uniform equation.The exact solution provides a perfect sample for data experiments,and can also be used as a reference frame to identify the performance of the algorithm.The experiments confirm that the pPINN algorithm can improve the prediction accuracy and training efficiency well,and reduce the training time to a large extent for simulating nonlinear waves of high-dimensional equations.

关键词： High-dimensional systems mixed sampling nonlinear wave pre-training physics-informed neural network

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Effect of wave nonlinearity on the Instantaneous Seabed Liquefaction

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China Ocean Engineering 2024年第1期38卷 93-103页

作者： WANG Zhao-jun SUI Ti-ti ZHANG Chi PAN Jun-ning Key Laboratory of Port Waterway and Sedimentation Engineering of the Ministry of TransportNanjing Hydraulic Research InstituteNanjing 210024China Key Laboratory of Coastal Disaster and Protection(Hohai University) Ministry of EducationNanjing 210098China The National Key Laboratory of Water Disaster Prevention Nanjing 210098China

The nonlinear variation of wave is commonly seen in nearshore area,and the resulting seabed response and liquefaction are of high concern to coastal engineers.In this study,an analytical formula considering the nonlinear wave skewness and asymmetry is adopted to provide wave pressure on the seabed surface.The liquefaction depth attenuation coefficient and width growth coefficient are defined to quantitatively characterize the nonlinear effect of wave on seabed liquefaction.Based on the 2D full dynamic model of wave-induced seabed response,a detailed parametric study is carried out in order to evaluate the influence of the nonlinear variation of wave loadings on seabed liquefaction.Further,new empirical prediction formulas are proposed to fast predict the maximum liquefaction under nonlinear wave.Results indicate that(1)Due to the influence of wave nonlinearity,the vertical transmission of negative pore water pressure in the seabed is hindered,and therefore,the amplitude decreases significantly.(2)In general,with the increase of wave nonlinearity,the liquefaction depth of seabed decreases gradually.Especially under asymmetric and skewed wave loading,the attenuation of maximum seabed liquefaction depth is the most significant among all the nonlinear wave conditions.However,highly skewed wave can cause the liquefaction depth of seabed greater than that under linear wave.(3)The asymmetry of wave pressure leads to the increase of liquefaction width,whereas the influence of skewedness is not significant.(4)Compared with the nonlinear waveform,seabed liquefaction is more sensitive to the variation of nonlinear degree of wave loading.

关键词： nonlinear wave seabed response seabed liquefaction numerical simulation liquefaction prediction

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A New Pursuit Mode of nonlinear wave Surface

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海洋工程：英文版 2004年第2期18卷 257-266页

作者：孙大鹏李玉成郭海滨 State Key Laboratory of Coastal and Offshore Engineering

The numerical mode of nonlinear wave transformation based on both the Laplace equation for water field and the Bernoulli equation for water surface is a kind of time-domain boundary problem with initial conditions. And the basis for establishing the numerical mode of nonlinear wave in time domain is to trace the position of wave free surface and to calculate the instantaneous surface height and surface potential function. This paper firstly utilizes the ‘0-1' combined BEM to separate the boundary by means of discretization of Green's integral equation based on the Laplace equation, then separates the free surface of wave with FEM and derives the FEM equation of wave surface that satisfies the nonlinear boundary conditions. By jointly solving the above BEM and FEM equations, the wave potential and surface height could be obtained with iteration in time domain. Thus a new kind of nonlinear numerical mode is established for calculating wave transformation. The wave test in the numerical wave tank shows that the numerical simulation with this mode is of high accuracy.

关键词： laplace equation nonlinear wave 0-1 combined type BEM FEM

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An Improved Numerical Simulation Mode of nonlinear wave with Consideration of High Order Terms

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China Ocean Engineering 2011年第4期25卷 687-697页

作者：孙大鹏包伟斌吴浩李玉成 The State Key Laboratory of Coastal and Offshore Engineering Dalian University of Technology Wenzhou Haiwan Marine Transport Project Consultation Design Co. LTD

SUN Da-peng BAO Wei-bin, WU Hao and LI Yu-cheng （ In this paper the 0-1 combined BEM is adopted to subdivide the computational domain boundary, and to discretize the Green＇s integral expression based on Laplace equation. The FEM is used to subdivide the wave surface and deduce the surface equation which satisfies the nonlinear boundary conditions on the surface. The equations with potential function and wave surface height as an unknown quantity by application of Taylor expansion approach can be solved by iteration within the time step. In m-time iteration within the computational process of time step （n-1）At to nat, the results of the previous iteration are taken as the initial value of the two-order unknown terms in the present iteration. Thus, an improved tracking mode of nonlinear wave surface is estabIished, and numerical results of wave tank test indicate that this mode is improved obviously and is more precise than the previous numerical model which ignored the two-order unknown terms of wave surface location and velocity potential function in comparison with the theoretical values.

