Research On The Application Of Physical Information Neural Network Algorithms In Numerical Solutions Of Differential Equations

With the development of modern science and technology,various complex nonlinear models are emerging in scientific research and practical applications.The calculation,simulation and analysis of nonlinear partial differential equation models have become one of the important research directions in modern science,engineering,management and other related fields.The solving algorithm of this kind of model is the key theoretical and technical problem of obtaining the information of the actual background problem of the model and its management.Traditionally,there are many researches on qualitative and various numerical solutions methods of general partial differential equation model,but it is generally recognized that these solutions have lots of limitations such as heavy workload and harsh application conditions.In particular,there are still many problems to be studied in terms of solving the individualized solutions required in application management and information acquisition.Various types of traveling wave solutions implied by partial differential equation models can be used to describe different fluctuation phenomena of information flow in practical problems.For example,financial capital fluctuations,traffic flow fluctuations and ocean waves can be described by soliton wave solutions and rogue solutions.Due to the special forms and properties of such solutions,the traditional methods are not necessarily suitable for solving such special tasks.Therefore,people are paying attention to the research and implementation of various new algorithms.At present,the physical information neural network algorithm is one of the effective emerging algorithms for solving such problems.The research and application of this algorithm is an interdisciplinary frontier subject of algorithms,(data)information processing,computer science and management disciplines,which has important theoretical and practical significance.Based on the above background,this paper explores the improvement of the"physical information neural network algorithm" which is developing well at present,and applies it to several types of partial differential equation model soliton wave solutions and rogue waves solutions that appear in traffic flow,financial management,biology and other disciplines.This paper aims to develop new algorithms,study their application in differential systems in information and management-related fields,and explore the ubiquity of soliton and rogue wave solutions in partial differential equation models.The main work and academic achievements of this paper are summarized as follows.First,several classical differential equation models and traditional algorithms in the fields of traffic flow and finance are reviewed.The development of " physical information neural network algorithm " and the research status at home and abroad are introduced in detail.Second,the mathematical principles of forward propagation and back propagation of deep neural networks are introduced.Third,a parameter identification method for identifying differential system parameters from existing data is explored,and the physical information parameter identification algorithm is successfully applied to solve the parameter identification problems such as optimal control,traffic flow model KdV-Burger equation,and mKdV equation.The Huxley equation of nerve regeneration is studied by using the improved parameter identification algorithm based on physical information,and a new method for solving biological population dynamics model that is different from traditional numerical methods is presented.In the course of research,different degrees of noise are added to the clean data,and it is found that after adding a small amount of noise,the parameters of the differential system can still be identified,which indicates that the algorithm has good robustness.By applying the algorithm to the parameter identification of differential system in different fields,it shows that the algorithm has wide adaptability and certain expansibility;Fourth,based on the basic "physical information neural network algorithm",a new neural network algorithm is proposed by introducing a new parameter method of automatically adjusting the slope,the algorithm effectively develops the basic algorithm,which has a wide range of applications and high efficiency.The improved algorithm is used to study the new soliton solution of Huxley equation in biology,the soliton wave and rogue wave solution of third-order Schrodinger equation in quantum mechanics;fifthly,as the application of the given algorithm,the improved algorithm is used to the financial soliton solution,the financial rogue wave solution and the numerical solution of the Black-Scholes model of Ivancevi option pricing model in finance,revealing the existence of such wave solutions for these models under certain initial conditions.It also shows the applicability and effectiveness of the algorithm in solving the financial differential models,and the obtained results have a good inspiration and reference for the research of practical application problems and the understanding of the nature of the problem.The work of this paper not only discusses the new development and promotion of "physical information neural network algorithm",but also studies the application of this method in different fields and different types of problems,and provides a reference algorithm and conclusion for solving differential models in practical mathematics,engineering and financial management.