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An Adaptive Physics-Informed Neural Network with Two-Stage Learning Strategy to Solve partial differential equations

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Numerical Mathematics(Theory,Methods and Applications) 2023年第2期16卷 298-322页

作者： Shuyan Shi Ding Liu Ruirui Ji Yuchao Han School of Automation and Information Engineering Xi‘an University of TechnologyXi‘an 710048P.R.China National&Local Joint Engineering Research Center of Crystal Growth Equipment and System Integration Xi‘an 710048P.R.China Xi’an University of Technology Shannxi Key Laboratory of Complex System Control and Intelligent Information ProcessingXi‘an 710048P.R.China

Physics-Informed Neural Network(PINN)represents a new approach to solve partial differential equations(PDEs).PINNs aim to solve PDEs by integrating governing equations and the initial/boundary conditions(I/BCs)into a loss function.However,the imbalance of the loss function caused by parameter settings usually makes it difficult for PINNs to converge,e.***.because they fall into local optima.In other words,the presence of balanced PDE loss,initial loss and boundary loss may be critical for the convergence.In addition,existing PINNs are not able to reveal the hidden errors caused by non-convergent boundaries and conduction errors caused by the PDE near the boundaries.Overall,these problems have made PINN-based methods of limited use on practical situations.In this paper,we propose a novel physics-informed neural network,i.***.an adaptive physics-informed neural network with a two-stage training process.Our algorithm adds spatio-temporal coefficient and PDE balance parameter to the loss function,and solve PDEs using a two-stage training process:pre-training and formal training.The pre-training step ensures the convergence of boundary loss,whereas the formal training process completes the solution of PDE by balancing various loss functions.In order to verify the performance of our method,we consider the imbalanced heat conduction and Helmholtz equations often appearing in practical situations.The Klein-Gordon equation,which is widely used to compare performance,reveals that our method is able to reduce the hidden errors.Experimental results confirm that our algorithm can effectively and accurately solve models with unbalanced loss function,hidden errors and conduction errors.The codes developed in this manuscript are publicy available at https://***/callmedrcom/ATPINN.

关键词： Physics informed neural networks partial differential equations two-stage learning scientific computing

LaNets:Hybrid Lagrange Neural Networks for Solving partial differential equations

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Computer Modeling in Engineering & Sciences 2023年第1期134卷 657-672页

作者： Ying Li Longxiang Xu Fangjun Mei Shihui Ying School of Computer Engineering and Science Shanghai UniversityShanghai200444China School of Science Shanghai UniversityShanghai200444China

We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.

关键词： Hybrid Lagrange neural networks interpolation polynomials deep learning numerical simulation partial differential equations

Implementing fractional Fourier transform and solving partial differential equations using acoustic computational metamaterials in space domain

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Acta Mechanica Sinica 2021年第9期37卷 1371-1377,I0001页

作者： Zengyao Lv Peng Liu Yuanshuai Ding Hangyu Li Yongmao Pei Department of Engineering Mechanics Center for Flexible Electronics TechnologyApplied Mechanics LaboratoryTsinghua UniversityBeijing100084China State Key Laboratory for Turbulence and Complex Systems Department of Mechanics and Engineering ScienceCollege of EngineeringPeking UniversityBeijing100871China Locomotive and Car Research Institute China Academy of Railway Sciences Corporation LimitedBeijing100081China

Metamaterials can control incident waves in the sub-wavelength range through the design of artificial structures, and realize the functions that natural materials cannot achieve. The study of metamaterials has important theoretical value and application prospects. In recent years, the proposal of computational metamaterials has opened up a brand-new direction for analog computing, providing high-throughput, energy-free computing methods for special computing tasks. However, the development of acoustic computing metamaterials is relatively preliminary, and it is necessary to develop design theories. There is no work to solve partial differential equations and realize fractional Fourier transform in spatial domain acoustic computing metamaterials. In this paper, the acoustic wave computational metamaterial is designed, and the simulation realizes the spatial domain fractional Fourier transform and partial differential equation calculation. It is expected that acoustic computational metamaterials will enable new capabilities in signal acquisition and processing, network computing, and drive new applications of sound wave.