关键词： Laplace equation nonlinear wave 0-1 combined type BEM FEM two order terms

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A Coupling Model of nonlinear wave and Sandy Seabed Dynamic Interaction

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China Ocean Engineering 2007年第1期21卷 77-89页

作者：程永舟王永学蒋昌波 State Key Laboratory of Coastal and Offshore Engineering Dalian University of Technology Department of River and Ocean Engineering Changsha University of Science and Technology

In the paper, a weak coupling numerical model is developed for the study of the nonlinear dynamic interaction between water waves and permeable sandy seabed. The wave field solveris based on the VOF （Volume of Fluid） method for continuity equation and the two-dimensional Reynolds Averaged Navier Stokes （RANS） equations with a k-ε closure. The free surface of cnoidal wave is traced through the PLIC-VOF （P/ecewise Linear/nterface Construction）. Blot＇s equations have been applied to solve the sandy seabed, and the u-p fmite dement formulations are derived by the application of the Galerkin weighted-residual procedure. The continuity of the pressure on the interface between fluid and porous medium domains is considered. Laboratory tests were performed to verify the proposed numerical model, and it is shown that the pore-water pressures and the wave heights computed by the VOF-FEM models are in good agreement with the experimental results. It is found that the proposed model is effective in predicting the seabed-nonlinear wave interaction and is able to handle the wave-breakwater-seabed interaction problem.

关键词： nonlinear wave sandy seabed pore water pressure PLIC-VOF method VOF-FEM model

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STUDY　OF　nonlinear　LONG　wave　APPOXIMATION　IN　UNIFORM　CHANNELS　VIA　HAMILTONIAN　STRUCTURE

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Journal of Hydrodynamics 1995年第1期7卷 66-76页

作者： Xu Xin-sheng Zhong Wan-xie Lu Yu-lin (Institute of Engineering Mechanics,Dalian University of Technology,Dalian 116024,P. R. China) Dalian Univ of Technology Dalian China

This peper studies the nonlinear wave theory in shallow water via the Hamiltonian structure. The principal is the surface wave evolution on water contained in uniform channels. The Ploper Hamiltonian appoximating scheme for the more general case of waves that undergo transverse variations in amplitude in the course of longitudinal propagations is constructed. Some solutions for channels with different cross-sections,especially,the rectangular cross-section, are presented to elucidate the main features of the approxiamting scheme for the problem of interest. The obtained results shows that the nonlinear approximation of wave evolution in channels depends not only on water depth but also on a parameter determined by the geometric shape Of the channel.

关键词： Hamiltonian structure nonlinear wave uniform channel.

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Three-Dimensional Boundary Element Method Applied to nonlinear wave Transformation

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China Ocean Engineering 1999年第2期14卷 163-170页

作者： Sun, DP Li, YC Teng, B Dalian Univ Technol State Key Lab Coastal & Offshore Engn Dalian 116023 Peoples R China

For higher accuracy in simulating the transformation of three dimensional waves, in consideration of the advantages of constant panels and linear elements, a combined boundary elements is applied in this research. The method can be used to remove the transverse vibration due to the accumulation of computational errors. A combined boundary condition of sponge layer and Sommerfeld radiation condition is used to remove the reflected waves from the computing domain. By following the water particle on the water surface, the third order Stokes wave transform is simulated by the numerical wave flume technique. The computed results are in good agreement with theoretical ones.

关键词： Laplace equation nonlinear wave 3D boundary element method combined elements

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High-Order Models of nonlinear and Dispersive wave in Water of Varying Depth with Arbitrary Sloping Bottom

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China Ocean Engineering 1997年第3期12卷 243-260页

作者： Hong Guangwen Professor, Coastal and Ocean Engineering Research Institute, Hohai University, Nanjing 210024, P. R. China. Coast. and Ocean Eng. Res. Institute Hohai University Nanjing 210024 China

High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).

关键词： nonlinear wave dispersive wave high order models Boussinesq-type equations varying depth arbitrary sloping bottom

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A numerical study of 1-D nonlinear P-wave propagation in solid

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Acta Seismologica Sinica(English Edition) 2004年第1期17卷 80-86页

作者： ZHENG Hai-shan(郑海山) ZHANG Zhong-jie(张中杰) YANG Bao-jun(杨宝俊) Institute of Geology and Geophysics Chinese Academy of Sciences Beijing 100101 China Institute of Geology and Geophysics Chinese Academy of Sciences Beijing 100101 China Department of Geophysics Jilin University Changchun 130026 China

Two central schemes of finite difference (FD) up to different accuracy orders of space sampling step Dx (Fourth order and Sixth order respectively) were used to study the 1-D nonlinear P-wave propagation in the nonlinear solid media by the numerical method. Distinctly different from the case of numerical modeling of linear elastic wave, there may be several difficulties in the numerical treatment to the nonlinear partial differential equation, such as the steep gradients, shocks and unphysical oscillations. All of them are the great obstacles to the stability and conver-gence of numerical calculation. Fortunately, the comparative study on the modeling of nonlinear wave by the two FD schemes presented in the paper can provide us with an easy method to keep the stability and convergence in the calculation field when the product of the absolute value of nonlinear coefficient and the value of u/x are small enough, namely, the value of bu/x is much smaller than 1. Several results are founded in the numerical study of nonlinear P-wave propagation, such as the waveform aberration, the generation and growth of harmonic wave and the energy redistribution among different frequency components. All of them will be more violent when the initial amplitude A0 is larger or the nonlinearity of medium is stronger. Correspondingly, we have found that the nonlinear P-wave propagation velocity will change with different initial frequency f of source wave or the wave velocity c (equal to the P-wave velocity in the same medium without considering nonlinearity).

关键词： nonlinear wave numerical simulation central schemes of finite difference

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nonlinear waveS AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

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Acta Mechanica Solida Sinica 2006年第1期19卷 1-8页

作者： Liu Zhifang Zhang Shanyuan Institute of Applied Mechanics Taiyuan University of Technology Taiyuan 030024 China

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.

关键词： nonlinear wave finite deformation Poisson effect Jacobi elliptic function

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