关键词： Fractional Fourier transform partial differential equations Acoustic computational metamaterials

differential characteristic set algorithm for the complete symmetry classification of partial differential equations

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Applied Mathematics and Mechanics(English Edition) 2009年第5期30卷 595-606页

作者：特木尔朝鲁白玉山 Department of Mathematics Shanghai Maritime University College of Science Inner Mongolia University of Technology

In this paper, we present a differential polynomial characteristic set algorithm for the complete symmetry classification of partial differential equations （PDEs） with some parameters. It can make the solution to the complete symmetry classification problem for PDEs become direct and systematic. As an illustrative example, the complete potential symmetry classifications of nonlinear and linear wave equations with an arbitrary function parameter are presented. This is a new application of the differential form characteristic set algorithm, i.e., Wu＇s method, in differential equations.

关键词： partial differential equations symmetry classification differential characteristic set

New Kamenev-type Oscillation Criteria for Half-linear partial differential equations

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Acta Mathematicae Applicatae Sinica 2012年第3期28卷 535-548页

作者： Ge-feng YANG Zhi-ting XU Cisco School of Informatics Guangdong University of Foreign StudiesGuangzhou510006China School of Mathematical Sciences South China Normal UniversityGuangzhou510631China

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping div（A（x）｜｜ u｜｜^p-2 u）＋〈b（x）,｜｜ u｜｜^p-2 u〉＋c（x）｜u｜^p-2u=0（E）under quite general conditions. These results are extensions of the recent results developed by Sun [Y.C. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 （2004） 341-351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.

关键词： oscillation half-linear partial differential equations Kamenev-type Damped equation

A New Approach to Nonlinear partial differential equations

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Communications in Nonlinear Science and Numerical Simulation 1997年第4期2卷 230-235页

作者： Jihuan HE (Shanghai University,Shanghai Institute of Applied Mathematics and Mechanics,Shanghai 2 0 0 0 72 ,China) Shanghai University Shanghai Institute of Applied Mathematics and Mechanics Shanghai 200072 China

In this paper,a novel method called variational iteration method is proposed to solve nonlinear partial differential equations without linearization or small perturbations.In this method,a correction functional is constructed by a general Lagrange multiplier,which can be identified via variational theory.An analytical solution can be obtained from its trial- function with possible unknown constants,which can be identified by imposing the boundary conditions,by successively iteration.

关键词： Nonlinear Problems partial differential equations Variational Iteration Method Variational Theory

A SOLVING METHOD FOR A SYSTEM OF partial differential equations WITH AN APPLICATION TO THE BENDING PROBLEM OF A THICK PLATE

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Applied Mathematics and Mechanics(English Edition) 1999年第11期20卷 1259-1265页

作者：尹益辉陈刚陈裕泽 Institute of Structural Mechanics CAEP Mianyang P R China

A theorem of solving a system of linear non-homogeneous differential equations through integrating and adding its basic solutions is put forward and proved, the mathematical role and physical nature of the theorem is interpreted briefly. As an example, the theorem is applied to solve the problem of thermo-force bending of a thick plate.

关键词： partial differential equations integrating method thick plate thermo-force bending

SINGULARLY PERTURBED METHODS IN THE THEORY OF OPTIMAL CONTROL OF SYSTEMS GOVERNED BY partial differential equations

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Applied Mathematics and Mechanics(English Edition) 1994年第8期15卷 713-719页

作者：田根宝林宗池 Mathematical Section Shanghai Railway College Shanghai Department of Mathematics Fujian Normal University Fuzhou

In this paper, the various problems associaled with the optimal control of systemsgoverned by partial differential equations are introduced by using singularly perturbedmethods for analysis based on stale equations,... 详细信息

关键词： optimal control perturbation techniques partial differential equations

Solving Inhomogeneous Linear partial differential equations

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Journal of partial differential equations 2010年第4期23卷 374-388页

作者： SCHWARZ Fritz Fraunhofer Gesellschaft Institut SCAI 53754 Sankt Augustin Germany.

Lagrange's variation-of-constantsmethod for solving linear inhomogeneous ordinary differential equations (ode's) is replaced by a method based on the Loewy decomposition of the corresponding homogeneous equation. It uses only properties of the equations and not of its solutions. As a consequence it has the advantage that it may be generalized for partial differential equations (pde's). It is applied to equations of second order in two independent variables, and to a certain system of third-order pde's. Therewith all possible linear inhomogeneous pde's are covered that may occur when third-order linear homogeneous pde's in two independent variables are solved.

关键词： partial differential equations linear differential equations inhomogeneous differential equations